Positive Solutions for Boundary Value Problems in the Half-Space with Sign-Changing Nonlinearities

Abstract

We study a class of semilinear elliptic equations of the form $-\Delta v=g+\mu f\left(.,v\right)$ in the half-space subject to suitable boundary conditions. The source term g is assumed to be a nonnegative function in a Kato-type class, while the reaction term f may change sign. Under appropriate assumptions on f, we prove, for sufficiently small values of the parameter $\mu$, the existence and uniqueness of a positive continuous solution that is bounded above and below by a multiple of the solution for the corresponding linear problem.

1. Introduction

Semilinear elliptic equations have been widely studied because they arise in many stationary models, including reaction–diffusion processes, nonlinear heat transfer, population dynamics, fluid mechanics, and geometric problems (see, for example, [1] and references therein). Over the years, a substantial amount of literature has been devoted to studying semilinear elliptic equations in half-spaces:

$${\mathbb{R}}_{+}^N=\left\{z=(z^{\prime },z_N)\in {\mathbb{R}}^N:z_N>0\right\}, N\ge 2.$$

In particular, researchers have been interested in studying questions of existence and nonexistence results (see, for example, [2,3,4,5,6,7] and references therein), Liouville-type theorems (see, for instance, [8]), symmetry properties of solutions (see [9]), and classification of solutions (see, for example, [10,11]).

In [12], the author considered the following superlinear elliptic problem:

$$\left\{\begin{array}{cc}\Delta v=\rho \left(z\right)v^{\gamma },\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ v>0,\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b,\hfill & \end{array}\right.$$

(1)

where $\gamma \ge 1$, $a, b\ge 0$ with $a+b>0$ and $\varphi$ belongs to $C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right)$, the set of nonnegative bounded continuous functions in ${\mathbb{R}}^{N-1}$. Existence and uniqueness of positive solutions were obtained using a potential-theoretic approach, under the assumption that the weighted function $\rho$ is nonnegative and satisfies suitable hypotheses related to the Kato class ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ introduced in [13] for $N\ge 3$ and in [14] for $N=2$ as follows:

Definition 1.

We say that the function φ belongs to the class ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ if

$$\begin{array}{ccc}\hfill \underset{r\to 0}{\lim } \left(\underset{z\in {\mathbb{R}}_{+}^N}{\sup }{\int }_{{\mathbb{R}}_{+}^N\cap B(z,r)}{\displaystyle \frac{t_N}{z_N}}G(z,t)\left|\phi \left(t\right)\right|dt\right)& =& 0,\hfill \\ \hfill \underset{M\to \infty }{\lim } \left(\underset{z\in {\mathbb{R}}_{+}^N}{\sup }{\int }_{{\mathbb{R}}_{+}^N\cap (\left|t\right|\ge M)}{\displaystyle \frac{t_N}{z_N}}G(z,t)\left|\phi \left(t\right)\right|dt\right)& =& 0.\hfill \end{array}$$

Here, $G(z,t)$ denotes the Green function for the Dirichlet Laplacian in ${\mathbb{R}}_{+}^N.$

This class is quite rich; it contains (see [15], Example 2) any function in $L^s\left({\mathbb{R}}_{+}^N\right)\cap L^1\left({\mathbb{R}}_{+}^N\right),$ with $s>\frac{N}{2}$ and $N\ge 3$.

Furthermore, it has been proved (see [13], Proposition 9) that ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ strictly contains the classical Kato class $K_N^{\infty }\left({\mathbb{R}}_{+}^N\right)(N\ge 3)$, introduced in [16,17].

In [2], the authors extended the existence results obtained in [12] to the problem

$$\left\{\begin{array}{cc}\Delta v=\mu f\left(.,v\right),\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ v>0,\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b,\hfill & \end{array}\right.$$

(2)

where $\mu >0,$ $a, b\ge 0$ with $a+b>0$ and $\varphi$ belongs to $C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right).$ The nonnegative reaction term f belongs to a class of functions wider than those of the form $\rho \left(z\right)v^{\gamma },$ $\gamma \ge 1$. Using perturbation arguments, the authors proved some existence results and asymptotic behavior for a small range of $\mu$ (see [2], Theorem 3.6).

In [15] (Theorem 1.4), existence results are also obtained for problem (2) with a nonnegative reaction terms of the form $f(.,v)=v\phi (.,v).$

Nevertheless, these results rely on restrictive assumptions concerning the sign of the nonlinearity or the monotonicity of the associated reaction term.

In this work, we are concerned with semilinear elliptic equations of the form

$$\left\{\begin{array}{cc}-\Delta v=g+\mu f\left(.,v\right),\hfill & \mathrm{in} {\mathbb{R}}_{+}^N, (\mathrm{in} \mathrm{the} \mathrm{sense} \mathrm{of} \mathrm{distributions})\hfill \\ v>0,\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b,\hfill & \end{array}\right.$$

(3)

where $\mu >0,$ $a, b\ge 0$ with $a+b>0$ and $\varphi$ belongs to $C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right).$

The source term g is assumed to be a nonnegative function in the Kato class ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$. The main difficulty is that the nonlinearity f is allowed to change sign. This prevents the direct use of arguments that rely on positivity or monotonicity of the reaction term. We prove existence, uniqueness, and sharp global estimates for positive solutions by combining potential estimates with a fixed point argument.

