Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation

Abstract

With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem $\begin{matrix} \sqrt{- Δ + m^{2}} u + V u = (W * F (u)) f (u) in R^{N}, \end{matrix}$ where V is a bounded potential, not necessarily continuous, and F the primitive of f. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.

Keywords

Variational methods exponential decay fractional Laplacian Hartree equations

1. Introduction

In this paper we consider a generalized pseudo-relativistic Hartree equation $\begin{matrix} (1.1) & \sqrt{- Δ + m^{2}} u + V u = (W * F (u)) f (u) in R^{N}, \end{matrix}$ where $N ⩾ 2$ , $F (t) = \int_{0}^{t} f (s) d s$ , assuming that the nonlinearity f is a $C^{1}$ function, non-negative in $[0, \infty)$ , that satisfies

${lim}_{t \to 0} \frac{| f (t) |}{t} = 0$ ;

${lim}_{t \to \infty} \frac{f (t)}{t^{θ - 1}} = 0$ for some $2 < θ < 2^{#} = \frac{2 N}{N - 1}$ ;

$\frac{f (t)}{t}$ is increasing for all $t > 0$ .

We also postulate

V is continuous and satisfies $V (y) + V_{0} ⩾ 0$ for every $y \in R^{N}$ and some constant $V_{0} \in (0, m)$ ;

$V_{\infty} = {lim}_{| y | \to \infty} V (y) > 0$ ;

$V (y) ⩽ V_{\infty}$ for all $y \in R^{N}$ , $V (y) \neq V_{\infty}$ ;

$0 ⩽ W = W_{1} + W_{2} \in L^{r} (R^{N}) + L^{\infty} (R^{N})$ is radial, with $r > \frac{N}{N (2 - θ) + θ}$ .

Therefore, we aim to generalize the results obtained by Coti Zelati and Nolasco [12] and Cingolani and Secchi [8]. In the last paper, the authors have studied the equation $\begin{matrix} \sqrt{- Δ + m^{2}} u + V u = (W * u^{θ}) | u |^{θ - 2} u, \end{matrix}$ supposing, additionally to our hypotheses, that the potential V is continuous and has a horizontal asymptote for $N ⩾ 3$ . Furthermore, the homogeneity of the equation is a key ingredient in the proofs presented.

In Section 7 of Cingolani and Secchi [8], dealing with the decay of the solution, the authors assume additionally that the hypothesis $W (y) \to 0$ when $| y | \to \infty$ is explicitly assumed. The present work covers also the case $\begin{matrix} W (x) = \frac{| x |^{k}}{1 + | x |^{k}} . \end{matrix}$ So, applying different methods, we generalize [8]. A careful reading of our paper will also show that it generalizes [12].

The equation $\begin{matrix} (1.2) & \{\begin{matrix} i \partial_{t} u = \sqrt{- Δ + m^{2}} + G (u) in R^{N}, \\ u (x, 0) = ϕ (x), x \in R^{N} \end{matrix} \end{matrix}$ where $N ⩾ 2$ , G is a nonlinearity of Hartree type, $m > 0$ denotes the mass of bosons in units, was used to describe the dynamics of pseudo-relativistic boson stars in astrophysics. See [6,11,16,23] for more details. For the study of semiclassical analysis of the non-relativistic Hartree equations we would like to quote the papers [7,9,18,26,30] and also the recent paper [28]. For the Hartree equation without external potential V, we cite [23] for radial ground state solution, [21] for uniqueness and nondegeneracy of ground state solutions, and [12,13] for the existence of positive and radially symmetric solutions. In [25] is treated some Hartree problem imposing that the external potential V is radial, while in [8] this condition is dropped. Some versions of the magnetic Hartree equation are considered in [10,28].

A partial generalization of the present paper (obtained long after its submission) to the fractional operator ${(- Δ + m^{2})}^{σ}$ can be seen in [3] (see also [2]). Here, the results about regularity and decay are not so strong.

By considering an extension problem from $R^{N}$ to $R_{+}^{N + 1}$ , an alternative definition of $\sqrt{- Δ + m^{2}}$ is well-known (see [12] or [5]), so that equation (1.1) can be written as $\begin{matrix} (1.3) & \{\begin{matrix} - Δ u + m^{2} u = 0, & in R_{+}^{N + 1}, \\ - \frac{\partial u}{\partial x} (0, y) = - V (y) u (0, y) + (W (y) * F (u (0, y))) f (u (0, y)) & in R^{N} . \end{matrix} \end{matrix}$ We summarize our results:

Theorem 1.
Suppose that conditions ( $f_{1}$ )–( $f_{3}$ ), ( $V_{1}$ ) and ( $W_{h}$ ) are valid. Then, problem ( 1.3 ) has a non-negative ground state solution $w \in H^{1} (R_{+}^{N + 1})$ .
Theorem 2.
Assuming that hypotheses already stated are satisfied by f, V and W, any solution v of problem ( 1.3 ) satisfies $\begin{matrix} v \in C^{1, α} (R_{+}^{N + 1}) \cap C^{2} (R_{+}^{N + 1}) \end{matrix}$ and therefore is a classical solution of ( 1.3 ).

As a simple remark, we observe that if $f \in C^{\infty}$ , then the solution is also $C^{\infty}$ .

We also prove that the ground state solution has exponential decay:
Theorem 3.
Let w be the ground state solution obtained in Theorem 1 . Then $w (x, y) > 0$ in $[0, \infty) \times R^{N}$ and, for any $α \in (V_{0}, m)$ there exists $C > 0$ such that $\begin{matrix} 0 < w (x, y) ⩽ C e^{- (m - α) \sqrt{x^{2} + | y |^{2}}} e^{α x} \end{matrix}$ for any $(x, y) \in [0, \infty) \times R^{N}$ . In particular, $\begin{matrix} 0 < w (0, y) ⩽ C e^{- δ | y |}, \forall y \in R^{N}, \end{matrix}$ where $0 < δ < m - V_{0}$ .

The natural setting for problem (1.3) is the Sobolev space $\begin{matrix} H^{1} (R_{+}^{N + 1}) = {u \in L^{2} (R_{+}^{N + 1}) : \iint_{R_{+}^{N + 1}} | \nabla u |^{2} d x d y < \infty} \end{matrix}$ endowed with the norm $\begin{matrix} ‖ u ‖^{2} = \iint_{R_{+}^{N + 1}} (| \nabla u |^{2} + u^{2}) d x d y . \end{matrix}$
Notation.
The norm in the space $R_{+}^{N + 1}$ will be denoted by $‖ \cdot ‖$ . For all $q \in [1, \infty]$ , we denote by $| \cdot |_{q}$ the norm in the space $L^{q} (R^{N})$ and by $‖ \cdot ‖_{q}$ the norm in the space $L^{q} (R_{+}^{N + 1})$ .

It is well-known that traces of functions $H^{1} (R_{+}^{N + 1})$ are in $H^{1 / 2} (R^{N})$ and that every function in $H^{1 / 2} (R^{N})$ is the trace of a function in $H^{1} (R_{+}^{N + 1})$ , see [29]. Denoting $γ : H^{1} (R_{+}^{N + 1}) \to H^{1 / 2} (R^{N})$ the linear function that associates the trace $γ (v) \in H^{1 / 2} (R^{N})$ of the function $v \in H^{1} (R_{+}^{N + 1})$ , then $ker γ = H_{0}^{1} (R_{+}^{N + 1})$ .

