Existence of Positive Ground States of Nonlocal Nonlinear Schrödinger Equations

Abstract

We investigate ground states of a (nonlocal) nonlinear Schrödinger equation which generalizes classical (fractional, relativistic, etc.) Schrödinger equations, so that we extend relevant results and study common properties of these equations in a uniform way. To obtain the existence of ground states, we first solve a minimization problem and then prove that the solution of the minimization problem is a ground state of the equation. After examining the regularity of the solutions to the equation, we demonstrate that any ground state is sign-definite.

1. Introduction

We intend to deal with the (nonlocal) nonlinear Schrödinger equation

$$-Au+Vu={\left|u\right|}^{p-2}u,$$

(1)

where A is the infinitesimal generator of a rotationally invariant Lévy process (up to some constant coefficient), and V is a potential function. Some assumptions on A, V and p are given later.

The reason why we study Equation (1) is as follows.

The nonlinear Schrödinger equation similar to

$$-\mathsf{\Delta }u+Vu={\left|u\right|}^{p-2}u,$$

(2)

which is driven by the infinitesimal generator of a standard Brownian motion (sped up by the factor $\sqrt{2}$), was studied extensively. Refer to, for example, [1,2,3,4,5].

Since the Brownian motion is a rotationally invariant stable Lévy process of index two, we may generalize Equation (2) to the equation

$${(-\mathsf{\Delta })}^{\alpha /2}u+Vu={\left|u\right|}^{p-2}u,$$

(3)

where $0<\alpha \le 2$, because of the fact that $-{(-\mathsf{\Delta })}^{\alpha /2}$ is the infinitesimal generator of a rotationally invariant stable Lévy process (up to some constant coefficient) of index $\alpha$. By the path integral approach, Laskin obtained the fractional Schrödinger equation [6,7]. Schrödinger equations involving fractional Laplacians like (3) were largely studied. See [8,9,10,11,12] for instance.

As a matter of course, discarding the ‘stable’ property, we attempt to study Equation (1). Zhang and Zhu [13] explored this equation driven by rotationally invariant Lévy processes of lower dimensions. Zhang and Zhou [14] researched this equation with nondegenerate diffusion terms and finite Lévy measures. Zhang [15] studied this equation for general rotationally invariant Lévy processes, but with a constant potential function.

Equation (1) is also the result of seeking the solution $\psi (t,x):=e^{\mathrm{i}t}u\left(x\right)$ to the equation

$$\mathrm{i}\frac{\partial \psi }{\partial t}=-A\psi +(V-1)u-{\left|\psi \right|}^{p-2}\psi .$$

Longhi [16] employed a linear fractional Schrödinger equation to study optics, thus one can use the nonlocal operator A appearing in Equation (1) to examine the same problems as in [16], since a fractional Laplacian is a special case of the nonlocal operator A. Dysthe and Trulsen [17] used classical nonlinear Schrödinger equations to inspect freak-waves. For the same reason, one may take advantage of the nonlinear Schrödinger equation involving the nonlocal operator A to study freak waves. Based on the above facts, it is necessary in order to investigate Equation (1).

We denote the symbol of A by ${\sigma }_A$ and make the following assumption on the operator A and the potential function V:

(A1)

$-{\sigma }_A\left(\zeta \right)\ge c{\left|\xi \right|}^{2s}$ for all $\zeta \in {\mathbb{R}}^N$ with $\left|\xi \right|\ge K$, where $s,c,K>0$.

(A2)

For any $q\in [2,+\infty )$, $(1+{\left|\zeta \right|}^{2s})·{(1-{\sigma }_A\left(\zeta \right))}^{-1}$ is an $L^q$-Fourier multiplier.

(V1)

The function V is continuous and periodic, i.e., $V\in C\left({\mathbb{R}}^N\right)$ and $V(x+T_j)=V\left(x\right)$ for some $T_j:=(0,0,\cdots ,\underset{j\mathrm{th}}{\underbrace{t_j}},\cdots ,0)\in {\mathbb{R}}^N$, $j=1,2,\cdots ,N$, and any $x\in {\mathbb{R}}^N$.

(V2)

The function V has a positive lower bound, i.e., $\underset{x\in {\mathbb{R}}^N}{min}V\left(x\right)>0$.

Remark 1.