The paper is organized as follows. Section 2 recalls the main properties of the Kato class ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ used in the sequel. Section 3 contains the existence, uniqueness, and global behavior results. Some application examples are given. Concluding remarks and possible directions for future investigation are also discussed.

Notations.

Throughout the paper, we use the following notations:

i. 

${\mathbb{R}}_{+}^N=\left\{z=(z^{\prime },z_N)\in {\mathbb{R}}^N:z_N>0\right\},$  $N\ge 2.$

ii. 

For $z\in {\mathbb{R}}_{+}^N,$ write $\overline{z}=(z^{\prime },-z_N).$

iii.

Denote by $\mathcal{B}\left({\mathbb{R}}_{+}^N\right)$ the set of Borel measurable functions on ${\mathbb{R}}_{+}^N$, and by ${\mathcal{B}}^{+}\left({\mathbb{R}}_{+}^N\right)$ its subset of nonnegative functions.

iv.

${\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right):=\{\phi \in {\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ with $\phi \ge 0\}.$

v. 

$C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right)$ denotes the set of nonnegative bounded continuous functions in ${\mathbb{R}}^{N-1}.$

$C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right)=\{\phi \in C\left({\mathbb{R}}_{+}^N\right):$ $\phi$ is bounded in ${\mathbb{R}}_{+}^N\}.$

$C_0\left({\mathbb{R}}_{+}^N\right)=\{\phi \in C\left({\mathbb{R}}_{+}^N\right):$ $\underset{z_N\to 0}{\lim }\phi \left(z\right)=0$ and $\underset{\left|z\right|\to \infty }{\lim }\phi \left(z\right)=0\}.$

$C_0\left(\overline{{\mathbb{R}}_{+}^N}\right)=\{\phi \in C\left(\overline{{\mathbb{R}}_{+}^N}\right):$  $\underset{\left|z\right|\to \infty }{\lim }\phi \left(z\right)=0\}.$

Note that $C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right),$ $C_0\left({\mathbb{R}}_{+}^N\right)$ and $C_0\left(\overline{{\mathbb{R}}_{+}^N}\right)$ are Banach spaces endowed with

$${∥\phi ∥}_{\infty }=\underset{z\in {\mathbb{R}}_{+}^N}{\sup }\left|\phi \left(z\right)\right|.$$

vi.

Let $G(z,t)$ be the Green function of the Laplace operator in ${\mathbb{R}}_{+}^N$ defined as the solution of the following problem:

$$\left\{\begin{array}{cc}-\Delta G(.,t)\left(z\right)={\delta }_t\left(z\right),& z\in {\mathbb{R}}_{+}^N,\\ G(z,t)=0,& z\in \partial {\mathbb{R}}_{+}^N,\end{array}\right.$$

where ${\delta }_t$ denotes the Dirac measure at $t$.

From [18] (Theorem 4.1.6), we have

$$G(z,t)=\left\{\begin{array}{cc}{\displaystyle \frac{\Gamma \left(\frac{N}{2}-1\right)}{4{\pi }^{\frac{N}{2}}}}\left[{\displaystyle \frac{1}{{\left|z-t\right|}^{N-2}}}-{\displaystyle \frac{1}{{\left|z-\overline{t}\right|}^{N-2}}}\right],& \mathrm{if} N\ge 3,\\ {\displaystyle \frac{1}{4\pi }}\log \left(1+{\displaystyle \frac{4z_2{t}_2}{{\left|z-t\right|}^2}}\right),& \mathrm{if} N=2.\end{array}\right.$$

(4)

vii. 

For $\phi \in {\mathcal{B}}^{+}\left({\mathbb{R}}_{+}^N\right),$ define $\mathcal{U}\phi$ by

$${\displaystyle \mathcal{U}\phi \left(z\right):={\int }_{{\mathbb{R}}_{+}^N}G(z,t)\phi \left(t\right)dt, z\in {\mathbb{R}}_{+}^N.}$$

viii. 

Throughout the paper, for a given function $\varphi$ in $C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right),$ let

$$H\varphi \left(z\right):={\displaystyle \frac{2}{{\sigma }_N}}{\int }_{{\mathbb{R}}^{N-1}}{\displaystyle \frac{z_N}{{\left|z-\xi \right|}^N}}\varphi \left(\xi \right)d\xi , \mathrm{for} z\in {\mathbb{R}}_{+}^N,$$

(5)

where ${\sigma }_N={\displaystyle \frac{N{\pi }^{\frac{N}{2}}}{\Gamma (\frac{N}{2}+1)}}.$

From [18] (Section 1.7), $H\varphi$ is the unique nonnegative bounded continuous function in the ${\mathbb{R}}_{+}^N$ solution of the problem

$$\left\{\begin{array}{cc}\Delta v=0,\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=0.\hfill & \end{array}\right.$$

(6)

