Existence and Uniqueness of Solutions to Non-Local Problems of Brézis–Oswald Type and Its Application

Abstract

The aim of this paper is to establish the existence and uniqueness of solutions to non-local problems involving a discontinuous Kirchhoff-type function via a global minimum principle of Ricceri. More precisely, we first obtain the uniqueness result of weak solutions to nonlinear fractional Laplacian problems of Brézis–Oswald type. We then demonstrate the existence of a unique positive solution to Kirchhoff-type problems driven by the non-local fractional Laplacian as its application. The main features of the present paper are the lack of the continuity of the Kirchhoff function in $[0,\infty )$ and the localization of a positive solution.

1. Introduction

In past decades, a great deal of attention was dedicated to the study of fractional Sobolev spaces and the corresponding non-local equations because of the relevance of applied or pure mathematical theories that are used to describe certain specific phenomena, such as fractional quantum mechanics, frame propagation, optimization, anomalous diffusion in plasma, the thin obstacle problem, game theory and Lévy processes, geophysical fluid dynamics, image processing and American options in finances; see [1,2,3,4,5] for comprehensive studies and details on this topic.

The present paper is dedicated to obtaining the existence, uniqueness and localization of solutions to a Kirchhoff-type problem driven by the non-local fractional Laplacian, as follows:

$$\left\{\begin{array}{cc}-m\left({\left[\phi \right]}_{s,2}^2\right)\mathcal{L}\phi \left(y\right)=h(y,\phi )\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \phi >0\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \phi =0\hfill & \mathrm{on}{\mathbb{R}}^N\setminus \mathsf{\Omega },\hfill \\ {\phi ]}_{s,2}^2\in J,\hfill \end{array}\right.$$

(1)

where $s\in (0,1)$; $2s<N$; $\mathsf{\Omega }\subset {\mathbb{R}}^N$ is a bounded open set with Lipschitz boundary $\partial \mathsf{\Omega }$; $J\subseteq (0,+\infty )$ is an open interval; ${\left[\phi \right]}_{s,2}^2:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y-z)dydz$; m is a discontinuous Kirchhoff-type function and a and h are non-negative functions that will be specified later. Here, the non-local operator $\mathcal{L}$ is defined pointwise as

$$\mathcal{L}\phi \left(y\right)=2{\int }_{{\mathbb{R}}^N}(\phi (y+z)+\phi (y-z)-2\phi \left(y\right))\mathfrak{K}\left(y\right)dz\mathrm{for}\mathrm{all}y\in {\mathbb{R}}^N,$$

where a kernel function $\mathfrak{K}:{\mathbb{R}}^N\setminus \left\{0\right\}\to (0,+\infty )$ satisfies the following hypotheses:

($\mathcal{L}1$)

$\ell \mathfrak{K}\in L^1\left({\mathbb{R}}^N\right)$, where $\ell \left(y\right)=min\{1,|y{|}^2\}$;

($\mathcal{L}2$)

There is a real number ${\eta }_0>0$ such that $\mathfrak{K}\left(y\right)\ge {\eta }_0{\left|y\right|}^{-(N+2s)}$ for almost all $y\in {\mathbb{R}}^N\setminus \left\{0\right\}$;

($\mathcal{L}3$)

$\mathfrak{K}\left(y\right)=\mathfrak{K}(-y)$ for all $y\in {\mathbb{R}}^N\setminus \left\{0\right\}$.

If $\mathfrak{K}\left(x\right)={\left|x\right|}^{-(N+2s)}$, $\mathcal{L}$ is the fractional Laplacian operator ${(-\mathsf{\Delta })}^s$ defined as

$$-{(-\mathsf{\Delta })}^s\phi \left(y\right)={\int }_{{\mathbb{R}}^N}{\displaystyle \frac{\phi (y+z)+\phi (y-z)-2\phi \left(y\right)}{{|y-z|}^{N+2s}}}dz,y\in {\mathbb{R}}^N.$$

In order to investigate an extension of the classical D’Alembert’s wave equation by taking into account changes in the length of the strings during vibrations, Kirchhoff in [6] originally suggested a stationary version of the equation:

$$\varrho {\displaystyle \frac{{\partial }^2\phi }{\partial {\eta }^2}}-\left({\displaystyle \frac{{\varrho }_0}{\hslash }}+{\displaystyle \frac{E}{2L}}{\int }_0^L\left|{\displaystyle \frac{\partial \phi }{\partial y}}\right|dy\right){\displaystyle \frac{{\partial }^2\phi }{\partial y^2}}=0,$$

where $\varrho ,{\varrho }_0,\hslash ,E$ and L are constants. The variational problems of Kirchhoff type have interested and attracted researchers from diverse areas of physics and have consequently been extensively investigated in recent years, for instance, refs. [7,8,9,10,11,12,13,14,15,16] and the references therein. Fiscella and Valdinoci [9] first presented a detailed discussion about the physical meaning underlying the fractional Kirchhoff model. In particular, they establish the existence of nontrivial solutions to a non-local elliptic problem, with nondegenerate Kirchhoff coefficients, that is a c ontinuous and increasing function with the nondegenerate condition ${inf}_{\tau \in [0,+\infty )}m\left(\tau \right)\ge {\tau }_0>0$, where ${\tau }_0$ is a constant, by taking into account the mountain pass theorem and a truncation argument. See also [13] for critical stationary Kirchhoff equations. As we know, since the increasing condition eliminates the non-monotonic case, the existence of at least two different solutions to Schrödinger–Kirchhoff-type problems with fractional p-Laplacian was reported in [14], where the nondegenerate Kirchhoff function m is continuous and satisfies the following assumption:

(M1)

For $0<s<1$, there is $\vartheta \in [1,{\displaystyle \frac{N}{N-sp}})$ such that $\vartheta M\left(\tau \right)\ge m\left(\tau \right)\tau$ for any $\tau \ge 0$, where $M\left(\tau \right):={\int }_0^{\tau }m\left(\sigma \right)d\sigma$.

The condition (M1) includes the typical example $m\left(\tau \right)=1+a{\tau }^{\vartheta }$ ($a\ge 0,\tau \ge 0$) as well as the non-monotonic case. Under a stronger condition than (M1), Huang and Deng [10] obtained the existence of a positive ground state solution to a Kirchhoff-type problem with critical exponential growth. In this respect, the nonlinear elliptic Kirchhoff problems with (M1) have been intensively studied by many researchers in recent years and have received considerable attention; see [10,14,16,17,18]. In view of these related papers, the functional $\mathsf{\Phi }:W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right)\to \mathsf{ℝ}$ corresponding to the primitive part in (1) is defined by

$$\mathsf{\Phi }\left(\phi \right)={\displaystyle \frac{1}{2}}M\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y-z)dydz\right)$$

for any $\phi \in W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right)$, where $W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right)$ is a solution space that will be defined later. Then, it follows from the fact that $m\in C\left(\right[0,+\infty \left)\right)$, $\mathsf{\Phi }\in C^1(W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right),\mathsf{ℝ})$, and its Fréchet derivative is given by

$$\langle {\mathsf{\Phi }}^{\prime }\left(\phi \right),\omega \rangle =m\left({\left[\phi \right]}_{s,2}^2\right){\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}(\phi \left(y\right)-\phi \left(z\right))(\omega \left(y\right)-\omega \left(z\right))\mathfrak{K}(y,z)dydz$$

for any $\phi ,\omega \in W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right)$. In particular, conditions $m\in C\left({\mathbb{R}}_0^{+}\right)$ and (M1) are essential in obtaining some topological properties of functionals $\mathsf{\Phi },{\mathsf{\Phi }}^{\prime }$ and the Palais–Smale-type compactness condition for an energy functional corresponding to (1), which plays an effective role in employing variational methods such as the mountain pass theorem, fountain theorem and Ekeland variational principle. However, the continuity of the nondegenerate Kirchhoff function m in $[0,\infty )$ removes many examples; for instance, let us consider the Kirchhoff functions

$$m\left(\tau \right)=tan\tau \mathrm{for}0<\tau <{\displaystyle \frac{\pi }{2}}$$

and

$$m\left(\tau \right)={(c-\tau )}^{-s}\mathrm{for}\tau \in (-\infty ,c),\mathrm{where}c>0,s\in (0,1).$$