(1)

It follows from ([18], Exercise 2.4.23) that ${\sigma }_A\le 0$ and $s\le 1$ and furthermore, from ([19], p. 17), that $s=1$ if and only if the process generated by A involves diffusion terms.

(2)

There are numerous operators satisfying (A1) and (A2) including classical Laplacian, fractional Laplacian, relativistic Schrödinger operators, etc. Refer to [15].

From here on, (A1), (A2), (V1), (V2) and $2<p<2_s^{*}$ are assumed to hold. Here, $2_s^{*}:=\left\{\begin{array}{cc}+\infty ,\hfill & \mathrm{if}N\le 2s,\hfill \\ \frac{2N}{N-2s},\hfill & \mathrm{if}N>2s\hfill \end{array}\right.$.

We introduce a suitable space $\mathfrak{X}$ of functions in which we look for ground states of (1). Let ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^N\right)$ be the space of tempered distributions on ${\mathbb{R}}^N$ and ‘$\widehat{}$’ be Fourier transform. We define the Hilbert space $\mathfrak{X}$ through

$$\mathfrak{X}:=\{u:u\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^N\right)\mathrm{and}{(-{\sigma }_A(·))}^{1/2}\widehat{u}(·)\mathrm{and}\sqrt{V}u\in L^2\left({\mathbb{R}}^N\right)\},$$

and the inner product on $\mathfrak{X}$

$$(\varphi ,\psi ):={\left(2\pi \right)}^{-N}{\left({(-{\sigma }_A(·))}^{\frac{1}{2}}\widehat{\varphi }(·),{(-{\sigma }_A(·))}^{\frac{1}{2}}\widehat{\psi }(·)\right)}_{L^2}+{\int }_{{\mathbb{R}}^N}V\left(x\right)\varphi \left(x\right)\psi \left(x\right)\mathrm{d}x,$$

(4)

and the induced norm $\parallel ·\parallel$.

Define a functional $E:\mathfrak{X}\to \mathbb{R}$ through

$$E\left(v\right):=\frac{1}{2}{\parallel v\parallel }^2-\frac{1}{p}{\int }_{{\mathbb{R}}^N}{\left|v\left(x\right)\right|}^p\mathrm{d}x.$$

(5)

We summarize the main results in Theorem 1.

Theorem 1.

(a)

(Existence) There is a nonzero function $w\in \mathfrak{X}$ such that

$$-Aw+Vw={\left|w\right|}^{p-2}w$$

in a distribution sense. Moreover, $E\left(w\right)=(\frac{1}{2}-\frac{1}{p}){\left(\underset{u\in \mathcal{M}}{inf}\parallel u\parallel \right)}^{\frac{2p}{p-2}}$, where

$$\mathcal{M}:=\left\{u:u\in {\mathfrak{X}\mathrm{and}\parallel u\parallel }_{L^p}=1\right\},$$

and

$$E\left(w\right)=inf\left\{E\left(u\right):u\in \mathfrak{X}\setminus {\left\{0\right\}\mathrm{and}\parallel u\parallel }^2={\parallel u\parallel }_{L^p}^p\right\}.$$

(6)

(b)

(Regularity) Any solution $u\in \mathfrak{X}$ to Equation (1) in a distribution sense is a function in $W^{2s,r}\left({\mathbb{R}}^N\right)$ for any $r\ge max\{2,\frac{2_s^{*}}{p-1}\}$. Furthermore, if $s^{\prime }\le s$ and $0\le \mu \le 2s^{\prime }-\frac{N}{r}<1$, then u belongs to $C^{0,\mu }\left({\mathbb{R}}^N\right)$ and if $s=1$, then u is in $C_{\mathrm{loc}}^{2,\mu }\left({\mathbb{R}}^N\right)$. As a consequence, we have $\underset{\left|x\right|\to \infty }{lim}u\left(x\right)=0$.

(c)

(Positivity) Any ground state of Equation (1) is sign-definite. (A solution to Equation (1) satisfying (6) is called a ground state (cf. [20], p. 71).)

2. Preliminary

This section serves as a preliminary. After defining the Banach space $W_A^{s,r}\left({\mathbb{R}}^N\right)$, we list some properties concerning it. A result of Brézis and Lieb is required in our argument.