Furthermore, if $\varphi$ is non-trivial, then by using (5) and the inequality $\left|z-\xi \right|\le (1+\left|z\right|)(1+\left|\xi \right|),$ we obtain, for some positive constant $c$,

$$c{\displaystyle \frac{z_N}{{(1+\left|z\right|)}^N}}\le H\varphi \left(z\right)\le {∥\varphi ∥}_{\infty }.$$

(7)

Remark 1

(see [19]). For $\phi \in {\mathcal{B}}^{+}\left({\mathbb{R}}_{+}^N\right),$ if $\mathcal{U}\phi \left(t_0\right)<\infty$ for some $t_0\in {\mathbb{R}}_{+}^N,$ then $\mathcal{U}\phi \in L_{\mathrm{loc}}^1\left({\mathbb{R}}_{+}^N\right)$. Conversely, if $\phi \in {\mathrm{L}}_{\mathrm{loc}}^1\left({\mathbb{R}}_{+}^N\right)$ and $\mathcal{U}\phi \in {\mathrm{L}}_{\mathrm{loc}}^1\left({\mathbb{R}}_{+}^N\right),$ then

$$-\Delta \left(\mathcal{U}\phi \right)=\phi , in {\mathbb{R}}_{+}^N.$$

2. Basic Properties of ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$

We recall some properties of functions in ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ that will be used in the analysis of problem (3). Their proofs can be found in [13,14,15].

Proposition 1.

For $\phi \in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right),$ we have the following:

(i) 

${\displaystyle {\Lambda }_{\phi }:=\underset{z,y\in {\mathbb{R}}_{+}^N}{\sup }{\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{G(z,t)G(t,y)}{G(z,y)}}\phi \left(t\right)dt<\infty .}$

(ii) 

${\displaystyle {\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{t_N}{{\left(\left|t\right|+1\right)}^N}}\phi \left(t\right)dt<\infty .}$

(iii) 

$t\to \mathcal{U}\phi \left(t\right)\in C_0\left({\mathbb{R}}_{+}^N\right).$

From [13] (Proposition 8) if $N\ge 3$ and [14] (Proposition 3.7) if $N=2,$ we have the following:

Remark 2.

Let $\alpha <2<\beta$ and $\phi \left(z\right)={\displaystyle \frac{1}{{(\left|z\right|+1)}^{\beta -\alpha }{z}_N^{\alpha }}},$ then $\phi \in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ and

$$\mathcal{U}\phi \left(z\right)\le C.\left\{\begin{array}{cc}{\displaystyle \frac{z_N^{2-\alpha }}{{(\left|z\right|+1)}^{N+2-2\alpha }}},\hfill & \mathit{if} 1<\alpha <2 \mathit{and} \beta \ge N+2-\alpha ,\hfill \\ {\displaystyle \frac{z_N}{{(\left|z\right|+1)}^N}}\log \left({\displaystyle \frac{2{(\left|z\right|+1)}^2}{z_N}}\right),\hfill & \begin{array}{c}\mathit{if} \alpha =1 \mathit{and} \beta \ge N+1,\hfill \\ \mathit{or}\hfill \\ \alpha <1 \mathit{and} \beta =N+1,\hfill \end{array}\hfill \\ {\displaystyle \frac{z_N}{{(\left|z\right|+1)}^N}},\hfill & \mathit{if} \alpha =1 \mathit{and} \beta >N+1,\hfill \\ {\displaystyle \frac{z_N^{\beta -N}}{{(\left|z\right|+1)}^{2\beta -N-2}}}\hfill & \mathit{if} N<\beta <\min (N+1,N+2-\alpha ),\hfill \end{array}\right.$$

for some $C>0.$

In particular, if $\alpha =1$ and $\beta >N+1,$ then

$${\displaystyle \frac{1}{C}}{\displaystyle \frac{z_N}{{\left(\left|z\right|+1\right)}^N}}\le \mathcal{U}\phi \left(z\right)\le C{\displaystyle \frac{z_N}{{\left(\left|z\right|+1\right)}^N}}.$$

(8)

Example 1

(see [15] (Proposition 2.11)). Let $s>{\displaystyle \frac{N}{2}}$ and $a\in L^s\left({\mathbb{R}}_{+}^N\right),$ then functions of the form

$$z⟶{\displaystyle \frac{a\left(z\right)}{{(\left|z\right|+1)}^{\beta -\alpha }{z}_N^{\alpha }}},$$

belong to ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ provided that $\alpha <2-{\displaystyle \frac{N}{s}}<\beta .$

For $\phi \in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right),$ we put

$${\mathcal{M}}_{\phi }:=\left\{\psi \in \mathcal{B}\left({\mathbb{R}}_{+}^N\right), \left|\psi \left(z\right)\right|\le \phi \left(z\right), \mathrm{for} \mathrm{all} z\in {\mathbb{R}}_{+}^N\right\}.$$

We shall use the following lemma for the continuity argument.

Lemma 1.

Let $\phi \in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right),$ then

(i) 

The family of functions

$$\mathcal{U}\left({\mathcal{M}}_{\phi }\right)=\left\{\mathcal{U}\psi : \psi \in {\mathcal{M}}_{\phi }\right\}$$

is relatively compact in $C_0\left({\mathbb{R}}_{+}^N\right).$

(ii) 

The family of functions

$${\mathcal{N}}_{\phi }=\{z\to {\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{t_N}{z_N}}G(z,t)\left|\psi \left(t\right)\right|dt:\psi \in {\mathcal{M}}_{\phi }\}$$

is relatively compact in $C_0\left(\overline{{\mathbb{R}}_{+}^N}\right).$

Proof. 