These examples cannot be addressed by any of the results known up to now. Very recently, Ricceri [15] provided a new approach that is different from previous related studies [7,8,9,10,13,14,16] in order to obtain the existence and uniqueness of positive solutions to the Laplacian problem with discontinuous Kirchhoff coefficients. Motivated by this study, the author in [11] recently extended the result in [15] to the p-Laplacian problems; see also the paper [12] for double-phase problems. The main tools for obtaining these consequences in [11,12] are the abstract global minimum principle in [15] and the uniqueness result for the Brézis–Oswald-type problem based on [19]. In particular, the inequalities of Dìaz–Saa type in [20,21] are crucial in order to obtain the uniqueness of a positive solution to problems considered in the papers [11,12]. Applications of these inequalities require the consideration of the classical Hopf’s boundary lemma to show the fact that the quotient between solutions belongs to $L^{\infty }$-space. However, since solutions to fractional-order equations are usually singular at the boundary, the Hopf’s boundary lemma is not maintained, and thus, it is difficult to work with the quotient between solutions. Hence, in contrast to the papers [11,12], the main difficulty of the present paper is to obtain that the Brézis–Oswald-type problem involving the fractional Laplacian operator admits a unique positive weak solution. To obtain this uniqueness result as our first main result, we overcome this difficulty by considering the truncation of the solutions, which is based on the papers [22,23].

As its application, the second aim of the present paper is to demonstrate the existence and uniqueness of positive solutions to non-local problems with a discontinuous Kirchhoff-type function. To the best of our knowledge, the uniqueness result of weak solutions to nonlinear fractional Laplacian problems of Brézis–Oswald type with the Kirchhoff coefficient has not been comprehensively studied, and we only know about the paper [7] as an analogous result for this type of equation. Very recently, Biagi and Vecchi [7] established the uniqueness result to a Brézis–Oswald-type Laplacian problem with degenerate Kirchhoff functions when $m:[0,\infty )\to [0,\infty )$ is a non-decreasing and continuous function, such that $m\left(\tau \right)>0$ for every $\tau >0$. However, our approach for obtaining the second main result is different from that of paper [7] as well as previous studies [8,9,10,13,14,16] because we consider the lack of the continuity of the Kirchhoff function m in $[0,\infty )$ and the localization of the solution. This is the novelty of this paper.

The outline of the present paper is structured as follows. In Section 2, we introduce some necessary preliminary knowledge of function spaces to be used this paper. In Section 3, we give the variational framework related to problem (1), and then Section 4 presents a demonstration of the existence and uniqueness of positive solutions under proper conditions.

2. Preliminaries

In this section, we briefly introduce some useful definitions and elementary properties of the fractional Sobolev spaces that will be utilized in this paper. Let $s\in (0,1)$ and let $2_s^{*}$ be the fractional critical Sobolev exponent, namely,

$$2_s^{*}:=\left\{\begin{array}{cc}{\displaystyle \frac{2N}{N-2s}}\hfill & \mathrm{if}2s<N,\hfill \\ +\infty \hfill & \mathrm{if}2s\ge N.\hfill \end{array}\right.$$

Let $\mathsf{\Omega }\subset {\mathbb{R}}^N$ be an open and bounded set with Lipschitz boundary. Let us define the fractional Sobolev space $H^s\left(\mathsf{\Omega }\right)$ as follows:

$$H^s\left(\mathsf{\Omega }\right):=\left\{\phi \in L^2\left(\mathsf{\Omega }\right):{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\phi \left(y\right)-\phi \left(z\right)|}^2}{{|y-z|}^{N+2s}}}dydz<+\infty \right\},$$

which is endowed with the Gagliardo norm

$${\parallel \phi \parallel }_{H^s\left(\mathsf{\Omega }\right)}:={\left({\parallel \phi \parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\left|\phi \right|}_{H^s\left({\mathbb{R}}^N\right)}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

where

$${\left|\phi \right|}_{H^s\left({\mathbb{R}}^N\right)}^2:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\phi \left(y\right)-\phi \left(z\right)|}^2}{{|y-z|}^{N+2s}}}dydz.$$

Then, $H^s\left(\mathsf{\Omega }\right)$ is a Hilbert space with the following inner product:

$${〈\phi ,\psi 〉}_{H^s\left(\mathsf{\Omega }\right)}={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{\left(\phi \right(y)-\phi (z\left)\right)\left(\psi \right(y)-\psi (z\left)\right)}{{|y-z|}^{N+2s}}}dydz+{\int }_{\mathsf{\Omega }}\phi \left(y\right)\psi \left(y\right)dy.$$

Also, the space $C_0^{\infty }\left({\mathbb{R}}^N\right)$ is dense in $H^s\left(\mathsf{\Omega }\right)$; for example, see [4,24]. The following consequence can be found in [4].

Lemma 1.

Let $s\in (0,1)$. Then, the continuous embedding $H^s\left(\mathsf{\Omega }\right)\hookrightarrow L^q\left(\mathsf{\Omega }\right)$ holds for any $q\in [1,2_s^{*}]$ if $2s<N$. In addition, for any $q\in [1,2_s^{*})$, the space $H^s\left(\mathsf{\Omega }\right)$ is compactly embedded in $L^q\left(\mathsf{\Omega }\right)$.

Let us define the fractional Sobolev space $W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right)$ as follows:

$$W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right):=\left\{\phi \in L^2\left(\mathsf{\Omega }\right):{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y-z)dydz<+\infty \right\},$$

where a kernel function $\mathfrak{K}:{\mathbb{R}}^N\setminus \left\{0\right\}\to (0,+\infty )$ satisfies ($\mathcal{L}$1)–($\mathcal{L}$3). By the assumption ($\mathcal{L}$1), the function

$$(y,z)\mapsto (\phi \left(y\right)-\phi \left(z\right)){\mathfrak{K}}^{{\displaystyle \frac{1}{2}}}(y-z)\in L^2\left({\mathbb{R}}^N\right)$$

for all $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$. Define the closed linear subspace X of $W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right)$ defined by

$$X:=\left\{\phi \in W_{\mathfrak{K}}^{s,2}\left(\mathsf{\Omega }\right):\phi \left(y\right)=0\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^N\setminus \mathsf{\Omega }\right\}$$

with respect to the norm

$${\parallel \phi \parallel }_X^2:={\parallel \phi \parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\left[\phi \right]}_{s,2}^2,$$

where

$${\left[\phi \right]}_{s,2}^2:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y-z)dydz.$$

In what follows, let $s\in (0,1)$ with $2s<N$ and let the kernel function $\mathfrak{K}:{\mathbb{R}}^N\setminus \left\{0\right\}\to (0,\infty )$ ensure the assumptions ($\mathcal{L}$1)–($\mathcal{L}$3). The following assertion comes from the papers [5,16].

Lemma 2.

If $\phi \in X$, then $\phi \in H^s\left(\mathsf{\Omega }\right)$. Moreover,

$${\parallel \phi \parallel }_{H^s\left(\mathsf{\Omega }\right)}\le max\{1,{\eta }_0^{-{\displaystyle \frac{1}{p}}}\}{\parallel \phi \parallel }_X,$$

where ${\eta }_0$ is given in($\mathcal{L}$2).