The immersions $\begin{array}{l} (1.4) & H^{1} (R_{+}^{N + 1}) ↪ L^{q} (R_{+}^{N + 1}) \\ (1.5) & H^{1 / 2} (R^{N}) ↪ L^{q} (R^{N}) \end{array}$ are continuous for any $q \in [2, 2^{}]$ and $[2, 2^{#}]$ respectively, where $\begin{matrix} (1.6) & 2^{} = \frac{2 (N + 1)}{N - 1} and 2^{#} = \frac{2 N}{N - 1} . \end{matrix}$

The space $H^{1 / 2} (R^{N})$ is defined by means of Fourier transforms; therefore, we can not change $R^{N}$ to a bounded open set $Ω \subset R^{N}$ . However (see [14]), $H^{1 / 2} (R^{N}) = W^{1 / 2, 2} (R^{N})$ and $W^{1 / 2, 2} (Ω)$ is well-defined for an open set $Ω \subset R^{N}$ . We recall its definition. Let $u : Ω \to R$ a measurable function and Ω a bounded open set (that, in the sequel, we suppose to have Lipschitz boundary). Denoting $\begin{matrix} {[u]}_{Ω}^{2} = \int_{Ω} \int_{Ω} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 1}} d x d y \end{matrix}$ and $\begin{array}{l} W^{1 / 2, 2} (Ω) & = {u \in L^{2} (R^{N}) : {[u]}_{Ω}^{2} < \infty}, \end{array}$ then $W^{1 / 2, 2} (Ω)$ is a reflexive Banach space (see, e.g., [14] and [15]) endowed with the norm $\begin{matrix} ‖ u ‖_{W^{1 / 2, 2} (Ω)} = | u |_{2} + {[u]}_{Ω} . \end{matrix}$

The proof of the next result can be found in [14, Theorem 4.54].
Theorem 4.
The immersion $W^{1 / 2, 2} (Ω) ↪ L^{q} (Ω)$ is compact for any $q \in [1, 2^{#})$ .

As usual, the immersion $W^{1 / 2, 2} (Ω) ↪ L^{2^{#}} (Ω)$ is continuous: see [14, Corollary 4.53]. We denote the norm in the space $L^{q} (Ω)$ by $| \cdot |_{L^{q} (Ω)}$ .
2. Preliminaries

Let us suppose that $u \in H^{1} (R^{N + 1}) \cap C_{0}^{\infty} (R_{+}^{N + 1})$ and $u (x, y) ⩾ 0$ . Let us proceed heuristically: since $\begin{matrix} {| u (0, y) |}^{t} = \int_{\infty}^{0} \frac{\partial}{\partial x} {| u (x, y) |}^{t} d x = \int_{\infty}^{0} t {| u (x, y) |}^{t - 2} u (x, y) \frac{\partial}{\partial x} u (x, y) d x, \end{matrix}$ it follows from Hölder’s inequality $\begin{array}{l} \int_{R^{N}} {| γ (u) |}^{t} & = \int_{R^{N}} {| u (0, y) |}^{t} d y ⩽ \int_{R^{N}} \int_{0}^{\infty} t {| u (x, y) |}^{t - 1} | \nabla u (x, y) | d x d y \\ ⩽ t {(\int_{R_{+}^{N + 1}} | u |^{2 (t - 1)})}^{1 / 2} {(\int_{R_{+}^{N + 1}} | \nabla u |^{2})}^{1 / 2} \\ (2.1) & ⩽ t ‖ u ‖_{2 (t - 1)}^{t - 1} ‖ \nabla u ‖_{2} . \end{array}$ So, in order to apply the immersion $H^{1} (R_{+}^{N + 1}) ↪ L^{q} (R_{+}^{N + 1})$ we must have $2 ⩽ 2 (t - 1) ⩽ \frac{2 (N + 1)}{N - 1}$ , that is, $\begin{matrix} (2.2) & 2 ⩽ t ⩽ \frac{2 N}{N - 1} = 2^{#} . \end{matrix}$ By density of $H^{1} (R^{N + 1}) \cap C_{0}^{\infty} (R_{+}^{N + 1})$ in $H^{1} (R_{+}^{N + 1})$ , the estimate (2.1) is valid for all $u \in H^{1} (R_{+}^{N + 1})$ .

Taking into account (1.4), Young’s inequality applied to (2.1) yields $\begin{array}{l} {| γ (u) |}_{t} & ⩽ ‖ u ‖_{2 (t - 1)}^{(t - 1) / t} {(t ‖ \nabla u ‖_{2})}^{1 / t} \\ ⩽ \frac{t - 1}{t} ‖ u ‖_{2 (t - 1)} + ‖ \nabla u ‖_{2} \\ (2.3) & ⩽ C_{t} ‖ u ‖, \end{array}$ where $C_{t}$ is a constant. We summarize: $\begin{matrix} (2.4) & | γ (u) | \in L^{t} (R^{N}), \forall t \in [2, 2^{#}] . \end{matrix}$

The inequality (2.3) will also be valuable in the special case $t = 2$ : $\begin{array}{l} {| γ (u) |}_{2}^{2} & ⩽ ‖ u ‖_{2} (2 ‖ \nabla u ‖_{2}) \\ (2.5) & ⩽ λ \iint_{R_{+}^{N + 1}} u^{2} + \frac{1}{λ} \iint_{R_{+}^{N + 1}} | \nabla u |^{2} \end{array}$ where $λ > 0$ is a parameter, the last inequality being a consequence of Young’s inequality.

Remark 2.1.
It follows from ( $f_{3}$ ) that f satisfies the Ambrosetti–Rabinowitz inequality $2 F (t) ⩽ f (t) t$ , for all $t > 0$ . Furthermore, it follows from ( $f_{1}$ ) and ( $f_{2}$ ) that, for any fixed $ξ > 0$ , there exists a constant $C_{ξ}$ such that $\begin{matrix} (2.6) & | f (t) | ⩽ ξ t + C_{ξ} t^{θ - 1}, \forall t ⩾ 0 \end{matrix}$ and analogously $\begin{matrix} (2.7) & | F (t) | ⩽ ξ t^{2} + C_{ξ} t^{θ} ⩽ C (t^{2} + t^{θ}), \forall t ⩾ 0 . \end{matrix}$ Observe that $γ (u) \in L^{θ} (R^{N})$ and $γ (u) \in L^{2} (R^{N})$ imply $F (γ (u)) \in L^{1} (R^{N})$ .
Proposition 2.1 (Hausdorff–Young).

Assume that, for $1 ⩽ p, q, s ⩽ \infty$ , we have $f \in L^{p} (R^{N})$ , $g \in L^{q} (R^{N})$ and $\begin{matrix} \frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{s} . \end{matrix}$ Then $\begin{matrix} | f * g |_{s} ⩽ | f |_{p} | g |_{q} . \end{matrix}$

We now enhance the result given by (2.4). Observe that $\frac{N}{N (2 - θ) + θ} ⩾ 1$ and $\frac{N}{N (2 - θ) + θ} = 1$ if, and only if $N = θ = 2$ .

The results in the sequel will be useful when addressing the regularity of the solution of problem (1.3).

Lemma 2.1.
Concerning hypothesis $(W_{h})$ we have:
if $r \in (\frac{N}{N (2 - θ) + θ}, \frac{2 N}{N (2 - θ) + θ}]$ , there exists $p \in [1, \frac{2 N}{(N - 1) θ}]$ such that $\begin{matrix} {| γ (u) |}^{θ} \in L^{p} (R^{N}) \end{matrix}$ and $\begin{matrix} \frac{1}{p} + \frac{1}{r} = 1 + \frac{N (2 - θ) + θ}{2 N} . \end{matrix}$ Furthermore, $F (γ (u)) \in L^{p} (R^{N})$ and $\begin{matrix} | W_{1} * F (γ (u)) | = : g \in L^{2 N / [N (2 - θ) + θ]} (R^{N}) . \end{matrix}$

if $r^{'}$ denotes the conjugate exponent of r and $r > \frac{2 N}{N (2 - θ) + θ}$ , then $F (γ (u)) \in L^{r^{'}} (R^{N})$ and $W_{1} * F (γ (u)) \in L^{\infty} (R^{N})$ .