The proof is given in [13] (Proposition 10 and Lemma 6) if $N\ge 3$ and in [14] (Proposition 3.5 and Lemma 5.1) if $N=2.$ □

3. Existence Results

We now study problem (3). We first establish several preliminary results.

Let $a,b\ge 0$ with $a+b>0$ and $\varphi \in C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right).$ Define

$$\theta \left(z\right):=1+z_N \mathrm{and} h\left(z\right):=aH\varphi \left(z\right)+b{z}_N, \mathrm{for} z\in {\mathbb{R}}_{+}^N,$$

(9)

where $H\varphi \left(z\right)$ is given by (5).

Lemma 2.

Let $\phi \in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right),$ then the family of functions

$${\mathcal{F}}_{\phi }=\{z\to {\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{\theta \left(t\right)}{\theta \left(z\right)}}G(z,t)\left|\psi \left(t\right)\right|dt:\psi \in {\mathcal{M}}_{\phi }\}$$

is relatively compact in $C_0\left(\overline{{\mathbb{R}}_{+}^N}\right).$

Proof. 

Observe that

$${\displaystyle \frac{\theta \left(t\right)}{\theta \left(z\right)}}={\displaystyle \frac{1+t_N}{1+z_N}}\le \max (1,{\displaystyle \frac{t_N}{z_N}})\le 1+{\displaystyle \frac{t_N}{z_N}}.$$

Therefore, the conclusion follows from Lemma 1.  □

Remark 3.

Note that h is the unique nonnegative continuous function in the ${\mathbb{R}}_{+}^N$ solution of the problem

$$\left\{\begin{array}{cc}\Delta v=0,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b.\hfill & \end{array}\right.$$

(10)

Indeed, since the function $z\to z_N$ is harmonic in ${\mathbb{R}}_{+}^N,$ then from (6) it is clear that $h\in C\left({\mathbb{R}}_{+}^N\right)$ and satisfies (10). The uniqueness follows from the maximum principle ([18], Theorem 3.1.6).

Lemma 3.

Let $\phi \in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ and $p\in {\mathcal{B}}^{+}\left(D\right).$ Set

$$\omega \left(z\right):=h\left(z\right)+\mathcal{U}p\left(z\right)=aH\varphi \left(z\right)+b{z}_N+\mathcal{U}p\left(z\right).$$

Then the following estimate holds:

$$\mathcal{U}\left(\omega \phi \right)\left(z\right)\le {\Lambda }_{\phi } \omega \left(z\right), \mathrm{for} z\in {\mathbb{R}}_{+}^N.$$

Proof. 

Applying Fatou’s lemma, we obtain

$${\displaystyle {\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{t_N}{z_N}}G(z,t)\phi \left(t\right)dt\le \underset{\left|\zeta \right|\to \infty }{\lim \inf }{\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{t_N}{z_N}}\frac{{\left|z-\zeta \right|}^N}{{\left|t-\zeta \right|}^N}G(z,t)\phi \left(t\right)dt.}$$

On the other hand, since

$$\underset{y\to \zeta \in \partial {\mathbb{R}}_{+}^N}{\lim }{\displaystyle \frac{G(t,y)}{G(z,y)}}={\displaystyle \frac{t_N}{z_N}}{\displaystyle \frac{{\left|z-\zeta \right|}^N}{{\left|t-\zeta \right|}^N}},$$

we deduce again by Fatou’s lemma

$$\begin{array}{ccc}\hfill {\displaystyle {\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{t_N}{z_N}}\frac{{\left|z-\zeta \right|}^N}{{\left|t-\zeta \right|}^N}G(z,t)\phi \left(t\right)dt}& \le & {\displaystyle \underset{y\to \zeta }{\lim \inf }{\int }_{{\mathbb{R}}_{+}^N}G(z,t)\frac{G(t,y)}{G(z,y)}\phi \left(t\right)dt}\hfill \\ & \le & {\Lambda }_{\phi }.\hfill \end{array}$$

Hence

$${\displaystyle {\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{t_N}{z_N}}G(z,t)\phi \left(t\right)dt\le {\Lambda }_{\phi }}$$

(11)

and

$${\displaystyle {\int }_{{\mathbb{R}}_{+}^N}\frac{t_N}{{\left|t-\zeta \right|}^N}G(z,t)\phi \left(t\right)dt\le {\Lambda }_{\phi }\frac{z_N}{{\left|z-\zeta \right|}^N}, \mathrm{for} \mathrm{all} \zeta \in {\mathbb{R}}^{N-1}.}$$

(12)

Integrating (12) with respect to $\zeta$ and using the Fubini–Tonelli theorem and (5), we obtain

$${\displaystyle {\int }_{{\mathbb{R}}_{+}^N}H\varphi \left(t\right)G(z,t)\phi \left(t\right)dt\le {\Lambda }_{\phi }H\varphi \left(z\right).}$$

(13)

Now using the Fubini–Tonelli theorem and Proposition 1 (i), we have

$${\int }_{{\mathbb{R}}_{+}^N}G(z,t) \phi \left(t\right) \mathcal{U}p\left(t\right) dt\le {\Lambda }_{\phi } \mathcal{U}p\left(z\right), \mathrm{for} z\in {\mathbb{R}}_{+}^N.$$

(14)

Combining estimates (11), (13) and (14), the desired estimate follows.  □

We now state the assumptions required for our existence result.