By virtue of Lemmas 1 and 2, we can instantly obtain the following assertion:

Lemma 3.

For any $\phi \in X$, there is a constant $C_0=C_0(N,2,s)>0$ such that for $1\le q\le 2_s^{*}$,

$$\begin{array}{cc}\hfill {\parallel \phi \parallel }_{L^q\left(\mathsf{\Omega }\right)}^2& \le C_0{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|\phi \left(y\right)-\phi \left(z\right)|}^2}{{|y-z|}^{N+2s}}}dydz\hfill \\ \hfill & \le {\displaystyle \frac{C_0}{{\eta }_0}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y-z)dydz,\hfill \end{array}$$

where ${\eta }_0$ is given in($\mathcal{L}$2). Accordingly, for any $q\in [1,2_s^{*}]$, the space X is continuously embedded in $L^q\left(\mathsf{\Omega }\right)$. Furthermore, the embedding

$$X\hookrightarrow L^q\left(\mathsf{\Omega }\right)$$

is compact for $q\in (1,2_s^{*})$.

3. Variational Setting and a Brezis–Oswald-Type Result

In this section, we demonstrate the existence and the uniqueness of a positive solution to the Brézis–Oswald-type problem:

$$\left\{\begin{array}{cc}-\mathcal{L}\phi \left(y\right)=\lambda h(y,\phi )\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \phi >0\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \phi =0\hfill & \mathrm{on}{\mathbb{R}}^N\setminus \mathsf{\Omega }.\hfill \end{array}\right.$$

(2)

To perform this, we introduce the variational setting corresponding to the problem (2). Subsequently, we present some basic topological properties for an energy functional.

Definition 1.

We say that $\phi \in X$ is a weak solution of (2) if

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}(\phi \left(y\right)-\phi \left(z\right))(\omega \left(y\right)-\omega \left(z\right))\mathfrak{K}(y-z)dydz=\lambda {\int }_{\mathsf{\Omega }}h(y,\phi )\omega dy\end{array}$$

(3)

for any $\omega \in X$.

Let us define the functional $\mathsf{\Phi }:X\to \mathbb{R}$ by

$$\mathsf{\Phi }\left(\phi \right):={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y,z)dy.$$

Then, it is not difficult to prove that the functional $\mathsf{\Phi }:X\to \mathbb{R}$ belongs to $C^1(X,\mathbb{R})$ and its Fréchet derivative is

$$〈{\mathsf{\Phi }}^{\prime }\left(\phi \right),\omega 〉={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}(\phi \left(y\right)-\phi \left(z\right))(\omega \left(y\right)-\omega \left(z\right))\mathfrak{K}(y,z)dydz$$

for any $\phi ,\omega \in X$, where $〈·,·〉$ denotes the pairing of X and its dual $X^{*}$; see [14].

Lemma 4.

Let $s\in (0,1)$ and $2s<N$. Then, the functional Φ is convex and has a weak lower semicontinuity on X.

Proof. 

It is obvious that $\mathsf{\Phi }$ is convex. Let $\left\{u_n\right\}$ be a sequence in X such that $u_n\rightharpoonup u$ in X. Since the functional $\mathsf{\Phi }$ is convex and $C^1$-functional on X, it follows that

$$\mathsf{\Phi }\left(u_n\right)\ge 〈{\mathsf{\Phi }}^{\prime }\left(u_n\right),u_n-u〉+\mathsf{\Phi }\left(u\right).$$

Then, the following is immediately determined:

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; inf}\mathsf{\Phi }\left(u_n\right)& \ge \mathsf{\Phi }\left(u\right)+\underset{n\to \infty }{lim\; inf}〈{\mathsf{\Phi }}^{\prime }\left(u_n\right),u_n-u〉\hfill \\ \hfill & \ge \mathsf{\Phi }\left(u\right).\hfill \end{array}$$

Therefore, the conclusion holds. □

Next, we now fix the standing conditions on the reaction term h:

(H1)

$h:\mathsf{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function;

(H2)

$h(·,\xi )\in L^{\infty }\left(\mathsf{\Omega }\right)$ for every $\xi \ge 0$, and there exists $c_1>0$ such that

$$\left|h(y,\xi )\right|\le c_1(1+\xi )$$

for almost all $y\in \mathsf{\Omega }$ and for any $\xi \ge 0$;

(H3)

${lim}_{\xi \to 0^{+}}{\displaystyle \frac{h(y,\xi )}{\xi }}=+\infty$ and ${lim}_{\xi \to +\infty }{\displaystyle \frac{h(y,\xi )}{\xi }}=0$;

(H4)

The function $\xi \mapsto {\displaystyle \frac{h(y,\xi )}{\xi }}$ is decreasing in $(0,+\infty )$.

Under condition (H1), let us define the functional $\mathsf{\Psi }:X\to \mathbb{R}$ by

$$\mathsf{\Psi }\left(\phi \right):={\int }_{\mathsf{\Omega }}H(y,\phi \left(y\right))dy$$

for any $\phi \in X$, where $H(y,\xi )={\int }_0^{\xi }h(y,t)dt$. Then, it is straightforward to prove that $\mathsf{\Psi }\in C^1(X,\mathbb{R})$, and its Fréchet derivative is

$$〈{\mathsf{\Psi }}^{\prime }\left(\phi \right),w〉={\int }_{\mathsf{\Omega }}h(y,\phi )wdy$$

for any $\phi ,w\in X$. Next, we define the functional $\mathcal{E}:X\to \mathbb{R}$ by

$$\begin{array}{c}\hfill \mathcal{E}\left(\phi \right)=\mathsf{\Phi }\left(\phi \right)-\lambda \mathsf{\Psi }\left(\phi \right).\end{array}$$

Then, it follows that the functional $\mathcal{E}$ belongs to $C^1(X,\mathbb{R})$ and its Fréchet derivative is

$$〈{\mathcal{E}}^{\prime }\left(\phi \right),w〉=〈{\mathsf{\Phi }}^{\prime }\left(\phi \right),w〉-\lambda 〈{\mathsf{\Psi }}^{\prime }\left(\phi \right),w〉\mathrm{for}\mathrm{any}\phi ,w\in X.$$

Evidently, the weak solutions of the problem (2) are exactly the critical points of the functional $\mathcal{E}$.

We show a technical Lemma which will be very practical hereinafter. For any ${\phi }_i\in X$ and $\epsilon >0$, define the truncation

$${\phi }_{i,\epsilon }:=\min \{{\phi }_i,{\epsilon }^{-1}\}\mathrm{for}i=1,2.$$

(4)

Lemma 5.

Let ${\phi }_1,{\phi }_2\in X$ with ${\phi }_1,{\phi }_2\ge 0$ and set

$$w:={\displaystyle \frac{{\phi }_{2,\epsilon }^2}{{\phi }_1+\epsilon }}-{\phi }_{1,\epsilon },$$

where ${\phi }_{1,\epsilon },{\phi }_{2,\epsilon }$ are as in (4). Then, we have $w\in X$.

Proof. 