Proof.
(i) We verify the values of r that satisfy the equality $\begin{matrix} \frac{1}{p} + \frac{1}{r} = 1 + \frac{N (2 - θ) + θ}{2 N} . \end{matrix}$ Observe that $r \in (\frac{N}{N (2 - θ) + θ}, \frac{2 N}{N (2 - θ) + θ}]$ if, and only if, $p \in [1, \frac{2 N}{(N - 1) θ})$ .

As consequence of (2.4) $| γ (u) |^{θ} \in L^{p} (R^{N})$ and thus $| γ (u) |^{2} \in L^{p} (R^{N})$ and (2.7) yields $F (γ (u)) \in L^{p} (R^{N})$ . So, $| W_{1} * F (γ (u)) | = g \in L^{2 N / [N (2 - θ) + θ]} (R^{N})$ follows from the Hausdorff–Young inequality.

( $ii$ ) Since $W_{1} \in L^{r} (R^{N})$ for $r = \frac{2 N}{N (2 - θ) + θ}$ and $r^{'} = \frac{r}{r - 1} = \frac{2 N}{(N - 1) θ}$ , applying (i) we conclude that $F (γ (u)) \in L^{r^{'}} (R^{N})$ and $W_{1} * F (γ (u)) \in L^{\infty} (R^{N})$ is consequence of Proposition 2.1. □
Corollary 2.1.
We have $| W * F (γ (u)) | ⩽ C + g$ with $g \in L^{2 N / [N (2 - θ) + θ]} (R^{N})$ .
Proof.
An immediately consequence of Lemma 2.1, since $W_{2} \in L^{\infty} (R^{N})$ . □

Following arguments in [12], we have:
Lemma 2.2.
For all $θ \in (2, \frac{2 N}{N - 1})$ , we have $| γ (u) |^{θ - 2} ⩽ 1 + g_{2}$ , where $g_{2} \in L^{N} (R^{N})$ .
Proof.
We have $\begin{matrix} {| γ (u) |}^{θ - 2} = {| γ (u) |}^{θ - 2} χ_{{| γ (u) | ⩽ 1}} + {| γ (u) |}^{θ - 2} χ_{{| γ (u) | > 1}} ⩽ 1 + g_{2}, \end{matrix}$ with $g_{2} = | γ (u) |^{θ - 2} χ_{{| γ (u) | > 1}}$ . If $(θ - 2) N ⩽ 2$ , then $\begin{matrix} \int_{R^{N}} {| γ (u) |}^{(θ - 2) N} χ_{{| γ (u) | > 1}} ⩽ \int_{R^{N}} {| γ (u) |}^{2} χ_{{| γ (u) | > 1}} ⩽ \int_{R^{N}} {| γ (u) |}^{2} < \infty . \end{matrix}$

When $2 < (θ - 2) N$ , then $(θ - 2) N \in (2, \frac{2 N}{N - 1})$ and $| γ (u) |^{θ - 2} \in L^{N} (R^{N})$ as outcome of (2.4). □
Lemma 2.3.
For all $θ \in (2, \frac{2 N}{N - 1})$ we have $h = g | γ (u) |^{θ - 2} \in L^{N} (R^{N})$ , where g is the function of Lemma 2.1 .
Proof.
Application of the Hölder inequality yields $\begin{matrix} \int_{R^{N}} {(g {| γ (u) |}^{θ - 2})}^{N} ⩽ {(\int_{R^{N}} g^{N α})}^{\frac{1}{α}} {(\int_{R^{N}} {({| γ (u) |}^{(θ - 2) N})}^{α^{'}})}^{\frac{1}{α^{'}}}, \end{matrix}$ if we define α so that $α N = 2 N / [N (2 - θ) + θ]$ . Thus, $α^{'} = 2 / [(N - 1) (θ - 2)]$ and we have $α^{'} N (θ - 2) = 2 N / [N - 1]$ . Since both integrals of the right-hand side of the last inequality are integrable, we are done. □

We now handle the existence of the “energy” functional. We denote by $L_{w}^{q} (R^{N})$ the weak $L^{q} (R^{N})$ space and by $| \cdot |_{q_{w}}$ its usual norm (see [22]). The next result is a generalized version of the Hardy-Littlewood-Sobolev inequality:
Proposition 2.2 (Lieb [22]).

Assume that $p, q, r \in (1, \infty)$ and $\begin{matrix} \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 2 . \end{matrix}$ Then, for some constant $N_{p, q, t} > 0$ and for any $f \in L^{p} (R^{N})$ , $g \in L^{r} (R^{N})$ and $h \in L_{w}^{q} (R^{N})$ , we have the inequality $\begin{matrix} \int_{R^{N}} \int_{R^{N}} f (t) h (t - s) g (s) d t d s ⩽ N_{p, q, t} | f |_{p} | g |_{r} | h |_{q_{w}} . \end{matrix}$

Lemma 2.4.
For a positive constant C holds $\begin{matrix} | \frac{1}{2} \int_{R^{N}} (W * F (γ (u))) F (γ (u)) | ⩽ C {(‖ u ‖^{2} + ‖ u ‖^{θ})}^{2} . \end{matrix}$
Proof.
Let us denote $\begin{matrix} Ψ (u) = \frac{1}{2} \int_{R^{N}} [W * F (γ (u))] F (γ (u)) . \end{matrix}$ Since $W = W_{1} + W_{2}$ , $\begin{array}{l} Ψ (u) & = \frac{1}{2} \int_{R^{N}} [W_{1} * F (γ (u))] F (γ (u)) + \frac{1}{2} \int_{R^{N}} [W_{2} * F (γ (u))] F (γ (u)) \\ (2.8) & = : J_{1} (u) + J_{2} (u) . \end{array}$

Let us suppose that $| γ (u) |^{θ} \in L^{t} (R^{N})$ for some $t ⩾ 1$ . Then $| γ (u) |^{2} \in L^{t} (R^{N})$ and $F (γ (u)) \in L^{t} (R^{N})$ (as consequence of (2.7)). Application of Proposition 2.2 yields $\begin{matrix} | J_{1} (u) | = | \frac{1}{2} \int_{R^{N}} W_{1} * F (γ (u)) F (γ (u)) | ⩽ N | W_{1} |_{r} {| F (γ (u)) |}_{t} {| F (γ (u)) |}_{t} . \end{matrix}$ Since $\frac{1}{r} + \frac{2}{t} = 2$ implies $t = \frac{2 r}{2 r - 1}$ , we have $\begin{array}{l} (2.9) & | J_{1} (u) | & ⩽ C {| F (γ (u)) |}_{\frac{2 r}{2 r - 1}} {| F (γ (u)) |}_{\frac{2 r}{2 r - 1}} ⩽ C^{'} {(‖ u ‖^{2} + ‖ u |^{θ})}^{2} < \infty, \end{array}$ (Observe that, in order to apply the immersion $H^{1} (R_{+}^{N + 1}) ↪ L^{q} (R_{+}^{N + 1})$ , we must have $t θ < 2 N / (N - 1)$ , that is, $r > N / [N (2 - θ) + θ]$ .)