Definition 1 

([21], Chapter 3 and [15], Definition 2.1). Let ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^N\right)$ be the space of tempered distributions on ${\mathbb{R}}^N$. For $s\in \mathbb{R}$ and $r\in (1,+\infty )$, the Banach space $W_A^{s,r}\left({\mathbb{R}}^N\right)$ is defined to be the set of all tempered distributions w for which $({(1-{\sigma }_A(·))}^{\frac{s}{2}}\widehat{w}(·))ˇ$ is a function in $L^q\left({\mathbb{R}}^N\right)$, i.e.,

$$W_A^{s,q}\left({\mathbb{R}}^N\right):=\{w:w\in {\mathcal{D}}^{\prime }\left({\mathbb{R}}^N\right)\mathrm{and}({(1-{\sigma }_A(·))}^{\frac{s}{2}}\widehat{w}(·))ˇ\in L^r\left({\mathbb{R}}^N\right)\}.$$

Here, ‘$\widehat{}$’ (‘$\stackrel{ˇ}{}$’) denotes the Fourier transform (inverse Fourier transform).

Some properties about $W_A^{s,r}\left({\mathbb{R}}^N\right)$ are stated as follows.

Lemma 1.

(i)

The following embeddings

$$W_A^{2,r}\left({\mathbb{R}}^N\right)\hookrightarrow W^{2s,r}\left({\mathbb{R}}^N\right),r>1,$$

$$W_A^{1,2}\left({\mathbb{R}}^N\right)\hookrightarrow L^r\left({\mathbb{R}}^N\right),N\le 2s\mathrm{and}r\ge 2,$$

$$W_A^{1,2}\left({\mathbb{R}}^N\right)\hookrightarrow L^r\left({\mathbb{R}}^N\right),N>2s\mathrm{and}2\le r\le 2_s^{*},$$

$$\mathfrak{X}\hookrightarrow W_A^{1,2}\left({\mathbb{R}}^N\right)$$

are continuous.

(ii)

If $2\le r<2_s^{*}$, every bounded sequence in $W_A^{1,2}\left({\mathbb{R}}^N\right)$ possesses a convergent subsequence in $L_{\mathrm{loc}}^r\left({\mathbb{R}}^N\right)$.

Proof. 

Those conclusions come from ([15], Lemma 2.2) except for $\mathfrak{X}\hookrightarrow W_A^{1,2}\left({\mathbb{R}}^N\right)$. The embedding $\mathfrak{X}\hookrightarrow W_A^{1,2}\left({\mathbb{R}}^N\right)$ is an immediate consequence of (V1), (V2) and the definitions of $\mathfrak{X}$ and $W_A^{1,2}\left({\mathbb{R}}^N\right)$. □

Lemma 2 

(Concentration compactness principle ([15], Lemma 2.3)). Let $a>0$ and $2\le r<2_s^{*}$. If ${\left\{u_n\right\}}_{n=1}^{\infty }$ is bounded in $W_A^{1,2}\left({\mathbb{R}}^N\right)$ and if

$$\underset{n\to \infty }{lim}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{B(y,a)}{\left|u_n\left(x\right)\right|}^r\mathrm{d}x=0,$$

then $u_n\to 0$ in $L^r\left({\mathbb{R}}^N\right)$.

We need the following lemma from Brézis and Lieb.

Lemma 3 

([20], Lemma 1.32). Let Ω be an open subset of ${\mathbb{R}}^N$ and ${\left\{u_n\right\}}_{n=1}^{\infty }\subset L^p(\mathsf{\Omega })$, where $1\le p<\infty$. If (1) ${\left\{u_n\right\}}_{n=1}^{\infty }$ is bounded in $L^p(\mathsf{\Omega })$ and (2) $u_n\to u$ a.e. on Ω as $n\to \infty$, then $\underset{n\to \infty }{lim}(\parallel u_n{\parallel }_{L^p(\mathsf{\Omega })}^p-\parallel u_n{-u\parallel }_{L^p(\mathsf{\Omega })}^p{)=\parallel u\parallel }_{L^p(\mathsf{\Omega })}^p$.

3. Proof of Theorem 1

We show a proof of Theorem 1 by solving a minimization problem and close the paper with a corollary of Theorem 1.

A property concerning the norm induced by the inner product (4) is needed.

Lemma 4. 

For $u\in \mathfrak{X}$ and $y:=(k_1{t}_1,k_2{t}_2,\cdots ,k_N{t}_N)$, where $k_j\in \mathbb{Z}$, $j=1,2,\cdots ,N$, set $v(·):=u(·+y)$, then $\parallel v\parallel =\parallel u\parallel$.