Let $\epsilon >0$ be fixed. Since the function $t\mapsto \min \left\{\right|t|,{\epsilon }^{-1}\}$ is 1-Lipschitz, we infer that

$$|{\phi }_{i,\epsilon }\left(y\right)-{\phi }_{i,\epsilon }\left(z\right)|\le |{\phi }_i\left(y\right)-{\phi }_i\left(z\right)|\mathrm{for}i=1,2,$$

(5)

which directly implies that ${\phi }_{i,\epsilon }\in X$. Since $\epsilon \le {\phi }_{1,\epsilon }+\epsilon$ and ${\phi }_{2,\epsilon }\le {\displaystyle \frac{1}{\epsilon }}$, taking (5) and the triangle inequality into account, we have

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(y\right)}{{\phi }_1\left(y\right)+\epsilon }}-{\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(z\right)}{{\phi }_1\left(z\right)+\epsilon }}|\hfill \\ \hfill & =|{\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(y\right)-{\phi }_{2,\epsilon }^2\left(z\right)}{{\phi }_1\left(y\right)+\epsilon }}+{\phi }_{2,\epsilon }^2\left(z\right)\left({\displaystyle \frac{1}{{\phi }_1\left(y\right)+\epsilon }}-{\displaystyle \frac{1}{{\phi }_1\left(z\right)+\epsilon }}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{2}{{\epsilon }^2}}|{\phi }_{2,\epsilon }\left(y\right)-{\phi }_{2,\epsilon }\left(z\right)|+{\displaystyle \frac{1}{{\epsilon }^2}}\left|{\displaystyle \frac{{\phi }_1\left(z\right)-{\phi }_1\left(y\right)}{({\phi }_1\left(y\right)+\epsilon )({\phi }_1\left(z\right)+\epsilon )}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{2}{{\epsilon }^2}}|{\phi }_2\left(y\right)-{\phi }_2\left(z\right)|+{\displaystyle \frac{1}{{\epsilon }^4}}\left|{\phi }_1\left(y\right)-{\phi }_1\left(z\right)\right|.\hfill \end{array}$$

Hence, the Gagliardo seminorm of w is finite. In addition, we know

$${\displaystyle \frac{{\phi }_{2,\epsilon }^2}{({\phi }_1+\epsilon )}}\le {\displaystyle \frac{{\phi }_2}{{\epsilon }^2}},$$

and so,

$${\int }_{\mathsf{\Omega }}{\left|w\right|}^{2}dy\le 2\left({\int }_{\mathsf{\Omega }}|{\displaystyle \frac{{\phi }_{2,\epsilon }^2}{{\phi }_1+\epsilon }}{|}^{2}dy+{\int }_{\mathsf{\Omega }}{\left|{\phi }_{1,\epsilon }\right|}^{2}dy\right)\le C\left(\epsilon \right)({\parallel {\phi }_2\parallel }_{L^2\left(\mathsf{\Omega }\right)}+{\parallel {\phi }_1\parallel }_{L^2\left(\mathsf{\Omega }\right)})<+\infty ,$$

where $C\left(\epsilon \right)$ is a positive constant. Consequently, this implies that $w\in X$. □

With the aid of Lemmas 4 and 5, we obtain the existence of at least one positive solution to problem (2).

Lemma 6.

If (H1)–(H3) hold, then problem (2) admits a positive solution for any $\lambda >0$.

Proof. 

Moreover, due to the subcritical growth of h, the functional $\mathsf{\Psi }$ is sequentially weakly continuous. In view of Lemma 4, we infer that the functional $\mathsf{\Phi }-\lambda \mathsf{\Psi }$ is sequentially weakly lower semicontinuous on X. Let us choose

$$ϵ\in \left(0,{\displaystyle \frac{C_0+{\eta }_0}{4\lambda C_0}}\right),$$

where ${\eta }_0$ and $C_0$ are given in Lemma 3. Since ${lim}_{\xi \to +\infty }{\displaystyle \frac{H(y,\xi )}{{\xi }^2}}=0$, there is $C\left(ϵ\right)>0$ such that

$$H(y,\xi )\le {ϵ\left|\xi \right|}^2+C\left(ϵ\right)$$

for any $\xi \in \mathbb{R}$. Thus, we have

$$\mathsf{\Psi }\left(u\right)\le ϵ{\int }_{\mathsf{\Omega }}{\left|\phi \left(y\right)\right|}^{2}dy+C\left(ϵ\right)\left|\mathsf{\Omega }\right|,$$

where $|·|$ denotes the Lebesgue measure on ${\mathbb{R}}^N$. From this and Lemma 3, we deduce that for any $\phi \in X$,

$$\begin{array}{cc}\hfill \mathsf{\Phi }\left(\phi \right)-\lambda \mathsf{\Psi }\left(\phi \right)& \ge {\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\phi \left(y\right)-\phi \left(z\right)\right|}^2\mathfrak{K}(y-z)dydz\hfill \\ \hfill & -ϵ\lambda {\int }_{\mathsf{\Omega }}{\left|\phi \left(x\right)\right|}^{2}dx-\lambda C\left(ϵ\right)\left|\mathsf{\Omega }\right|\hfill \\ \hfill & \ge \left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{{\eta }_0}{4C_0}}\right){\parallel \phi \parallel }_X^2-\lambda ϵ{\int }_{\mathsf{\Omega }}{\left|\phi \left(y\right)\right|}^{2}dy-\lambda C\left(ϵ\right)\left|\mathsf{\Omega }\right|\hfill \\ \hfill & \ge \left({\displaystyle \frac{C_0+{\eta }_0}{4C_0}}-\lambda ϵ\right){\parallel \phi \parallel }_X^2-\lambda C\left(ϵ\right)\left|\mathsf{\Omega }\right|.\hfill \end{array}$$

Hence, according to the choice of $ϵ$, we have

$$\underset{\parallel \phi \parallel \to +\infty }{lim}(\mathsf{\Phi }\left(\phi \right)-\lambda \mathsf{\Psi }\left(\phi \right))=+\infty .$$

Let the modified energy functional $\tilde{\mathcal{E}}:X\to \mathbb{R}$ be defined by

$$\tilde{\mathcal{E}}\left(\phi \right):={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^2\mathfrak{K}(y-z)dydz-\lambda {\int }_{{\mathbb{R}}^N}{H}^{+}(y,\phi )dy,\phi \in X,$$

where

$$H^{+}(y,\tau ):={\int }_0^{\tau }{h}^{+}(y,\xi )d\xi \mathrm{and}{h}^{+}(y,\tau ):=\left\{\begin{array}{cc}h(y,\tau ),\hfill & \tau \ge 0,\hfill \\ 0,\hfill & \tau <0\hfill \end{array}\right.$$

for almost all $y\in {\mathbb{R}}^N$ and all $\tau \in \mathbb{R}$. Then, it is clear from the above argument and Lemma 4 that the functional $\tilde{\mathcal{E}}$ is also coercive and sequentially weakly lower semicontinuous on X. This implies that there exists ${\phi }_0\in X$ such that

$$\tilde{\mathcal{E}}\left({\phi }_0\right)=inf\{\tilde{\mathcal{E}}\left(\phi \right):\phi \in X\}.$$

First, we prove that it is possible to suppose that ${\phi }_0\ge 0$. To this purpose, let us suppose that ${\phi }_0$ is sign-changing. Owing to Lemma 5, one has ${\phi }_0^{+}\in X$, and so, $\tilde{\mathcal{E}}\left({\phi }_0\right)\le \tilde{\mathcal{E}}\left({\phi }_0^{+}\right)$, where ${\phi }_0^{+}:=max\{{\phi }_0,0\}$. Since $\tilde{\mathcal{E}}\left(\phi \right)=\mathcal{E}\left(\phi \right)$ when $\phi \left(y\right)\ge 0$ for almost all $y\in \mathsf{\Omega }$, we assert

$$\begin{array}{cc}\hfill \tilde{\mathcal{E}}\left({\phi }_0^{+}\right)& =\mathcal{E}\left({\phi }_0^{+}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|{\phi }_0^{+}\left(y\right)-{\phi }_0^{+}\left(z\right)\right|}^2\mathfrak{K}(y,z)dydz-\lambda {\int }_{\mathsf{\Omega }}H(y,{\phi }_0^{+})dy\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|{\phi }_0\left(y\right)-{\phi }_0\left(z\right)\right|}^2\mathfrak{K}(y,z)dydz-\lambda {\int }_{\mathsf{\Omega }}H(y,{\phi }_0^{+})dy\hfill \\ \hfill & =\tilde{\mathcal{E}}\left({\phi }_0\right).\hfill \end{array}$$