In the case $W_{2} \in L^{\infty} (R^{N})$ we can take $t = 1$ , therefore $\begin{array}{l} | J_{2} (u) | & = | \frac{1}{2} \int_{R^{N}} [W_{2} * F (γ (u))] F (γ (u)) | ⩽ C {({| γ (u) |}_{2}^{2} + {| γ (u) |}_{θ}^{θ})}^{2} \\ (2.10) & ⩽ C^{″} {(‖ u ‖^{2} + ‖ u ‖^{θ})}^{2} . \end{array}$

From (2.10) and (2.9) results the claim. □
Lemma 2.5.
The functional $\begin{array}{l} I (u) & = \frac{1}{2} \iint_{R_{+}^{N + 1}} (| \nabla u |^{2} + m^{2} u^{2}) + \frac{1}{2} \int_{R^{N}} V (y) {[γ (u (y))]}^{2} \\ - \frac{1}{2} \int_{R^{N}} (W * F (γ (u))) F (γ (u)) \\ = : I_{1} (u) + I_{2} (u) - Ψ (u) \end{array}$ is well-defined.
Proof.
Of course $\begin{array}{l} I_{1} (u) = \frac{1}{2} \iint_{R_{+}^{N + 1}} (| \nabla u |^{2} + m^{2} u^{2}) ⩽ \frac{k}{2} ‖ u ‖^{2} < \infty, \end{array}$ if we take $k = max {1, m^{2}}$ . Since hypothesis ( $V_{1}$ ) implies $| V (y) | < C$ , we have $\begin{array}{l} | I_{2} (u) | = | \frac{1}{2} \int_{R^{N}} V (y) {[γ (u (y))]}^{2} | ⩽ \frac{C}{2} \int_{R^{N}} {| γ (u) |}^{2} = C^{'} {| γ (u) |}_{2}^{2} ⩽ C^{″} ‖ u ‖^{2} . \end{array}$ Taking into account Lemma 2.4, the proof is complete. □

Since the derivative of the energy functional is given by $\begin{array}{l} I^{'} (u) \cdot φ & = \iint_{R_{+}^{N + 1}} [\nabla u \cdot \nabla φ + m^{2} u φ] + \int_{R^{N}} V (y) γ (u) γ (φ) \\ (2.11) & - \int_{R^{N}} (W * F (γ (u))) f (γ (u)) γ (φ), \forall φ \in H^{1} (R_{+}^{N + 1}), \end{array}$ we see that critical points of I are weak solutions (1.3).

Because we are looking for a positive solution, we suppose that $f (t) = 0$ for $t < 0$ . Proposition 2.3.
The quadratic form $\begin{matrix} u \mapsto \frac{1}{2} \iint_{R_{+}^{N + 1}} (| \nabla u |^{2} + m^{2} u^{2}) + \frac{1}{2} \int_{R^{N}} V (y) {[γ (u (y))]}^{2} \end{matrix}$ defines an norm in the space $H^{1} (R_{+}^{N + 1})$ , which is equivalent to the norm $‖ \cdot ‖$ .
Proof.
We keep up with the notation already introduced and note that $I_{2} (u) ⩾ - (1 / 2) V_{0} \int_{R^{N}} | γ (u) |^{2}$ . Furthermore, as consequence of (2.5), we have $\begin{array}{l} (2.12) & \int_{R^{N}} {| γ (u) |}^{2} ⩽ m \iint_{R_{+}^{N + 1}} | u |^{2} + \frac{1}{m} \iint_{R_{+}^{N + 1}} | \nabla u |^{2} . \end{array}$ Therefore, $\begin{array}{l} I_{1} (u) + I_{2} (u) & ⩾ \frac{1}{2} \iint_{R_{+}^{N + 1}} (| \nabla u |^{2} + m^{2} u) - \frac{V_{0} m}{2} \iint_{R_{+}^{N + 1}} | u |^{2} - \frac{V_{0}}{2 m} \iint_{R_{+}^{N + 1}} | \nabla u |^{2} \\ = \frac{1}{2} (1 - \frac{V_{0}}{m}) \iint_{R_{+}^{N + 1}} | \nabla u |^{2} + \frac{1}{2} m (m - V_{0}) \iint_{R_{+}^{N + 1}} | u |^{2} . \end{array}$ Defining $K = min {\frac{1}{2} (1 - \frac{V_{0}}{m}), \frac{1}{2} m (m - V_{0})} > 0$ , we conclude that $\begin{matrix} I_{1} (u) + I_{2} (u) ⩾ K ‖ u ‖^{2} . \end{matrix}$ By applying (2.12) it easily follows that $\begin{array}{l} I_{1} (u) + I_{2} (u) & ⩽ \frac{1}{2} (1 + \frac{V_{0}}{m}) \iint_{R_{+}^{N + 1}} | \nabla u |^{2} + \frac{1}{2} (m^{2} + V_{\infty} m) \iint_{R_{+}^{N + 1}} | u |^{2} \\ (2.13) & ⩽ C ‖ u ‖^{2} \end{array}$ for a constant $C > 0$ . We are done. □

3. Mountain pass geometry and Nehari manifold

Lemma 3.1.
I satisfies the mountain pass theorem geometry. More precisely,
There exist $ρ, δ > 0$ such that $I |_{S} ⩾ δ > 0$ for all $u \in S$ , where $\begin{matrix} S = {u \in H^{1} (R_{+}^{N + 1}) : ‖ u ‖ = ρ} . \end{matrix}$

For each $u_{0} \in H^{1} (R_{+}^{N + 1})$ such that ${(u_{0})}_{+} \neq 0$ , there exists $τ \in R$ , satisfying $‖ τ u_{0} ‖ > ρ$ and $I (τ u_{0}) < 0$ .

Proof.
Since we have already showed that $\begin{array}{l} (3.1) & I_{1} (u) + I_{2} (u) ⩾ K ‖ u ‖^{2} \end{array}$ and so $I (u) ⩾ K ‖ u ‖^{2} - Ψ (u) ⩾ K ‖ u ‖^{2} - C {(‖ u ‖^{2} + ‖ u ‖^{θ})}^{2}$ , we obtain (i) by choosing $ρ > 0$ small enough.

In order to prove ( $ii$ ), fix $u_{0} \in H^{1} (R_{+}^{N + 1}) ∖ {0}$ such that $u_{0} ⩾ 0$ . For all $t > 0$ consider the function $g_{u_{0}} : (0, \infty) \to R$ defined by $\begin{matrix} g_{u_{0}} (t) = Ψ (\frac{t u_{0}}{‖ u_{0} ‖}) \end{matrix}$ where, as before, $\begin{matrix} Ψ (u) = \frac{1}{2} \int_{R^{N}} (W * F (γ (u))) F (γ (u)) . \end{matrix}$

An easy calculation shows that $\begin{array}{l} g_{u_{0}}^{'} (t) & = \frac{2}{t} \int_{R^{N}} (W * F (γ (\frac{t u_{0}}{‖ u_{0} ‖}))) \frac{f}{2} (γ (\frac{t u_{0}}{‖ u_{0} ‖})) γ (\frac{t u_{0}}{‖ u_{0} ‖}) ⩾ \frac{4}{t} g_{u_{0}} (t), \end{array}$ the last inequality being a consequence of the Ambrosetti–Rabinowitz inequality. Observe that $g_{u_{0}}^{'} (t) > 0$ for $t > 0$ .