Proof. 

By the definition of the norm, we have

$$\begin{array}{cc}\hfill {\parallel v\parallel }^2& ={\left(2\pi \right)}^{-N}{\left({(-{\sigma }_A(·))}^{\frac{1}{2}}\widehat{v}(·),{(-{\sigma }_A(·))}^{\frac{1}{2}}\widehat{v}(·)\right)}_{L^2}+{\int }_{{\mathbb{R}}^N}V\left(x\right)v{\left(x\right)}^2\mathrm{d}x\hfill \\ & ={\left(2\pi \right)}^{-N}{\int }_{{\mathbb{R}}^N}{(-{\sigma }_A\left(\xi \right))}^{\frac{1}{2}}\widehat{v}\left(\xi \right)\overline{{(-{\sigma }_A\left(\xi \right))}^{\frac{1}{2}}\widehat{v}\left(\xi \right)}\mathrm{d}\xi +{\int }_{{\mathbb{R}}^N}V\left(x\right)v{\left(x\right)}^2\mathrm{d}x\hfill \\ & ={\left(2\pi \right)}^{-N}{\int }_{{\mathbb{R}}^N}{(-{\sigma }_A\left(\xi \right))}^{\frac{1}{2}}\widehat{u}\left(\xi \right)\overline{{(-{\sigma }_A\left(\xi \right))}^{\frac{1}{2}}\widehat{u}\left(\xi \right)}\mathrm{d}\xi +{\int }_{{\mathbb{R}}^N}V\left(x\right)u{\left(x\right)}^2\mathrm{d}x\hfill \\ & \mathrm{since}\widehat{v}\left(\xi \right)=exp(\mathrm{i}y·\xi )\widehat{u}\left(\xi \right)\mathrm{and},\mathrm{by}\left(1\right),V(x-y)=V\left(x\right)\hfill \\ & ={\parallel u\parallel }^2,\hfill \end{array}$$

which completes the proof. □

Lemma 5. 

Let $\mathcal{M}:=\left\{u:u\in {\mathfrak{X}\mathrm{and}\parallel u\parallel }_{L^p}=1\right\}$. The minimization problem

$$\underset{u\in \mathcal{M}}{inf}\parallel u\parallel$$

(7)

has a solution.

Proof. 

• Let $\phi$ be a nonzero, rapidly decreasing function. We have $\phi /{\parallel \phi \parallel }_{L^p\left({\mathbb{R}}^N\right)}\in \mathcal{M}$, thus $\mathcal{M}\ne \varnothing$.
• Let ${\left\{v_n\right\}}_{n=1}^{\infty }\subset \mathcal{M}$ be a minimizing sequence, i.e.,

  $$\parallel v_n\parallel \to \underset{u\in \mathcal{M}}{inf}\parallel u\parallel \mathrm{as}n\to \infty .$$

Then, ${\left\{v_n\right\}}_{n=1}^{\infty }$ is a bounded sequence in $\mathcal{M}\subset \mathfrak{X}\subset W_A^{1,2}\left({\mathbb{R}}^N\right)$.

Take $r:=\left|T\right|+1$, where $T:=(t_1,t_2,\cdots ,t_N)$. It follows from Lemma 2 that there is a subsequence of ${\left\{v_n\right\}}_{n=1}^{\infty }$, again denoted by ${\left\{v_n\right\}}_{n=1}^{\infty }$, such that

$${\int }_{B(y_n,r)}{v}_n{\left(x\right)}^2\mathrm{d}x>\epsilon$$

for a sequence ${\left\{y_n\right\}}_{n=1}^{\infty }$ with $y_n\in {\mathbb{R}}^N$ and a positive number $\epsilon$.