Therefore, ${\phi }_0^{+}$ is a non-negative solution to problem (2). Now, for simplicity, let us write ${\phi }_0$ instead of ${\phi }_0^{+}$. We claim that ${\phi }_0>0$. To perform this, it suffices to prove that $\tilde{\mathcal{E}}\left({\phi }_0\right)<0$. Now, in light of Proposition 9 in [25], fix any non-negative function $\varphi \in X$, with $\varphi =0$ on $\partial \mathsf{\Omega }$, such that

$${\gamma }_1{\int }_{\mathsf{\Omega }}{\left|\varphi \left(y\right)\right|}^{2}dy={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\varphi \left(y\right)-\varphi \left(z\right)\right|}^2\mathfrak{K}(y-z)dydz,$$

(6)

where ${\gamma }_1$ is a positive eigenvalue that can be characterized as

$${\gamma }_1=\underset{\{\varphi \in X:\parallel \varphi |{|}_{L^2\left(\mathsf{\Omega }\right)}=1\}}{min}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\varphi \left(y\right)-\varphi \left(z\right)\right|}^2\mathfrak{K}(y-z)dydz.$$

Let ${\alpha }_0\in L^{\infty }\left(\mathsf{\Omega }\right)$ with ${\alpha }_0>0$ and let ${\kappa }_0\in (0,{\parallel {\alpha }_0\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)})$ be fixed. Then, the set

$${\mathsf{\Omega }}_{{\kappa }_0}:=\{y\in \mathsf{\Omega }:{\alpha }_0\left(y\right)\ge {\kappa }_0\}$$

(7)

has a positive measure. Furthermore, fix $\mathfrak{N}>0$ so that

$$\mathfrak{N}>{\displaystyle \frac{{\gamma }_1{\int }_{\mathsf{\Omega }}{\left|\varphi \left(y\right)\right|}^{2}dy}{\lambda {\kappa }_0{\int }_{{\mathsf{\Omega }}_{{\kappa }_0}}{\left|\varphi \left(y\right)\right|}^{2}dy}}.$$

From the second condition in (H3), there exists ${\xi }_0>0$ such that

$${\displaystyle \frac{H(y,\xi )}{{\xi }^2}}\ge {\displaystyle \frac{{\alpha }_0\left(y\right)\mathfrak{N}}{2}}$$

for all $\xi \in (0,{\xi }_0]$. Then, for sufficiently small $\epsilon >0$, one has

$$\begin{array}{cc}\hfill \lambda {\int }_{\mathsf{\Omega }}{\displaystyle \frac{H(y,\epsilon \varphi )}{{\epsilon }^2}}dy& \ge {\displaystyle \frac{\lambda \mathfrak{N}}{2}}{\int }_{\mathsf{\Omega }}{\alpha }_0\left(y\right){\left|\varphi \left(y\right)\right|}^{2}dy\hfill \\ \hfill & \ge {\displaystyle \frac{\lambda \mathfrak{N}{\kappa }_0}{2}}{\int }_{{\mathsf{\Omega }}_{{\kappa }_0}}{\left|\varphi \left(y\right)\right|}^{2}dy\hfill \\ \hfill & >{\displaystyle \frac{{\gamma }_1}{2}}{\int }_{\mathsf{\Omega }}{\left|\varphi \left(y\right)\right|}^{2}dy\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\varphi \left(y\right)-\varphi \left(z\right)\right|}^2\mathfrak{K}(y-z)dydz={\displaystyle \frac{1}{2}}{\left[\varphi \right]}_{s,2}^2.\hfill \end{array}$$

(8)

Therefore, using (8), we conclude that

$${\left[\varphi \right]}_{s,2}^2-2\lambda {\int }_{{\mathbb{R}}^N}{\displaystyle \frac{H(y,\epsilon \varphi )}{{\epsilon }^2}}<0$$

for any $\epsilon >0$ that is small enough, which is $\mathcal{E}\left(\epsilon \varphi \right)<0$, and thus, the conclusion holds. □

We are ready to demonstrate our first main consequence. The basic idea of the following assertion comes from the papers [22,23].

Theorem 1.

If (H1)–(H4) hold, then problem (2) possesses a unique positive solution for any $\lambda >0$.

Proof. 

Let ${\phi }_1,{\phi }_2$ be two positive weak solutions of (2). For any $\epsilon >0$, we define the truncations ${\phi }_{i,\epsilon }$ as in (4). Let us define the functions

$${\omega }_{1,\epsilon }:={\displaystyle \frac{{\phi }_{2,\epsilon }^2}{{\phi }_1+\epsilon }}-{\phi }_{1,\epsilon }$$

and

$${\omega }_{2,\epsilon }:={\displaystyle \frac{{\phi }_{1,\epsilon }^2}{{\phi }_2+\epsilon }}-{\phi }_{2,\epsilon }.$$

By utilizing Lemma 5, we arrive to the conclusion that ${\omega }_{i,\epsilon }\in X$ for $i=1,2$. Considering the above weak Formulation (3) of ${\phi }_i$, by choosing $\omega ={\omega }_{i,\epsilon }$ for $i=1,2$, we infer

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}& ({\phi }_1\left(y\right)-{\phi }_1\left(z\right))({\omega }_{1,\epsilon }\left(y\right)-{\omega }_{1,\epsilon }\left(z\right))\mathfrak{K}(y-z)dydz\hfill \\ \hfill & =\lambda {\int }_{\mathsf{\Omega }}h(y,{\phi }_1){\omega }_{1,\epsilon }\left(y\right)dy\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}& ({\phi }_2\left(y\right)-{\phi }_2\left(z\right))({\omega }_{2,\epsilon }\left(y\right)-{\omega }_{2,\epsilon }\left(z\right))\mathfrak{K}(y-z)dydz\hfill \\ \hfill & =\lambda {\int }_{\mathsf{\Omega }}h(y,{\phi }_2){\omega }_{2,\epsilon }\left(y\right)dy.\hfill \end{array}$$

(10)

Adding the above two Equations (9) and (10), and using the fact that

$${\phi }_i\left(y\right)-{\phi }_i\left(z\right)=({\phi }_i+\epsilon )\left(y\right)-({\phi }_i+\epsilon )\left(z\right)\mathrm{for}i=1,2,$$

we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\left(({\phi }_1+\epsilon )\left(y\right)-({\phi }_1+\epsilon )\left(z\right)\right)\left({\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(y\right)}{{\phi }_1\left(y\right)+\epsilon }}-{\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(z\right)}{{\phi }_1\left(z\right)+\epsilon }}\right)\mathfrak{K}(y,z)dydz\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}({\phi }_1\left(y\right)-{\phi }_1\left(z\right))({\phi }_{1,\epsilon }\left(y\right)-{\phi }_{1,\epsilon }\left(z\right))\mathfrak{K}(y,z)dydz\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\left(({\phi }_2+\epsilon )\left(y\right)-({\phi }_2+\epsilon )\left(z\right)\right)\left({\displaystyle \frac{{\phi }_{1,\epsilon }^2\left(y\right)}{{\phi }_2\left(y\right)+\epsilon }}-{\displaystyle \frac{{\phi }_{1,\epsilon }^2\left(z\right)}{{\phi }_2\left(z\right)+\epsilon }}\right)\mathfrak{K}(y,z)dydz\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}({\phi }_2\left(y\right)-{\phi }_2\left(z\right))({\phi }_{2,\epsilon }\left(y\right)-{\phi }_{2,\epsilon }\left(z\right))\mathfrak{K}(y,z)dydz\hfill \\ \hfill & =\lambda \left({\int }_{\mathsf{\Omega }}h(y,{\phi }_1)\left({\displaystyle \frac{{\phi }_{2,\epsilon }^2}{{\phi }_1+\epsilon }}-{\phi }_{1,\epsilon }\right)+h(y,{\phi }_2)\left({\displaystyle \frac{{\phi }_{1,\epsilon }^2}{{\phi }_2+\epsilon }}-{\phi }_{2,\epsilon }\right)dy\right).\hfill \end{array}$$