Thus, we obtain $\begin{array}{l} ln g_{u_{0}} (t) |_{1}^{τ ‖ u_{0} ‖} ⩾ 4 ln t |_{1}^{τ ‖ u_{0} ‖} \Rightarrow \frac{g_{u_{0}} (τ ‖ u_{0} ‖)}{g_{u_{0}} (1)} ⩾ {(τ ‖ u_{0} ‖)}^{4}, \end{array}$ proving that $\begin{array}{l} (3.2) & Ψ (τ u_{0}) = g_{u_{0}} (τ ‖ u_{0} ‖) ⩾ D {(τ ‖ u_{0} ‖)}^{4} . \end{array}$ for a constant $D > 0$ .

It follows from (2.13) that $\begin{array}{l} I (τ u_{0}) & ⩽ C τ^{2} ‖ u_{0} ‖^{2} - D τ^{4} ‖ u_{0} ‖^{4} . \end{array}$ Thus, it suffices to take τ large enough. □

The existence of a Palais–Smale sequence $(u_{n}) \subset H^{1} (R_{+}^{N + 1})$ such that $\begin{matrix} I^{'} (u_{n}) \to 0 and I (u_{n}) \to c, \end{matrix}$ where $\begin{matrix} c = inf_{α \in Γ} max_{t \in [0, 1]} I (α (t)), \end{matrix}$ and $Γ = {α \in C^{1} ([0, 1], H^{1} (R_{+}^{N + 1})) : α (0) = 0, α (1) < 0}$ results from the mountain pass theorem without the PS condition.

We now consider the Nehari manifold $\begin{array}{l} N & = {u \in H^{1} (R_{+}^{N + 1}) ∖ {0} : I^{'} (u) \cdot u = 0} . \end{array}$ It is not difficult to see that $N$ is a manifold in $H^{1} (R_{+}^{N + 1}) ∖ {0}$ .

The next result, which follows immediately from our estimates, proves that $N$ is a closed manifold in $H^{1} (R_{+}^{N + 1})$ :
Lemma 3.2.
There exists $β > 0$ such that $‖ u ‖ ⩾ β$ for all $u \in N$ .

An alternative characterization of c is obtained by a standard method: for $u_{+} \neq 0$ , consider the function $Φ (t) = I_{1} (t u) + I_{2} (t u) - Ψ (t u)$ , preserving the notation of Lemma 3.1. The proof of Lemma 3.1 assures that $Φ (t u) > 0$ for t small enough, $Φ (t u) < 0$ for t large enough and $g_{u}^{'} (t) > 0$ if $t > 0$ . Therefore, ${max}_{t ⩾ 0} Φ (t)$ is achieved at a unique $t_{u} = t (u) > 0$ and $Φ^{'} (t u) > 0$ for $t < t_{u}$ and $Φ^{'} (t u) < 0$ for $t > t_{u}$ . Furthermore, $Φ^{'} (t_{u} u) = 0$ implies that $t_{u} u \in N$ .

The map $u \mapsto t_{u}$ ( $u \neq 0$ ) is continuous and $c = c^{}$ , where $\begin{matrix} c^{} = inf_{u \in H^{1} (R_{+}^{N + 1}) ∖ {0}} max_{t ⩾ 0} I (t u) . \end{matrix}$ For details, see [27, Section 3] or [17].

Standard arguments prove the next affirmative:
Lemma 3.3.
Let $(u_{n}) \subset H^{1} (R_{+}^{N + 1})$ be a sequence such that $I (u_{n}) \to c$ and $I^{'} (u_{n}) \to 0$ , where $\begin{matrix} c = inf_{u \in H^{1} (R_{+}^{N + 1}) ∖ {0}} max_{t ⩾ 0} I (t u) . \end{matrix}$ Then $(u_{n})$ is bounded and (for a subsequence) $u_{n} ⇀ u$ in $H^{1} (R_{+}^{N + 1})$ .
Lemma 3.4.
Let $U ⫅ R^{N}$ be any open set. For $1 < p < \infty$ , let $(f_{n})$ be a bounded sequence in $L^{p} (U)$ such that $f_{n} (x) \to f (x)$ a.e. Then $f_{n} ⇀ f$ .

The proof of Lemma 3.4 can be found, e.g., in [20, Lemme 4.8, Chapitre 1].
4. The limit problem

In this section we consider a variant of problem (1.3), changing the potential $V (y)$ for $V_{\infty}$ .

Theorem 5.
Assuming $(f_{1})$ , $(f_{2})$ , $(f_{3})$ and $(W_{h})$ , problem $\begin{matrix} (P_{\infty}) & \{\begin{matrix} - Δ u + m^{2} u = 0 in R_{+}^{N + 1} \\ - \frac{\partial u}{\partial x} = - V_{\infty} u + [W * F (u)] f (u), (x, y) \in {0} \times R^{N} ≃ R^{N}, \end{matrix} \end{matrix}$ has a non-negative ground state solution.
Proof.
Let $(u_{n})$ be the minimizing sequence given by Lemma 3.1. Then, there exist $R, δ > 0$ and a sequence $(z_{n}) \subset R^{N}$ such that $\begin{matrix} (4.1) & \underset{n \to \infty}{lim inf} \int_{B_{R} (z_{n})} {| γ (u_{n}) |}^{2} ⩾ δ . \end{matrix}$ If false, a result of Lions (see [24]) guarantees that $γ (u_{n}) \to 0$ in $L^{q} (R^{N})$ for $2 < q < 2^{}$ , thus implying that $\begin{matrix} \int_{R^{N}} (W F (γ (u_{n}))) f (γ (u_{n})) γ (u_{n}) \to 0, \end{matrix}$ contradicting Lemma 3.2.

We define $\begin{matrix} v_{n} (x, y) = u_{n} (x, y - z_{n}) . \end{matrix}$ From (4.1) we derive that $\begin{matrix} \int_{B_{R} (0)} {| γ (v_{n}) |}^{2} ⩾ \frac{δ}{2} . \end{matrix}$ We observe that the energy functional $\begin{array}{l} I_{\infty} (u) & = \frac{1}{2} \iint_{R_{+}^{N + 1}} (| \nabla u |^{2} + m^{2} u^{2}) + \frac{1}{2} \int_{R^{N}} V_{\infty} {| γ (u) |}^{2} \\ - \frac{1}{2} \int_{R^{N}} [W * F (γ (u))] F (γ (u)) \end{array}$ and its derivative as well are invariant with respect to any translation in $R^{N}$ . Therefore, it also holds that $\begin{matrix} I_{\infty}^{'} (v_{n}) \to 0 and I_{\infty} (v_{n}) \to c_{\infty}, \end{matrix}$ where $\begin{matrix} c_{\infty} = inf_{u \in H^{1} (R_{+}^{N + 1}) ∖ {0}} max_{t ⩾ 0} I_{\infty} (t u) . \end{matrix}$ (Observe that all reasoning in Section 3 is valid for $I_{\infty}$ and its minimizing sequence.)

Since $(v_{n})$ is bounded (see Lemma 3.3) it follows that $v_{n} ⇀ v$ . A standard argument shows that we can suppose $v_{n} (x, y) \to v (x, y)$ a.e. in $(R_{+}^{N + 1})$ , $v_{n} \to v$ in $L_{loc}^{s} (R_{+}^{N + 1})$ for all $s \in [1, 2^{})$ , $γ (v_{n} (x, y)) \to γ (v (x, y))$ a.e. in $(R^{N})$ and $γ (v_{n}) \to γ (v)$ in $L_{loc}^{q} (R^{N})$ , for all $q \in [1, 2^{#})$ .

We will show that $v \in N_{\infty} = {u \in H^{1} (R^{N + 1_{+}}) ∖ {0} : I_{\infty}^{'} (u) \cdot u = 0}$ .