H1. 

g belongs to ${\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right).$

H2. 

$f(z,0)=0,$ for all $z\in {\mathbb{R}}_{+}^N.$

H3. 

There exists $q\in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right),$ such that

$$\left|f(z,y)-f(z,t)\right|\le q\left(z\right)\left|y-t\right|, \mathit{for} \mathit{all} z\in {\mathbb{R}}_{+}^N \mathit{and} y,t\in \mathbb{R}.$$

Example 2.

Assume that there exist $C>0$ and $\alpha <2<\beta$ such that $0\le q\left(z\right)\le {\displaystyle \frac{C}{{(\left|z\right|+1)}^{\beta -\alpha }{z}_N^{\alpha }}}$ for all $z\in {\mathbb{R}}_{+}^N.$ Then from Remark 2, q belongs to ${\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ and assumptions (H2) and (H3) are satisfied with

$$f(z,t)=q\left(z\right)\sin t.$$

Remark 4.

Assume (H1) and let

$$\overline{\omega }\left(z\right):=h\left(z\right)+\mathcal{U}g\left(z\right)=aH\varphi \left(z\right)+b{z}_N+\mathcal{U}g\left(z\right),\hspace{1em}z\in {\mathbb{R}}_{+}^N.$$

(15)

Since by Proposition 1 (iii), we have $\mathcal{U}g\in C_0\left({\mathbb{R}}_{+}^N\right),$ then $\overline{\omega }\in C\left(\overline{{\mathbb{R}}_{+}^N}\right)$ and from Remarks 1–3, we conclude that

$$\left\{\begin{array}{cc}-\Delta \overline{\omega }=g,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \overline{\omega }>0,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }\overline{\omega }\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{\overline{\omega }\left(z\right)}{z_N}}=b.\hfill & \end{array}\right.$$

(16)

Note that

$$\begin{array}{ccc}\hfill \overline{\omega }\left(z\right)& \le & a{∥H\varphi ∥}_{\infty }+b{z}_N+{∥\mathcal{U}g∥}_{\infty }\hfill \\ & \le & \tilde{c}\theta \left(z\right),\hfill \end{array}$$

(17)

with $\tilde{c}:=\max (a{∥H\varphi ∥}_{\infty }+{∥\mathcal{U}g∥}_{\infty },b).$

The next lemma gives the integral formulation of problem (3).

Lemma 4.

Assume that hypotheses (H1)–(H3) hold. Let $v\in C\left(\overline{{\mathbb{R}}_{+}^N}\right)$ satisfying $\left|v\left(z\right)\right|\le c_0\overline{\omega }\left(z\right)$ on ${\mathbb{R}}_{+}^N$ for some $c_0>0.$ Then v solves problem (3) if and only if

$$v\left(z\right)=\overline{\omega }\left(z\right)+\mu \mathcal{U}f(.,v)\left(z\right), \mathit{for} z\in {\mathbb{R}}_{+}^N.$$

(18)

Proof. 

Assume that v satisfies Equation (18). From hypotheses and (17), we obtain

$$\begin{array}{ccc}\hfill \left|f(t,v(t\left)\right)\right|& =& \left|f(t,v(t\left)\right)-f(t,0)\right|\hfill \\ & \le & q\left(t\right)\left|v\left(t\right)\right|\hfill \\ & \le & c_{0}q\left(t\right)\overline{\omega }\left(t\right)\hfill \\ & \le & c_0\tilde{c}q\left(t\right)\theta \left(t\right).\hfill \end{array}$$

Consequently by writing

$$\mathcal{U}f(.,v)\left(z\right)=\theta \left(z\right){\int }_{{\mathbb{R}}_{+}^N}{\displaystyle \frac{G(z,t)}{\theta \left(z\right)}}f(t,v\left(t\right))dt,$$

and using Lemma 2, we deduce that $z\to \mathcal{U}f(.,v)\left(z\right)\in C\left(\overline{{\mathbb{R}}_{+}^N}\right)$ and

$$\underset{z_N\to 0}{\lim }\mathcal{U}f(.,v)\left(z\right)=0 \mathrm{and}\underset{z_N\to \infty }{\lim }{\displaystyle \frac{\mathcal{U}f(.,v)\left(z\right)}{z_N}}=0.$$

By means of Remark 1, we obtain

$$(-\Delta )\mathcal{U}\left(\mu f\right(·,v\left)\right)=\mu f(·,v).$$

Combining this with Remark 4, it follows that v solves problem (3).