(11)

Now, applying the fact that $t\to \min \left\{\right|t|,{\epsilon }^{-1}\}$ is 1-Lipschitz, we determine by (5) that

$$\left(({\phi }_1+\epsilon )\left(y\right)-({\phi }_1+\epsilon )\left(z\right)\right)\left({\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(y\right)}{{\phi }_1\left(y\right)+\epsilon }}-{\displaystyle \frac{{\phi }_{2,\epsilon }^2\left(z\right)}{{\phi }_1\left(z\right)+\epsilon }}\right)\le {\left|{\phi }_2\left(y\right)-{\phi }_2\left(z\right)\right|}^2$$

and

$$\left(({\phi }_2+\epsilon )\left(y\right)-({\phi }_2+\epsilon )\left(z\right)\right)\left({\displaystyle \frac{{\phi }_{1,\epsilon }^2\left(z\right)}{{\phi }_2\left(z\right)+\epsilon }}\left(y\right)-{\displaystyle \frac{{\phi }_{1,\epsilon }^2\left(z\right)}{{\phi }_2\left(z\right)+\epsilon }}\right)\le {\left|{\phi }_1\left(y\right)-{\phi }_1\left(z\right)\right|}^2.$$

Since ${\phi }_{i,\epsilon }\to {\phi }_i$ as $\epsilon \to 0$ for $i=1,2$, by passing to the limit in (11) and by employing the Fatou Lemma in the first and third terms of (11) and the Dominated Convergence Theorem in all the other terms, we derive that

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}({\phi }_1\left(y\right)-{\phi }_1\left(z\right))\left({\displaystyle \frac{{\phi }_2^2\left(y\right)}{{\phi }_1\left(y\right)}}-{\displaystyle \frac{{\phi }_2^2\left(z\right)}{{\phi }_1\left(z\right)}}\right)\mathfrak{K}(y,z)dydz\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|{\phi }_1\left(y\right)-{\phi }_1\left(z\right)\right|}^2\mathfrak{K}(y,z)dydz\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}({\phi }_2\left(y\right)-{\phi }_2\left(z\right))\left({\displaystyle \frac{{\phi }_1^2\left(y\right)}{{\phi }_2\left(y\right)}}-{\displaystyle \frac{{\phi }_1^2\left(z\right)}{{\phi }_2\left(z\right)}}\right)\mathfrak{K}(y,z)dydz\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|{\phi }_2\left(y\right)-{\phi }_2\left(z\right)\right|}^2\mathfrak{K}(y,z)dydz\hfill \\ \hfill & \ge \lambda \left({\int }_{\mathsf{\Omega }}h(y,{\phi }_1)\left({\displaystyle \frac{{\phi }_2^2}{{\phi }_1}}-{\phi }_1\right)+h(y,{\phi }_2)\left({\displaystyle \frac{{\phi }_1^2}{{\phi }_2}}-{\phi }_2\right)dy\right)\hfill \\ \hfill & =-\lambda {\int }_{\mathsf{\Omega }}\left({\displaystyle \frac{h(y,{\phi }_1)}{{\phi }_1}}-{\displaystyle \frac{h(y,{\phi }_2)}{{\phi }_2}}\right)({\phi }_1^2-{\phi }_2^2)dy.\hfill \end{array}$$

(12)

From a simple computation, one has

$$({\phi }_1\left(y\right)-{\phi }_1\left(z\right))\left({\displaystyle \frac{{\phi }_2^2\left(y\right)}{{\phi }_1\left(y\right)}}-{\displaystyle \frac{{\phi }_2^2\left(z\right)}{{\phi }_1\left(z\right)}}\right)\le {\left|{\phi }_2\left(y\right)-{\phi }_2\left(z\right)\right|}^2$$

and

$$\left({\phi }_2\left(y\right)-{\phi }_2\left(z\right)\right)\left({\displaystyle \frac{{\phi }_{1,\epsilon }^2\left(y\right)}{{\phi }_2\left(y\right)}}-{\displaystyle \frac{{\phi }_{1,\epsilon }^2\left(z\right)}{{\phi }_2\left(z\right)}}\right)\le {\left|{\phi }_1\left(y\right)-{\phi }_1\left(z\right)\right|}^2.$$

This, together with relation (12), yields

$${\int }_{\mathsf{\Omega }}\left({\displaystyle \frac{h(y,{\phi }_1)}{{\phi }_1}}-{\displaystyle \frac{h(y,{\phi }_2)}{{\phi }_2}}\right)({\phi }_1^2-{\phi }_2^2)dy\ge 0.$$

Hence, since the function $\xi \mapsto {\displaystyle \frac{h(y,\xi )}{\xi }}$ is decreasing in $(0,+\infty )$, we can conclude that ${\phi }_1={\phi }_2$. Therefore, we ensure that problem (2) has a unique positive solution. □

4. Application

In this section, as an application of Theorem 1, we demonstrate the existence result of a unique nontrivial positive solution to a Kirchhoff-type problem driven by the non-local fractional Laplacian. To perform this, we present the abstract theorem provided by B. Ricceri [15].

Definition 2.

We say that $\phi \in X$ is a weak solution of problem (1) if

$$\begin{array}{c}\hfill m\left({\left[\phi \right]}_{s,2}^2\right){\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}(\phi \left(y\right)-\phi \left(z\right))(w\left(y\right)-w\left(z\right))\mathfrak{K}(y-z)dydz={\int }_{\mathsf{\Omega }}h(y,\phi )wdy\end{array}$$

for any $w\in X$.

Definition 3.

Let E be a topological space. Then, a function $g:E\to \mathbb{R}$ is said to be inf-compact (resp. sup-compact) provided that, for each $t\in \mathbb{R}$, the set $g^{-1}\left((-\infty ,t]\right)$ (resp. $g^{-1}\left([t,+\infty )\right)$ is compact.

The following abstract global minimum principle in [15] is crucial to obtain our main result.

Theorem 2.

Let E be a topological space, and let ${\mathsf{\Phi }}_0:E\to \mathbb{R}$, with ${\mathsf{\Phi }}_0^{-1}\left(0\right)\ne \varnothing$ and $\mathsf{\Psi }:E\to \mathbb{R}$ be two functions such that the function $\gamma {\mathsf{\Phi }}_0-\mathsf{\Psi }$ is a lower semicontinuous inf-compact and has a unique global minimum for each $\gamma >0$. In addition, suppose that Ψ has no global maxima in E. Moreover, let $J\subseteq (0,+\infty )$ be an open interval and $m:J\to \mathbb{R}$ be an increasing function satisfying $m\left(J\right)=(0,+\infty )$. Then, there is a unique $\tilde{v}\in E$ such that ${\mathsf{\Phi }}_0\left(\tilde{v}\right)\in J$ and

$$m\left({\mathsf{\Phi }}_0\left(\tilde{v}\right)\right){\mathsf{\Phi }}_0\left(\tilde{v}\right)-\mathsf{\Psi }\left(\tilde{v}\right)=\underset{u\in E}{inf}(m\left({\mathsf{\Phi }}_0\left(\tilde{v}\right)\right){\mathsf{\Phi }}_0\left(u\right)-\mathsf{\Psi }\left(u\right)).$$

We are in position to obtain our second main result.