For all $φ \in C_{0}^{\infty} (R_{+}^{N + 1})$ , let us consider $ψ_{n} = (v_{n} - v) φ \in H^{1} (R_{+}^{N + 1})$ . We have $\begin{array}{l} ⟨ I_{\infty}^{'} (v_{n}), ψ_{n} ⟩ & = \iint_{R_{+}^{N + 1}} \nabla v_{n} \cdot \nabla ψ_{n} + \iint_{R_{+}^{N + 1}} m^{2} v_{n} ψ_{n} + \int_{R^{N}} V_{\infty} γ (v_{n}) γ (ψ_{n}) \\ - \int_{R^{N}} (W F (γ (v_{n})) f (γ (v_{n})) γ (ψ_{n}) \\ (4.2) & = J_{1} + J_{2} + J_{3} - J_{4} . \end{array}$ We start considering $\begin{array}{l} J_{4} = \int_{R^{N}} (W * F (γ (v_{n})) f (γ (v_{n})) γ (ψ_{n}) . \end{array}$ Because ${lim}_{n \to \infty} ⟨ I_{\infty}^{'} (v_{n}), (v_{n} - v) φ ⟩ = 0$ , it follows from [1, Lemma 3.5] that $J_{4} \to 0$ when $n \to \infty$ and thus is easily verified that $J_{2} + J_{3} - J_{4} \to 0$ when $n \to \infty$ . We now consider $J_{1}$ : $\begin{array}{l} J_{1} & = \iint_{R_{+}^{N + 1}} \nabla v_{n} \cdot \nabla ((v_{n} - v) φ) \\ = \iint_{R_{+}^{N + 1}} \nabla v_{n} \cdot φ \nabla (v_{n} - v) + \iint_{R_{+}^{N + 1}} \nabla v_{n} \cdot (v_{n} - v) \nabla φ \\ = \iint_{R_{+}^{N + 1}} {| \nabla (v_{n} - v) |}^{2} φ + φ \nabla v \cdot \nabla (v_{n} - v) + \nabla v_{n} \cdot (v_{n} - v) \nabla φ . \end{array}$ We infer that $\begin{array}{l} lim_{n \to \infty} \iint_{R_{+}^{N + 1}} {| \nabla (v_{n} - v) |}^{2} φ & = - lim_{n \to \infty} \iint_{R_{+}^{N + 1}} φ \nabla v \cdot \nabla (v_{n} - v) \\ - lim_{n \to \infty} \iint_{R_{+}^{N + 1}} (v_{n} - v) \nabla v_{n} \cdot \nabla φ . \end{array}$ Since $\begin{matrix} lim_{n \to \infty} \iint_{R_{+}^{N + 1}} φ \nabla v \cdot \nabla (v_{n} - v) = 0 and lim_{n \to \infty} \iint_{R_{+}^{N + 1}} (v_{n} - v) \nabla v_{n} \cdot \nabla φ = 0 \end{matrix}$ (because $\nabla v_{n}$ is bounded), we deduce that $\begin{matrix} \nabla v_{n} \to \nabla v a.e. in R_{+}^{N + 1} . \end{matrix}$ Thus $\begin{matrix} I_{\infty}^{'} (v) v = 0 \end{matrix}$ and $v \in N_{\infty}$ .

We now turn our attention to the positivity of v. Seeing that $\begin{matrix} \iint_{R_{+}^{N + 1}} (\nabla v \cdot \nabla φ + m^{2} v φ) + \int_{R^{N}} V_{\infty} γ (v) γ (φ) = \int_{R^{N}} [W * F (γ (v))] f (γ (v)) γ (φ) \end{matrix}$ and choosing $φ = v^{-}$ , the left-hand side of the equality is positive (by the definition of $I_{\infty}$ and equation (3.1) applied to $I_{\infty}$ ), since $J_{1} + J_{2} + J_{3} = I_{1} + I_{2} ⩾ K ‖ v ‖^{2}$ ), while $Ψ (v) = J_{4} ⩽ 0$ . We are done. □

5. Proof of Theorem 1

In order to consider the general case of the potential $V (y)$ , we state a well-known result due to M. Struwe:

Lemma 5.1 (Splitting Lemma).

Let $(v_{n}) \subset H^{1} (R_{+}^{N + 1})$ be such that $\begin{matrix} I (u_{n}) \to c, I^{'} (u_{n}) \to 0 \end{matrix}$ and $u_{n} ⇀ u$ weakly on $H^{1} (R_{+}^{N + 1})$ . Then $I^{'} (u_{0}) = 0$ and we have either

$u_{n} \to u$ strongly on $H^{1} (R_{+}^{N + 1})$ ;

there exist $k \in N$ , $(y_{n}^{j}) \in R^{N}$ such that $| y_{n}^{j} | \to \infty$ for $j \in {1, \dots, k}$ and nontrivial solutions $u^{1}, \dots, u^{k}$ of problem ( P ∞ ) so that $\begin{matrix} I (u_{n}) \to I (u_{0}) + \sum_{j = 1}^{k} I_{\infty} (u_{j}) \end{matrix}$ and $\begin{matrix} ‖ u_{n} - u_{0} - \sum_{j = 1}^{k} u^{j} (\cdot - y_{n}^{j}) ‖ \to 0 . \end{matrix}$

Lemma 5.2.
The functional I satisfies ${(PS)}_{c}$ for any $0 ⩽ c < c_{\infty}$ .
Proof.
Let us suppose that $(u_{n})$ satisfies $\begin{matrix} I (u_{n}) \to c < c_{\infty} and I^{'} (u_{n}) \to 0 . \end{matrix}$ We can suppose that the sequence $(u_{n})$ is bounded, according to Lemma 3.3. Therefore, for a subsequence, we have $u_{n} ↪ u_{0}$ in $H^{1} (R_{+}^{N + 1})$ . It follows from the Splitting Lemma (Lemma 5.1) that $I^{'} (u_{0}) = 0$ . Since $\begin{matrix} I^{'} (u_{0}) \cdot u_{0} & = \iint_{R_{+}^{N + 1}} (| \nabla u_{0} |^{2} + m^{2} u_{0}^{2}) + \int_{R^{N}} V (y) {| γ (u_{0}) |}^{2} \\ - \int_{R^{N}} [W * F (γ (u_{0}))] f (γ (u_{0})) γ (u_{0}) \end{matrix}$ and $\begin{matrix} I (u_{0}) & = \frac{1}{2} \iint_{R_{+}^{N + 1}} (| \nabla u_{0} |^{2} + m^{2} u_{0}^{2}) + \frac{1}{2} \int_{R^{N}} V (y) {| γ (u_{0}) |}^{2} \\ - \frac{1}{2} \int_{R^{N}} [W * F (γ (u_{0}))] F (γ (u_{0})), \end{matrix}$ we conclude that $\begin{array}{l} I (u_{0}) & = \frac{1}{2} \int_{R^{N}} [W * F (γ (u_{0}))] (f (γ (u_{0})) γ (u_{0}) - 2 F (γ (u_{0}))) \\ + \frac{1}{2} \int_{R^{N}} [W * F (γ (u_{0}))] F (γ (u_{0})) \\ (5.1) & > \frac{1}{2} \int_{R^{N}} [W * F (γ (u_{0}))] (f (γ (u_{0})) γ (u_{0}) - 2 F (γ (u_{0}))) > 0 \end{array}$ as consequence of the Ambrosetti–Rabinowitz condition.