Conversely, assume that v is a solution of problem (3). Define $v_0\left(z\right):v\left(z\right)-\overline{\omega }\left(z\right)-\mu \mathcal{U}f(.,v)\left(z\right)$. Then

$$\left\{\begin{array}{cc}-\Delta v_0=0,\hfill & \mathrm{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }{v}_0\left(z\right)=0,\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v_0\left(z\right)}{z_N}}=0.\hfill & \end{array}\right.$$

From the maximum principle (see [18] (Theorem 3.1.6)), it follows that $v_0\equiv 0$, and therefore v satisfies (18).  □

We are now in a position to state the main result of the paper.

Theorem 1.

Suppose that (H1)–(H3) are fulfilled. Then, there exists a constant ${\mu }^{*}>0$ such that, for any $\mu \in (0,{\mu }^{*}),$ problem (3) admits a unique continuous solution v satisfying on ${\mathbb{R}}_{+}^N,$

$${\displaystyle \frac{2({\mu }^{*}-\mu )}{2{\mu }^{*}-\mu }}\overline{\omega }\left(z\right)\le v\left(z\right)\le {\displaystyle \frac{2{\mu }^{*}}{2{\mu }^{*}-\mu }}\overline{\omega }\left(z\right),$$

(19)

where $\overline{\omega }\left(z\right)=aH\varphi \left(z\right)+b{z}_N+\mathcal{U}g\left(z\right).$

In particular,

(i) 

If $a=0,$ then for some positive constant c

$${\displaystyle \frac{1}{c}}{z}_N\le v\left(z\right)\le c(1+z_N).$$

(20)

(ii) 

If $b=0$ and ϕ is non-trivial, then for some positive constant c

$${\displaystyle \frac{1}{c}}{\displaystyle \frac{z_N}{{\left(\left|z\right|+1\right)}^N}}\le v\left(z\right)\le c(aH\varphi \left(z\right)+\mathcal{U}g\left(z\right)).$$

(21)

Here the positive constant c depends only on $N$, $a$, $b$, $\varphi$, $g$, q and $\mu .$

Proof. 

Assume (H1)–(H3). For $z\in {\mathbb{R}}_{+}^N$, define

$${\omega }_0\left(z\right):={\displaystyle \frac{\overline{\omega }\left(z\right)}{\theta \left(z\right)}},$$

where $\overline{\omega }\left(z\right)=aH\varphi \left(z\right)+b{z}_N+\mathcal{U}g\left(z\right)$ and $\theta \left(z\right):=1+z_N.$

Using (H1), Remark 4 and Lemma 3, we have

$${\omega }_0\in C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right) \mathrm{and} {\displaystyle \frac{1}{\theta \left(z\right)}} \mathcal{U}\left(q\theta {\omega }_0\right)\left(z\right)\le {\Lambda }_q {\omega }_0\left(z\right).$$

(22)

Let

$$\mathcal{D}=\{u\in C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right): \underset{z\in {\mathbb{R}}_{+}^N}{\sup }{\displaystyle \frac{\left|u\left(z\right)\right|}{{\omega }_0\left(z\right)}}<\infty \}.$$

Observe that

$$u\in \mathcal{D}\iff u={\omega }_0\phi , \mathrm{where} \phi \in C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right).$$

Hence $\mathcal{D}$ is a Banach space equipped with the following ${\omega }_0$-norm:

$${∥u∥}_{{\omega }_0}:=\underset{z\in {\mathbb{R}}_{+}^N}{\sup }{\displaystyle \frac{\left|u\left(z\right)\right|}{{\omega }_0\left(z\right)}}.$$

Consequently $(\mathcal{D},d)$ is a complete metric space equipped with the distance defined by

$$d(u_1,u_2):={∥u_1-u_2∥}_{{\omega }_0}.$$

Set ${\mu }^{*}:={\displaystyle \frac{1}{2{\Lambda }_q}}>0.$ For $\mu \in (0,{\mu }^{*}),$ let

$$\alpha :={\displaystyle \frac{2({\mu }^{*}-\mu )}{2{\mu }^{*}-\mu }} \mathrm{and} \beta :={\displaystyle \frac{2{\mu }^{*}}{2{\mu }^{*}-\mu }}.$$

Define

$$\mathcal{S}:=\{u\in \mathcal{D}, \alpha {\omega }_0\left(z\right)\le u\left(z\right)\le \beta {\omega }_0\left(z\right), \forall z\in {\mathbb{R}}_{+}^N\}.$$

Since $\mathcal{S}$ is a closed subset of $\mathcal{D}$, then $(\mathcal{S},d)$ is a complete metric space.