Choose some $y_n^{\prime }\in B(y_n,r)$ with $y_n^{\prime }=(k_1{t}_1,k_2{t}_2,\cdots ,k_N{t}_N)$, where $k_j\in \mathbb{Z}$, $j=1,2,\cdots ,N$. Thus, we have

$${\int }_{B(y_n^{\prime },2r)}{v}_n{\left(x\right)}^2\mathrm{d}x>\epsilon .$$

For $n=1,2,\cdots$, set $v_n^{\prime }(·):=v_n(·+y_n^{\prime })$. Then,

$${\int }_{B(0,2r)}{v}_n^{\prime }{\left(x\right)}^2\mathrm{d}x>\epsilon ,$$

(8)

and by Lemma 4

$${\left\{v_n^{\prime }\right\}}_{n=1}^{\infty }\mathrm{is}\mathrm{also}\mathrm{a}\mathrm{minimizing}\mathrm{sequence}\mathrm{of}\left(7\right).$$

(9)

Note that ${\left\{v_n^{\prime }\right\}}_{n=1}^{\infty }$ is bounded in $\mathfrak{X}$. Thus, there is a subsequence of ${\left\{v_n^{\prime }\right\}}_{n=1}^{\infty }$, again denoted by ${\left\{v_n^{\prime }\right\}}_{n=1}^{\infty }$, such that

$$v_n^{\prime }\rightharpoonup v\mathrm{in}\mathfrak{X}$$

(10)

for some $v\in \mathfrak{X}$, and by Lemma 1

$$v_n^{\prime }\to v\mathrm{in}{L}_{\mathrm{loc}}^p\left({\mathbb{R}}^N\right),$$

(11)

and

$$v_n^{\prime }\to v\mathrm{a}.\mathrm{e}.\mathrm{on}{\mathbb{R}}^N.$$

(12)

The sequence ${\left\{v_n^{\prime }\right\}}_{n=1}^{\infty }$ is bounded in $L^p\left({\mathbb{R}}^N\right)$ by the boundness of it in $\mathfrak{X}$ and Lemma 1, which, together with (12), implies that

$$\underset{n\to \infty }{lim}\parallel v_n^{\prime }{-v\parallel }_{L^p\left({\mathbb{R}}^N\right)}^p=1-{\parallel v\parallel }_{L^p\left({\mathbb{R}}^N\right)}^p$$

(13)

by Lemma 3 and $\parallel v_n^{\prime }{\parallel }_{L^p\left({\mathbb{R}}^N\right)}=1$.

3.

Setting $S:=\underset{u\in \mathcal{M}}{inf}\parallel u\parallel$, we have

$$\parallel u\parallel \ge {S\parallel u\parallel }_{L^p\left({\mathbb{R}}^N\right)}\mathrm{for}\mathrm{any}u\in \mathfrak{X}.$$

(14)

Then, we obtain

$$\begin{array}{cc}\hfill S^2& =\underset{n\to \infty }{lim}{\parallel v_n^{\prime }\parallel }^2\mathrm{by}(9)\hfill \\ \hfill & =\underset{n\to \infty }{lim}{\parallel v_n^{\prime }-v+v\parallel }^2\hfill \\ \hfill & =\underset{n\to \infty }{lim}\parallel v_n^{\prime }{-v\parallel }^2+2\underset{n\to \infty }{lim}(v,v_n^{\prime }-v)+{\parallel v\parallel }^2\hfill \\ \hfill & =\underset{n\to \infty }{lim}\parallel v_n^{\prime }{-v\parallel }^2+{\parallel v\parallel }^2\mathrm{by}(10)\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \ge S^2({(1-\parallel v\parallel }_{L^p\left({\mathbb{R}}^N\right)}^p{)}^{2/p}+{\parallel v\parallel }_{L^p\left({\mathbb{R}}^N\right)}^2)\mathrm{by}(13)\mathrm{and}(14).\hfill \end{array}$$

(16)

4.

It follows from (8) and (11) that $v\ne 0$, which, together with (15), implies $S>0$, and then, by (16), we obtain ${(1-\parallel v\parallel }_{L^p\left({\mathbb{R}}^N\right)}^p{)}^{2/p}+{\parallel v\parallel }_{L^p\left({\mathbb{R}}^N\right)}^2\le 1$. Note that ${(1-s)}^{2/p}+s^{2/p}\ge 1$ for $s\in [0,1]$ and ${(1-s)}^{2/p}+s^{2/p}=1$ iff $s=0$ or $s=1$. Then, we have ${\parallel v\parallel }_{L^p\left({\mathbb{R}}^N\right)}=1$, i.e., $v\in \mathcal{M}$.

5.