Theorem 3.

Suppose that there is an open interval $J\subseteq (0,+\infty )$ such that the restriction of m to J is increasing and $m\left(J\right)=(0,+\infty )$. Let $h:\mathsf{\Omega }\times [0,+\infty )\to (0,+\infty )$ be a function such that conditions (H1)–(H4) are satisfied and $h(y,0)=0$ for almost all $y\in \mathsf{\Omega }$. Then, problem (1) admits a unique weak solution $\tilde{v}$, which is the unique global minimum in X of the functional

$$\begin{array}{cc}\hfill \phi \mapsto & {\displaystyle \frac{1}{2}}m\left({\left[\tilde{v}\right]}_{s,2}^2\right){\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\phi \left(y\right)-\phi \left(z\right)\right|}^2\mathfrak{K}(y-z)dydz-{\int }_{\mathsf{\Omega }}\left({\int }_0^{{\phi }^{+}\left(y\right)}h(y,\xi )d\xi \right)dy,\hfill \end{array}$$

where ${\phi }^{+}:=max\{\phi ,0\}$.

Proof. 

To start with, extend h to $\mathbb{R}$ by putting $h(y,\xi )=0$ for all $\xi <0$ and for almost all $y\in \mathsf{\Omega }$. To apply Theorem 2, put $E=X$, ${\mathsf{\Phi }}_0:=2\mathsf{\Phi }$ and define $\mathsf{\Psi }$ by

$$\mathsf{\Psi }\left(\phi \right):=2{\int }_{\mathsf{\Omega }}H(y,{\phi }^{+}\left(y\right))dy$$

for any $\phi \in X$, where $H(y,\xi )={\int }_0^{\xi }h(y,t)dt$. The functional $\mathsf{\Psi }$ is $C^1$ with derivatives, given by

$$〈{\mathsf{\Psi }}^{\prime }\left(\phi \right),\omega 〉=2{\int }_{\mathsf{\Omega }}h(y,\phi \left(y\right))\omega \left(y\right)dy$$

for all $\phi ,\omega \in X$. Fix $\gamma >0$. Then, let us choose

$$ϵ\in \left(0,{\displaystyle \frac{\gamma (C_0+{\eta }_0)}{2C_0}}\right).$$

From (H3), there is $C\left(ϵ\right)>0$, such that

$$H(y,\xi )\le {\displaystyle \frac{ϵ}{2}}{\left|\xi \right|}^2+{\displaystyle \frac{C\left(ϵ\right)}{2}}$$

for any $\xi \in \mathbb{R}$. Thus, we have

$$\mathsf{\Psi }\left(u\right)\le ϵ{\int }_{\mathsf{\Omega }}{\left|\phi \left(y\right)\right|}^{2}dy+C\left(ϵ\right)\left|\mathsf{\Omega }\right|.$$

From this and Lemma 3, one has

$$\begin{array}{cc}\hfill \gamma {\mathsf{\Phi }}_0\left(\phi \right)-\mathsf{\Psi }\left(\phi \right)& \ge \gamma {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\phi \left(y\right)-\phi \left(z\right)\right|}^2\mathfrak{K}(y-z)dydz\hfill \\ \hfill & -ϵ{\int }_{\mathsf{\Omega }}{\left|\phi \left(x\right)\right|}^{2}dx-C\left(ϵ\right)\left|\mathsf{\Omega }\right|\hfill \\ \hfill & \ge \gamma \left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{{\eta }_0}{2C_0}}\right){\parallel \phi \parallel }_X^2-ϵ{\int }_{\mathsf{\Omega }}{\left|\phi \left(y\right)\right|}^{2}dy-C\left(ϵ\right)\left|\mathsf{\Omega }\right|\hfill \\ \hfill & \ge \left({\displaystyle \frac{\gamma (C_0+{\eta }_0)}{2C_0}}-ϵ\right){\parallel \phi \parallel }_X^2-C\left(ϵ\right)\left|\mathsf{\Omega }\right|.\hfill \end{array}$$

Hence, in accordance with the choice of $ϵ$, we obtain

$$\underset{\parallel \phi \parallel \to +\infty }{lim}(\gamma {\mathsf{\Phi }}_0\left(\phi \right)-\mathsf{\Psi }\left(\phi \right))=+\infty$$

for any $\phi \in X$. This, together with the Eberlein–Smulyan theorem, and the reflexivity of X yield that the sequentially weakly lower semicontinuous functional $\gamma {\mathsf{\Phi }}_0-\mathsf{\Psi }$ is weakly inf-compact. Now, we claim that functional $\gamma {\mathsf{\Phi }}_0-\mathsf{\Psi }$ admits a unique global minimum in X. In fact, the critical points of $\gamma {\mathsf{\Phi }}_0-\mathsf{\Psi }$ are exactly the weak solutions of the problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathcal{L}\phi \left(y\right)={\displaystyle \frac{1}{\gamma }}h(y,\phi )\hfill & \mathrm{in} \mathsf{\Omega }\hfill \\ u=0\hfill & \mathrm{on} \partial \mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(13)

where we say that $\phi \in X$ is a weak solution of (13) if

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}(\phi \left(y\right)-\phi \left(z\right))(\omega \left(y\right)-\omega \left(z\right))\mathfrak{K}(y-z)dydz={\displaystyle \frac{1}{\gamma }}{\int }_{\mathsf{\Omega }}h(y,\phi )\omega dy$$

for any $\omega \in X$. Let ${\phi }_1,{\phi }_2\in X$ be two positive solutions of problem (13). Then, it follows from the same arguments as in Theorem 1 that ${\phi }_1={\phi }_2$, and therefore, problem (13) admits a unique positive solution. Now, fix any non-negative function $\phi \in X$ with $\phi =0$ on $\partial \mathsf{\Omega }$, such that

$${\gamma }_1{\int }_{\mathsf{\Omega }}{\left|\phi \left(y\right)\right|}^{p}dy={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\phi \left(y\right)-\phi \left(z\right)\right|}^p\mathfrak{K}(y-z)dydz,$$

where ${\gamma }_1$ is given in (6). Let ${\alpha }_0\in L^{\infty }\left(\mathsf{\Omega }\right)$ with ${\alpha }_0>0$ and let ${\kappa }_0\in (0,{\parallel {\alpha }_0\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)})$ and $\overline{\mathfrak{N}}>0$ be fixed, so that

$$\overline{\mathfrak{N}}>{\displaystyle \frac{\gamma {\gamma }_1{\int }_{\mathsf{\Omega }}{\left|\phi \left(y\right)\right|}^{2}dy}{{\kappa }_0{\int }_{{\mathsf{\Omega }}_{{\kappa }_0}}{\left|\phi \left(y\right)\right|}^{2}dy}},$$

where ${\mathsf{\Omega }}_{{\kappa }_0}$ is given in (7). From the second condition in (H3), there exists ${\xi }_0>0$, such that

$$H(y,\xi )\ge {\displaystyle \frac{{\alpha }_0\left(y\right)\overline{\mathfrak{N}}}{2}}{\xi }^2$$

for all $\xi \in (0,{\xi }_0]$. Now, set $v=\mu \phi$, where $\mu ={\displaystyle \frac{{\xi }_0}{{sup}_{\overline{\mathsf{\Omega }}}\phi }}$; one thus has

$$\begin{array}{cc}\hfill \mathsf{\Psi }\left(v\right)=2{\int }_{\mathsf{\Omega }}H(y,v)dy& \ge \overline{\mathfrak{N}}{\int }_{\mathsf{\Omega }}{\alpha }_0\left(y\right){\left|v\left(y\right)\right|}^{2}dy\hfill \\ \hfill & \ge \overline{\mathfrak{N}}{\kappa }_0{\int }_{{\mathsf{\Omega }}_{{\kappa }_0}}{\left|v\left(y\right)\right|}^{2}dy\hfill \\ \hfill & >\gamma {\gamma }_1{\int }_{\mathsf{\Omega }}{\left|v\left(y\right)\right|}^{2}dy\hfill \\ \hfill & =\gamma {\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|v\left(y\right)-v\left(z\right)\right|}^2\mathfrak{K}(y-z)dydz=\gamma {\mathsf{\Phi }}_0\left(v\right).\hfill \end{array}$$

(14)

This implies that 0 is not a global minimum for $\gamma {\mathsf{\Phi }}_0-\mathsf{\Psi }$. Accordingly, the global minimum of this functional is consistent with its only non-zero critical point.