For $u\in \mathcal{S},$ define $\mathcal{T}$ by

$$\mathcal{T}u\left(z\right)={\omega }_0\left(z\right)+{\displaystyle \frac{\mu }{\theta \left(z\right)}}\mathcal{U}\left(f(.,\theta u(.))\right)\left(z\right), \mathrm{for} z\in {\mathbb{R}}_{+}^N.$$

(23)

We claim that $\mathcal{T}\left(\mathcal{S}\right)\subset \mathcal{S}.$ Indeed, using (H2) and (H3), we obtain

$$\begin{array}{ccc}\hfill \left|f(t,\theta (t\left)u\right(t\left)\right)\right|& =& \left|f(t,\theta (t\left)u\right(t\left)\right)-f(t,0)\right|\hfill \\ & \le & \beta {\omega }_0\left(t\right)\theta \left(t\right)q\left(t\right)\hfill \\ & \le & \beta {∥{\omega }_0∥}_{\infty }\theta \left(t\right)q\left(t\right).\hfill \end{array}$$

(24)

Since ${\omega }_0\in C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right)$ and $q\in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right)$, Lemma 2 implies that $z\mapsto {\displaystyle \frac{1}{\theta \left(z\right)}}\mathcal{U}f(·,\theta u)\left(z\right)\in C_0\left(\overline{{\mathbb{R}}_{+}^N}\right).$ Hence $\mathcal{T}u\in C_{\mathbf{b}}\left({\mathbb{R}}_{+}^N\right)$. Using (23) with the estimates in (22) and (24) we obtain

$$-\mu \beta {\Lambda }_q{\omega }_0\left(z\right)\le {\displaystyle \frac{\mu }{\theta \left(z\right)}}\mathcal{U}\left(f(.,\theta u(.))\right)\left(z\right)\le \mu \beta {\Lambda }_q{\omega }_0\left(z\right).$$

This implies that $\mathcal{T}\left(\mathcal{S}\right)\subset \mathcal{S}.$

Let $u_1,u_2\in \mathcal{S}.$ Using the Lipschitz condition $\left(H3\right)$ together with (22), we obtain

$$\begin{array}{ccc}\hfill \left|\mathcal{T}{u}_1\left(z\right)-\mathcal{T}{u}_2\left(z\right)\right|& \le & {\displaystyle \frac{\mu }{\theta \left(z\right)}}{\int }_{{\mathbb{R}}_{+}^N}G(z,t)\left|f(t,\theta \left(t\right)u_1\left(t\right))-f(t,\theta \left(t\right)u_2\left(t\right))\right| dt\hfill \\ & \le & {\displaystyle \frac{\mu }{\theta \left(z\right)}}{\int }_{{\mathbb{R}}_{+}^N}G(z,t)q\left(t\right)\theta \left(t\right)\left|u_1\left(t\right)-u_2\left(t\right)\right| dt\hfill \\ & \le & {\displaystyle \frac{\mu }{\theta \left(z\right)}}{\int }_{{\mathbb{R}}_{+}^N}G(z,t)q\left(t\right)\theta \left(t\right){\omega }_0\left(t\right){\displaystyle \frac{\left|u_1\left(t\right)-u_2\left(t\right)\right|}{{\omega }_0\left(t\right)}}dt\hfill \\ & \le & {\displaystyle \frac{\mu d(u_1,u_2)}{\theta \left(z\right)}}\mathcal{U}\left(q\theta {\omega }_0\right)\left(z\right)\hfill \\ & \le & \mu {\Lambda }_{q}d(u_1,u_2){\omega }_0\left(z\right).\hfill \end{array}$$

So

$$d(\mathcal{T}{u}_1,\mathcal{T}{u}_2)\le \mu {\Lambda }_{q}d(u_1,u_2).$$

Since $\mu {\Lambda }_q<{\displaystyle \frac{1}{2}}$, $\mathcal{T}$ becomes a contraction mapping. According to the Banach fixed point theorem, there is a unique point $u\in \mathcal{S}$ such that $u=\mathcal{T}u.$

Put $v\left(z\right):=u\left(z\right)\theta \left(z\right)$. Then $v\in C\left({\mathbb{R}}_{+}^N\right)$ and

$$v\left(z\right)=\overline{\omega }\left(z\right)+\mu \mathcal{U}f(.,v)\left(z\right),\hspace{1em}z\in {\mathbb{R}}_{+}^N.$$

Since $\left|v\left(z\right)\right|\le \beta \overline{\omega }\left(z\right)$ on ${\mathbb{R}}_{+}^N,$ then it follows from Lemma 4 that v is the unique solution of problem (3) satisfying (19).

Estimates (20) and (21) follow directly from (19), together with (17) and (7).  □

Remark 5.

By Remark 4, $\overline{\omega }$ solves the linear problem (16). Estimate (19) shows that, for the admissible range of μ, the solution of the perturbed problem (19) has the same global behavior as $\overline{\omega }.$

Example 3.

Let $N\ge 3$, $s>{\displaystyle \frac{N}{2}}$ and $q\in L^s\left({\mathbb{R}}_{+}^N\right)\cap L^1\left({\mathbb{R}}_{+}^N\right)$. By Theorem 1, there exists ${\mu }^{*}>0$ such that for $\mu \in (0,{\mu }^{*}),$ the problem

$$\left\{\begin{array}{cc}-\Delta v={\displaystyle \frac{1}{{(\left|z\right|+1)}^3}}+\mu q\left(z\right)\sin \left(v\left(z\right)\right),\hfill & \mathit{in} {\mathbb{R}}_{+}^N, \hfill \\ v>0,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=0,\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b>0,\hfill & \end{array}\right.$$

possesses a unique solution $v\in C\left({\mathbb{R}}_{+}^N\right)$ satisfying

$${\displaystyle \frac{1}{c}}{z}_N\le v\left(z\right)\le c(1+z_N),$$

for some $c>0.$

Indeed, by [15] (Example 2), q belongs to ${\mathcal{K}}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ and by Remark 2, the function $z\to {\displaystyle \frac{1}{{(\left|z\right|+1)}^3}}\in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right).$ Hypotheses (H2) and (H3) are fulfilled with $f(z,v)=q\left(z\right)\sin \left(v\left(z\right)\right).$

Example 4.