If $u_{n} ↛ u$ in $H^{1} (R_{+}^{N + 1})$ , by applying again the Splitting Lemma we guarantee the existence of $k \in N$ and nontrivial solutions $u^{1}, \dots, u^{k}$ of problem ( P ∞ ) satisfying $\begin{matrix} lim_{n \to \infty} I (u_{n}) = c = I (u_{0}) + \sum_{j = 1}^{k} I_{\infty} (u^{j}) ⩾ k c_{\infty} ⩾ c_{\infty} \end{matrix}$ contradicting our hypothesis. We are done. □

We prove the next result by adapting the proof given in Furtado, Maia e Medeiros [19]: Lemma 5.3.
Suppose that $V (y)$ satisfies $(V_{3})$ . Then $\begin{matrix} 0 < c < c_{\infty}, \end{matrix}$ where c is characterized in Lemma 3.3 .
Proof.
Let $\bar{u} \in N_{\infty}$ be the weak solution of ( P ∞ ) given by Theorem 5 and $t_{\bar{u}} > 0$ be the unique number such that $t_{\bar{u}} \bar{u} \in N$ . We claim that $t_{\bar{u}} < 1$ . Indeed, $\begin{array}{l} \int_{R^{N}} [W * F (γ (t_{\bar{u}} \bar{u}))] f (γ (t_{\bar{u}} \bar{u})) γ (t_{\bar{u}} \bar{u}) \\ = t_{\bar{u}}^{2} \iint_{R_{+}^{N + 1}} (| \nabla \bar{u} |^{2} + m^{2} {\bar{u}}^{2}) + t_{\bar{u}}^{2} \int_{R^{N}} V (y) {| γ (\bar{u}) |}^{2} \\ < t_{\bar{u}}^{2} \iint_{R_{+}^{N + 1}} (| \nabla \bar{u} |^{2} + m^{2} {\bar{u}}^{2}) + t_{\bar{u}}^{2} \int_{R^{N}} V_{\infty} {| γ (\bar{u}) |}^{2} \\ = t_{\bar{u}}^{2} \int_{R^{N}} [W * F (γ (\bar{u}))] f (γ (\bar{u})) γ (\bar{u}) \\ = t_{\bar{u}}^{2} (\int_{R^{N}} [W * F (γ (\bar{u}))] f (γ (\bar{u})) γ (\bar{u}) + \int_{R^{N}} [W * F (γ (t_{\bar{u}} \bar{u}))] f (γ (\bar{u})) γ (\bar{u}) \\ - \int_{R^{N}} [W * F (γ (t_{\bar{u}} \bar{u}))] f (γ (\bar{u})) γ (\bar{u})) \end{array}$ thus yielding $\begin{array}{l} 0 & > \int_{R^{N}} [W * F (γ (t_{\bar{u}} \bar{u}))] (\frac{f (γ (t_{\bar{u}} \bar{u}))}{γ (t_{\bar{u}} \bar{u})} - \frac{f (γ (\bar{u}))}{γ (\bar{u})}) t_{\bar{u}}^{2} γ {(\bar{u})}^{2} \\ + t_{\bar{u}}^{2} \int_{R^{N}} [W * (F (γ (t_{\bar{u}} \bar{u})) - F (γ (\bar{u})))] f (γ (u)) γ (u) . \end{array}$

If $t_{\bar{u}} ⩾ 1$ , since $f (s) / s$ is increasing, the first integral is non-negative and, since F is increasing, the second integral as well. We conclude that $t_{\bar{u}} < 1$ .

Lemma 3.3 and its previous comments show that $\begin{matrix} c ⩽ max_{t ⩾ 0} I (t \bar{u}) = I (t_{\bar{u}} \bar{u}) = \frac{1}{2} \int_{R^{N}} [W * F (γ (t_{\bar{u}} \bar{u}))] (f (γ (t_{\bar{u}} \bar{u})) γ (t_{\bar{u}} \bar{u}) - F (γ (t_{\bar{u}} \bar{u}))) . \end{matrix}$ Since $\begin{matrix} g (t) = \frac{1}{2} \int_{R^{N}} [W * F (γ (t \bar{u}))] (f (γ (t \bar{u})) γ (t \bar{u}) - F (γ (t \bar{u}))) \end{matrix}$ is a strictly increasing function, we conclude that $\begin{matrix} c = g (t_{\bar{u}}) < g (1) = \frac{1}{2} \int_{R^{N}} [W * F (γ (\bar{u}))] (f (γ (\bar{u})) γ (\bar{u}) - F (γ (\bar{u}))) = c_{\infty}, \end{matrix}$ proving our result. □
Proof of Theorem 1.
Let $(u_{n})$ be the minimizing sequence given by Lemma 3.1. It follows from Lemmas 5.2 and 5.3 that $u_{n} \to u$ such that $I (u) = c$ and $I^{'} (u) = 0$ .

We now turn our attention to the positivity of u. Seeing that $\begin{matrix} \iint_{R_{+}^{N + 1}} (\nabla u \cdot \nabla φ + m^{2} u φ) + \int_{R^{N}} V (y) γ (u) γ (φ) = \int_{R^{N}} [W * F (γ (u))] f (γ (u)) γ (φ) \end{matrix}$ and choosing $φ = w^{-}$ , the left-hand side of the equality is positive (by the definition of $I (u)$ and equation (3.1), since $I_{1} + I_{2} ⩾ K ‖ w ‖^{2}$ ), while $Ψ (u) ⩽ 0$ . The proof is complete. □

6. Proof of Theorem 2

The proof of the next result adapts arguments in [4] and [12].

Proposition 6.1.
For all $β > 0$ it holds $\begin{array}{l} {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2} & ⩽ 2 C_{2^{#}}^{2} C_{β} [(| V |_{\infty} + C C_{1} (2 + M)) {| γ {(v_{+})}^{1 + β} |}_{2}^{2} \\ + C_{1} | g |_{2 N / [N (2 - θ) + θ]} {| γ {(v_{+})}^{1 + β} |}_{2^{#} (2 / θ)}^{2}], \end{array}$ where $C_{β} = max {m^{- 2}, (1 + \frac{β}{2})}$ , C, $C_{1}$ , $\tilde{C}$ and $M = M (β)$ are positive constants and $g = | W_{1} * F (γ (v)) |$ is the function given by Lemma 2.1 .
Proof.
Choosing $φ = φ_{β, T} = v v_{T}^{2 β}$ in (2.11), where $v_{T} = min {v_{+}, T}$ and $β > 0$ , we have $0 ⩽ φ_{β, T} \in H^{1} (R_{+}^{N + 1})$ and $\begin{array}{l} \iint_{R_{+}^{N + 1}} \nabla v \cdot \nabla φ_{β, T} + m^{2} v φ_{β, T} \\ (6.1) & = - \int_{R^{N}} V (y) γ (v) γ (φ_{β, T}) + \int_{R^{N}} (W * F (γ (v))) f (γ (v)) γ (φ_{β, T}), \end{array}$ Since $\nabla φ_{β, T} = v_{T}^{2 β} \nabla v + 2 β v v_{T}^{2 β - 1} \nabla v_{T}$ , the left-hand side of (6.1) is given by $\begin{array}{l} (6.2) & \iint_{R_{+}^{N + 1}} \nabla v \cdot (v_{T}^{2 β} \nabla v + 2 β v v_{T}^{2 β - 1} \nabla v_{T}) + m^{2} v (v v_{T}^{2 β}) \\ (6.3) & = \iint_{R_{+}^{N + 1}} v_{T}^{2 β} [| \nabla v |^{2} + m^{2} v^{2}] + 2 β \iint_{D_{T}} v_{T}^{2 β} | \nabla v |^{2}, \end{array}$ where $D_{T} = {(x, y) \in (0, \infty) \times R^{N} : v_{T} (x, y) ⩽ T}$ .