It follows from the weak semicontinuity of $\parallel ·\parallel$ that v is a solution to (7). □

Recall the functional $E:\mathfrak{X}\to \mathbb{R}$,

$$E\left(v\right)=\frac{1}{2}{\parallel v\parallel }^2-\frac{1}{p}{\int }_{{\mathbb{R}}^N}{\left|v\left(x\right)\right|}^p\mathrm{d}x,$$

and define the Nehari manifold $\mathcal{N}$ by

$$\mathcal{N}:=\left\{v:v\in \mathfrak{X}\setminus \left\{0\right\}\mathrm{and}{E}^{\prime }\left(v\right)v=0\right\}=\left\{v:v\in \mathfrak{X}\setminus {\left\{0\right\}\mathrm{and}\parallel v\parallel }^2={\parallel v\parallel }_{L^p}^p\right\}.$$

We are in a position to provide a proof of Theorem 1.

Proof of Theorem 1. 

• Let v be a solution to (7), $S:=\underset{u\in \mathcal{M}}{inf}\parallel u\parallel$ and $w:=S^{\frac{2}{p-2}}v$. It follows from the Lagrange multiplier rule that

  $$2(v,\psi )=\lambda p{({\left|v\right|}^{p-2}v,\psi )}_{L^2}$$

  for some number $\lambda$ and any $\psi \in \mathfrak{X}$. In particular, taking $\psi =v$, we find $\lambda =2S^2/p$. Furthermore, $(v,\psi )=S^2{({\left|v\right|}^{p-2}v,\psi )}_{L^2}$, i.e., $-Av+Vv=S^2{\left|v\right|}^{p-2}v$, and $-Aw+Vw={\left|w\right|}^{p-2}w$. We have proved that w is a solution to Equation (1).
• By Lemma 5, we obtain

  $$\frac{\parallel u\parallel }{{\parallel u\parallel }_{L^p}}\ge \parallel v\parallel =S\mathrm{for}\mathrm{any}u\in \mathfrak{X}\setminus \left\{0\right\}.$$

Thus, if $u\in \mathcal{N}$,

$${\parallel u\parallel }^2\ge S^{\frac{2p}{p-2}}={\parallel w\parallel }^2$$

by the definitions of $\mathcal{N}$ and w.

Noting that $w\in \mathcal{N}$ by the definition of w, we have

$$\underset{u\in \mathcal{N}}{inf}E\left(u\right)=\underset{u\in \mathcal{N}}{inf}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{p}}\right){\parallel u\parallel }^2=\left(\frac{1}{2}-\frac{1}{p}\right){\parallel w\parallel }^2=\left(\frac{1}{2}-\frac{1}{p}\right)S^{\frac{2p}{p-2}},$$

which, along with step 1, implies (a).

3.

Let u be a solution to (1). With the help of the embedding $\mathfrak{X}\hookrightarrow W_A^{1,2}\left({\mathbb{R}}^N\right)$ in Lemma 1 and the bootstrapping procedure, we have $u\in H^{2s,q}\left({\mathbb{R}}^N\right)$ for $q\ge max\{2,2_s^{*}/(p-1)\}$ (cf. [15], Theorem 3.1). Ref. [22], Theorem 7.63 and Schauder’s estimate complete the proof of (b).

4.

Let $\phi$ be a ground state of Equation (1). Then, we have, by step 2, $\parallel \phi \parallel =S^{\frac{p}{p-2}}$. Setting $\varphi :=S^{\frac{2}{2-p}}\phi$, we see that $\varphi \in \mathcal{M}$ as ${\parallel \varphi \parallel }_{L^p}=S^{\frac{2}{2-p}}{\parallel \phi \parallel }_{L^p}=S^{\frac{2}{2-p}}{\parallel \phi \parallel }^{\frac{2}{p}}=S^{\frac{2}{2-p}}·S^{\frac{p}{p-2}·\frac{2}{p}}=1$, where we have used ${\parallel \phi \parallel }^2={\parallel \phi \parallel }_{L^p}^p$ for $\phi \in \mathcal{N}$.