Let $a>0$ and $\varphi \in C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right).$ Let $g\in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right)$ and assume that $0\le q\left(z\right)\le {\displaystyle \frac{C}{{(\left|z\right|+1)}^{\beta -\alpha }{z}_N^{\alpha }}},$ for some $C>0$ and $\alpha <2<\beta .$

Then by Theorem 1, for an admissible range of $\mu ,$ the problem

$$\left\{\begin{array}{cc}-\Delta v=g+\mu q\left(z\right)\tanh \left(v\left(z\right)\right),\hfill & \mathit{in} {\mathbb{R}}_{+}^N, \hfill \\ v>0,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=0,\hfill & \end{array}\right.$$

possesses a unique solution $v\in C\left({\mathbb{R}}_{+}^N\right)$ such that

$${\displaystyle \frac{1}{c}}(aH\varphi \left(z\right)+\mathcal{U}g\left(z\right))\le v\left(z\right)\le c(aH\varphi \left(z\right)+\mathcal{U}g\left(z\right)).$$

Observe that since $g\in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right)$, then hypothesis (H1) is immediate. Hypotheses (H2) and (H3) are fulfilled with $f(z,v)=q\left(z\right)\tanh \left(v\left(z\right)\right).$

Example 5.

Let $a,b\ge 0$ with $a+b>0$ and $q\in {\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right).$

Then by Theorem 1, for an admissible range of $\mu ,$ the problem

$$\left\{\begin{array}{cc}-\Delta v=\mu q\left(z\right)\arctan v\left(z\right),\hfill & \mathit{in} {\mathbb{R}}_{+}^N, \hfill \\ v>0,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a,\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b,\hfill & \end{array}\right.$$

possesses a unique solution $v\in C\left({\mathbb{R}}_{+}^N\right)$ with

$${\displaystyle \frac{1}{c}}(a+b{z}_N)\le v\left(z\right)\le c(a+b{z}_N).$$

(25)

In this case, $g\equiv 0$ and hypotheses (H2) and (H3) are fulfilled with $f(z,v)=q\left(z\right)\arctan v\left(z\right).$

The estimates in (25) follows from (19), (17) and that $H\mathbf{1}\left(z\right)\equiv 1.$

Example 6.

Let $a,b\ge 0$ with $a+b>0$ and $\varphi \in C_{\mathbf{b}}^{+}\left({\mathbb{R}}^{N-1}\right).$ For $\beta >N+1,$ let $g\left(z\right):={\displaystyle \frac{1}{{(\left|z\right|+1)}^{\beta -1}{z}_N}}$ and $q\left(z\right):={\displaystyle \frac{1}{{(\left|z\right|+1)}^{\beta }}}.$

Then by Theorem 1, for an admissible range of $\mu ,$ the problem

$$\left\{\begin{array}{cc}-\Delta v={\displaystyle \frac{1}{{(\left|z\right|+1)}^{\beta -1}{z}_N}}+{\displaystyle \frac{\mu }{{(\left|z\right|+1)}^{\beta }}}(\cos v\left(z\right)-1),\hfill & \mathit{in} {\mathbb{R}}_{+}^N, \hfill \\ v>0,\hfill & \mathit{in} {\mathbb{R}}_{+}^N,\hfill \\ \underset{z_N\to 0}{\lim }v\left(z\right)=a\varphi \left(z^{\prime }\right),\hfill & \\ \underset{z_N\to \infty }{\lim }{\displaystyle \frac{v\left(z\right)}{z_N}}=b,\hfill & \end{array}\right.$$

possesses a unique solution $v\in C\left({\mathbb{R}}_{+}^N\right)$ such that

$${\displaystyle \frac{1}{c}}(aH\varphi \left(z\right)+b{z}_N+{\displaystyle \frac{z_N}{{(\left|z\right|+1)}^N}})\le v\left(z\right)\le c(aH\varphi \left(z\right)+b{z}_N+{\displaystyle \frac{z_N}{{(\left|z\right|+1)}^N}}).$$

(26)

Indeed, by Remark 2, the functions g and q belong to ${\mathcal{K}}_{+}^{\infty }\left({\mathbb{R}}_{+}^N\right).$ Hypotheses (H2) and (H3) are clearly satisfied with $f(z,v)=q\left(z\right)\left(\cos v\right(z)-1).$

Estimates in (26) follow from (19) and (8).

4. Conclusions

In this work, we have studied semilinear elliptic equations in the half-space, subject to mixed boundary conditions both at infinity and on the boundary hyperplane. This research extends previous studies by allowing the reaction term to change sign. By exploiting properties of the convenient Kato class, we have proven the existence, uniqueness, and sharp global estimates for positive solutions. Future research may involve studying similar problems for more general elliptic operators, fractional Laplacians, and systems of elliptic equations posed in bounded or unbounded domains.