Lastly, let us prove that $\mathsf{\Psi }$ has no global maxima. Suppose on the contrary that $\widehat{\phi }\in X$ is a global maximum of $\mathsf{\Psi }$. Clearly, $\mathsf{\Psi }\left(\widehat{\phi }\right)>0$. As a consequence, the set

$${\mathsf{\Omega }}_{+}:=\{y\in \mathsf{\Omega }:h(y,\widehat{\phi }\left(y\right))>0\}$$

has a positive measure. Let a closed set $\mathsf{\Lambda }\subset {\mathsf{\Omega }}_{+}$ of positive measure be fixed and let $\omega \in X$ be such that $\omega \ge 0$ and $\omega \left(y\right)=1$ for all $y\in \mathsf{\Lambda }$. Then, we have

$${\int }_{\mathsf{\Omega }}h(y,\widehat{\phi }\left(y\right))\omega \left(y\right)dy\ge {\int }_{\mathsf{\Lambda }}h(y,\widehat{\phi }\left(y\right))dy>0,$$

and so, ${\mathsf{\Psi }}^{\prime }\left(\widehat{\phi }\right)\ne 0$, which is ridiculous.

Hence, each hypothesis of Theorem 2 is ensured. As a result, there is a unique $\tilde{v}\in X$, with ${\left[\tilde{v}\right]}_{s,2}^2\in J$, such that

$$\begin{array}{cc}\hfill & m\left({\left[\tilde{v}\right]}_{s,2}^2\right){\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\tilde{v}\left(y\right)-\tilde{v}\left(z\right)\right|}^p\mathfrak{K}(y,z)dydz-2{\int }_{\mathsf{\Omega }}H(y,{\tilde{v}}^{+}\left(y\right))dy\hfill \\ \hfill & =\underset{\phi \in X}{inf}\left\{m\left({\left[\tilde{v}\right]}_{s,2}^2\right){\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{\left|\phi \left(y\right)-\phi \left(z\right)\right|}^2\mathfrak{K}(y,z)dydz-2{\int }_{\mathsf{\Omega }}H(y,{\phi }^{+}\left(y\right))dy\right\}\hfill \end{array}$$

Eventually, as seen above, the function $\tilde{v}$ is the unique positive weak solution for problem (1). □

5. Conclusions

This paper is dedicated to deriving the existence and uniqueness of positive solutions to the nonlinear elliptic equation of Kirchhoff type driven by the non-local fractional Laplacian by employing the abstract global minimum principle in [15] and the uniqueness result for the Brézis–Oswald-type problem based on [19]. From what we know, the uniqueness result of a positive weak solution to nonlinear fractional Laplacian problems of Brézis–Oswald type with a Kirchhoff coefficient has not been much studied, and to our knowledge, only the study of [7] is related to this type of problem. As mentioned in the introduction, our result is different from that of paper [7] because we deal with the lack of the continuity of the Kirchhoff function m in $[0,\infty )$ and the localization of the solution. However, our condition (H3) can be counted as a special case of that of [7,23] because the nonlinear term h satisfies the following assumptions:

$${\mathfrak{a}}_0\left(y\right)=\underset{\xi \to 0^{+}}{lim}{\displaystyle \frac{h(y,\xi )}{\xi }}\mathrm{and}{\mathfrak{a}}_{\infty }\left(y\right)=\underset{\xi \to +\infty }{lim}{\displaystyle \frac{h(y,\xi )}{\xi }}.$$

Let us define ${\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)$ and ${\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_{\infty }\right)$ as

$${\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)=\underset{\phi \in X}{inf}\left\{{\left[\phi \right]}_{s,2}^2-{\int }_{\mathsf{\Omega }}{\mathfrak{a}}_0{\left|\phi \left(y\right)\right|}^2{dy:\parallel \phi \parallel }_{L^2\left(\mathsf{\Omega }\right)}=1\right\}$$

and

$${\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)=\underset{\phi \in X}{inf}\left\{{\left[\phi \right]}_{s,2}^2-{\int }_{\mathsf{\Omega }}{\mathfrak{a}}_{\infty }{\left|\phi \left(y\right)\right|}^2{dy:\parallel \phi \parallel }_{L^2\left(\mathsf{\Omega }\right)}=1\right\}.$$

If ${\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)<0<{\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)$ instead of (H3) is satisfied, then it follows from analogous arguments as in [23] that problem (2) possesses a unique positive solution for any $\lambda >0$; that is, Theorem 1 holds. Consequently, taking Theorem 2 into account, obvious modifications of the proof of Theorem 3 yield the same assertion concerning problem (1) when ${\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)<0<{\lambda }_1\left(\mathcal{L}-{\mathfrak{a}}_0\right)$ instead of (H3) is assumed.

Moreover, a new research direction is to study the non-local fractional p-Laplacian equations with a discontinuous Kirchhoff coefficient, as follows:

$$\left\{\begin{array}{cc}m\left({\left[\phi \right]}_{s,p}^p\right){\mathcal{L}}_p\phi \left(y\right)=h(y,\phi )\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \phi >0\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ \phi =0\hfill & \mathrm{on}\partial \mathsf{\Omega },\hfill \\ {\left[\phi \right]}_{s,p}^p\in J,\hfill \end{array}\right.$$

(15)

where $p\in (1,+\infty )$ with $sp<N$, ${\left[\phi \right]}_{s,p}^p:={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^p\mathfrak{K}(y,z)dydz$ and ${\mathcal{L}}_p$ is a non-local operator defined pointwise as

$${\mathcal{L}}_p\phi \left(y\right)=2{\int }_{{\mathbb{R}}^N}{|\phi \left(y\right)-\phi \left(z\right)|}^{p-2}(\phi \left(y\right)-\phi \left(z\right))\mathfrak{K}(y,z)dz\mathrm{for}\mathrm{almost}\mathrm{all}y\in {\mathbb{R}}^N,$$

where a kernel function $\mathfrak{K}:{\mathbb{R}}^N\times {\mathbb{R}}^N\to (0,+\infty )$ has the following conditions:

(${\mathcal{L}}_{p}1$)

$\ell \mathfrak{K}\in L^1({\mathbb{R}}^N\times {\mathbb{R}}^N)$, where $\ell (y,z)=min\{1,|y-z{|}^p\}$;

(${\mathcal{L}}_{p}2$)

There are constants ${\gamma }_0>0$ and ${\gamma }_1>0$ with $1<{\gamma }_1$, such that ${\gamma }_1\le \mathfrak{K}(y,z){|y-z|}^{N+sp}\le {\gamma }_0$ for almost all $(y,z)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$ with $y\ne z$.;

(${\mathcal{L}}_{p}3$)

$\mathfrak{K}(z,y)=\mathfrak{K}(y,z)$ for all $(y,z)\in {\mathbb{R}}^N\times {\mathbb{R}}^N$.

As far as we know, there are no consequences for the existence, uniqueness and localization of positive solutions to problem (15).