For any $u\in \mathfrak{X}\setminus \left\{0\right\}$, we find

$$\tilde{u}:={\left(\frac{{\parallel u\parallel }^2}{{\parallel u\parallel }_{L^p}^p}\right)}^{1/(p-2)}u\in \mathcal{N},$$

and then we obtain

$$\parallel \tilde{u}\parallel \ge \parallel \phi \parallel =S^{2/(p-2)}\parallel \varphi \parallel ,\mathrm{as}\phi \mathrm{is}\mathrm{a}\mathrm{ground}\mathrm{state}\mathrm{of}\mathrm{Equation}(),$$

i.e.,

$${\left(\frac{\parallel u\parallel }{{\parallel u\parallel }_{L^p}}\right)}^{\frac{p}{p-2}}\ge S^{\frac{2}{p-2}}\parallel \varphi \parallel .$$

From taking the infimum in the above inequality over $u\in \mathfrak{X}\setminus \left\{0\right\}$, it follows that $S^{\frac{p}{p-2}}\ge S^{\frac{2}{p-2}}\parallel \varphi \parallel$, i.e., $\parallel \varphi \parallel \le S$. In summary, $\varphi$ is a solution to (7).

Noting that for $\psi \in \mathfrak{X}$,

$${\parallel \psi \parallel }^2=\parallel \sqrt{V}{\psi \parallel }_{L^2}^2+a{\parallel \nabla \psi \parallel }_{L^2}^2+{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N\setminus \left\{0\right\}}{\left(\psi \left(x\right)-\psi (x+y)\right)}^2\nu \left(\mathrm{d}y\right)\mathrm{d}x$$

for some positive number a and Lévy measure $\nu$. In light of [23], Theorem 7.8 and the proof of [23], Theorem 7.13, we have

Claim

Let $f,g$ be real functions in $\mathfrak{X}$. Then,

$${∥\sqrt{f^2+g^2}∥}^2\le {∥f∥}^2+{∥g∥}^2.$$

Thanks to the above claim, we see that $\left|\varphi \right|$ is also a solution to (7). By steps 1–2, $\left|\phi \right|$ is also a ground state of Equation (1). Therefore, we assume, without loss of generality, that $\phi \ge 0$. If $\nu =0$, the maximum principle of elliptic equations tells us that $\phi >0$. Suppose that $\nu \ne 0$, and let $x_0$ be a point such that $\phi \left(x_0\right)=0$. Then, we have $A\phi \left(x_0\right)>0$ and reach the contradiction that $0=\phi {\left(x_0\right)}^{p-1}=-A\phi \left(x_0\right)+V\left(x_0\right)\phi \left(x_0\right)<0$. This contradiction tells us that $\phi >0$. Now, we obtain (c). □

Corollary 1.

(i)

Assume that $0<u\in \mathfrak{X}$ is a solution to equation $-Au+Vu={\left|u\right|}^{p-2}u$, and $x_0\in {\mathbb{R}}^N$ is a maximizer of u. We have $u\left(x_0\right)\ge V{\left(x_0\right)}^{\frac{1}{p-2}}$.

(ii)

Any solution to (7) is sign-definite.

Proof. 

Ad (i).

With the help of the positive maximum principle ([24], Proposition 1.5 and [18], Theorem 3.5.2), we have $Au\left(x_0\right)\le 0$. Consequently,

$$u{\left(x_0\right)}^{p-1}-V\left(x_0\right)u\left(x_0\right)=-Au\left(x_0\right)\ge 0.$$

Thus, the inequality $u\left(x_0\right)\ge V{\left(x_0\right)}^{\frac{1}{p-2}}$ holds.

Ad (ii).

Let v be a solution to (7). We found in the proof of Theorem 1 that $S^{\frac{2}{p-2}}v$ is a ground state of Equation (1), and then any ground state of Equation (1) is either positive or negative by (c) of Theorem 1. Thus, the proof is complete. □

4. Conclusions

We studied a nonlocal, nonlinear Schrödinger Equation (1). The equation extends classical (fractional, relativistic, etc.) Schrödinger equations. We recalled the definition of Banach space $W_A^{s,r}(\mathbb{R})$ and then gave some embedding properties and a concentration compactness principle in regard to $W_A^{s,r}(\mathbb{R})$. With these preparations in hand, we solved a minimization problem which implied a weak solution to Equation (1). In virtue of the embedding properties concerning $W_A^{s,r}\left(\mathbb{R}\right)$, the regularity of the solution was examined. Moreover, the solution was in fact a ground state to Equation (1), which was also sign-definite. In the literature, these equations were researched separately. Noting their common features, we studied them in a uniform way. In this direction, one may consider Schrödinger equations driven by general Feller processes. As a coin consists of two sides, we can ignore their specific characteristics. Nevertheless, one can employ Equation (1) and its developing form to study problems arising from physics [16,17].