Now we express (6.3) in terms of $‖ v v_{T}^{β} ‖^{2}$ . For this, we note that $\nabla (v v_{T}^{β}) = v_{T}^{β} \nabla v + β v v_{T}^{β - 1} \nabla v_{T}$ . Therefore, $\begin{matrix} \iint_{R_{+}^{N + 1}} {| \nabla (v v_{T}^{β}) |}^{2} = \iint_{R_{+}^{N + 1}} v_{T}^{2 β} | \nabla v |^{2} + (2 β + β^{2}) \iint_{D_{T}} v_{T}^{2 β} | \nabla v |^{2}, \end{matrix}$ thus yielding $\begin{array}{l} {‖ v v_{T}^{β} ‖}^{2} & = (\iint_{R_{+}^{N + 1}} v_{T}^{2 β} | \nabla v |^{2} + (2 β + β^{2}) \iint_{D_{T}} v_{T}^{2 β} | \nabla v |^{2}) + \iint_{R_{+}^{N + 1}} {(v v_{T}^{β})}^{2} \\ = \iint_{R_{+}^{N + 1}} v_{T}^{2 β} (| \nabla v |^{2} + | v |^{2}) + 2 β (1 + \frac{β}{2}) \iint_{D_{T}} v_{T}^{2 β} | \nabla v |^{2} \\ (6.4) & ⩽ C_{β} [\iint_{R_{+}^{N + 1}} v_{T}^{2 β} (| \nabla v |^{2} + m^{2} | v |^{2}) + 2 β \iint_{D_{T}} v_{T}^{2 β} | \nabla v |^{2}], \end{array}$ where $C_{β} = max {m^{- 2}, (1 + \frac{β}{2})}$ . Gathering (6.1), (6.3) and (6.4), we obtain $\begin{array}{l} {‖ v v_{T}^{β} ‖}^{2} & ⩽ C_{β} [- \int_{R^{N}} V γ {(v)}^{2} γ {(v_{T})}^{2 β} \\ (6.5) & + \int_{R^{N}} (W * F (γ (v))) f (γ (v)) γ (v) γ {(v_{T})}^{2 β}] . \end{array}$

We now start to consider the right-hand side of (6.5). Since $| f (t) | ⩽ C_{1} (| t | + | t |^{θ - 1})$ , Corollary 2.1 shows that it can be written as $\begin{array}{l} ⩽ C_{β} [| V |_{\infty} \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + \int_{R^{N}} (C + g) | f (γ (v)) | | γ (v) | γ {(v_{T})}^{2 β}] \\ ⩽ C_{β} [| V |_{\infty} \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + C \int_{R^{N}} C_{1} (| γ (v) | + {| γ (v) |}^{θ - 1}) | γ (v) | γ {(v_{T})}^{2 β} \\ + C_{1} \int_{R^{N}} g (| γ (v) | + {| γ (v) |}^{θ - 1}) | γ (v) | γ {(v_{T})}^{2 β}] \\ ⩽ C_{β} [(| V |_{\infty} + C C_{1}) \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + C C_{1} \int_{R^{N}} {| γ (v) |}^{θ - 2} γ {(v)}^{2} γ {(v_{T})}^{2 β} \\ (6.6) & + C_{1} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2} + C_{1} \int_{R^{N}} g {| γ (v) |}^{θ - 2} γ {(v v_{T}^{β})}^{2}] . \end{array}$

Applying Lemmas 2.2 and 2.3, inequality (6.6) becomes $\begin{array}{l} ⩽ C_{β} [(| V |_{\infty} + C C_{1}) \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + C C_{1} \int_{R^{N}} (1 + g_{2}) γ {(v v_{T}^{β})}^{2} \\ + C_{1} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2} + C_{1} \int_{R^{N}} h γ {(v v_{T}^{β})}^{2}] \\ ⩽ C_{β} [(| V |_{\infty} + 2 C C_{1}) \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + C C_{1} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2} + C C_{1} \int_{R^{N}} G γ {(v v_{T}^{β})}^{2}], \end{array}$ where $G = g_{2} + h \in L^{N} (R^{N})$ , admitting that $C C_{1} ⩾ C_{1}$ .

Because $| γ (u) |_{2^{#}} ⩽ C_{2^{#}} ‖ u ‖$ for all $u \in H^{1} (R_{+}^{N + 1})$ , the last inequality is equivalent to $\begin{array}{l} {| γ (v v_{T}^{β}) |}_{2^{#}}^{2} & ⩽ C_{2^{#}}^{2} C_{β} [(| V |_{\infty} + 2 C C_{1}) \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + C C_{1} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2} \\ (6.7) & + C C_{1} \int_{R^{N}} G γ {(v v_{T}^{β})}^{2}] . \end{array}$

Let us consider the last integral in the right-hand side of (6.7). For all $M > 0$ , define $A_{1} = {G ⩽ M}$ and $A_{2} = {G > M}$ . Then, whereas $G \in L^{N} (R^{N})$ , $\begin{array}{l} \int_{R^{N}} G γ {(v v_{T}^{β})}^{2} & ⩽ M \int_{A_{1}} γ {(v v_{T}^{β})}^{2} + {(\int_{A_{2}} G^{N})}^{\frac{1}{N}} {(\int_{A_{2}} γ {(v v_{T}^{β})}^{2 \frac{N}{N - 1}})}^{\frac{N - 1}{N}} \\ ⩽ M \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + ϵ (M) {(\int_{R^{N}} γ {(v v_{T}^{β})}^{2^{#}})}^{\frac{N - 1}{N}}, \end{array}$ and $ϵ (M) = {(\int_{A_{2}} G^{N})}^{1 / N} \to 0$ when $M \to \infty$ .

If M is taken so that $ϵ (M) C_{2^{#}}^{2} C_{β} C C_{1} < 1 / 2$ , we have $\begin{array}{l} {| γ (v v_{T}^{β}) |}_{2^{#}}^{2} & ⩽ 2 C_{2^{#}}^{2} C_{β} [(| V |_{\infty} + C C_{1} (2 + M)) \int_{R^{N}} γ {(v v_{T}^{β})}^{2} \\ (6.8) & + C C_{1} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2}] . \end{array}$

The Hölder inequality guarantees that $\begin{array}{l} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2} ⩽ | g |_{2 N / [N (2 - θ) + θ]} {(\int_{R^{N}} γ {(v v_{T}^{β})}^{2 α^{'}})}^{1 / α^{'}}, \end{array}$ where $\begin{matrix} α^{'} = \frac{\frac{2 N}{N (2 - θ) + θ}}{\frac{2 N}{N (2 - θ) + θ} - 1} = \frac{2 N}{(N - 1) θ} = \frac{2^{#}}{θ} . \end{matrix}$ Thus, $\begin{array}{l} \int_{R^{N}} g γ {(v v_{T}^{β})}^{2} ⩽ | g |_{2 N / [N (2 - θ) + θ]} {| γ (v v_{T}^{β}) |}_{2^{#} (2 / θ)}^{2} \end{array}$ and substitution on the right-hand side of (6.8) yields $\begin{array}{l} {| γ (v v_{T}^{β}) |}_{2^{#}}^{2} & ⩽ 2 C_{2^{#}}^{2} C_{β} [(| V |_{\infty} + C C_{1} (2 + M)) {| γ (v v_{T}^{β}) |}_{2}^{2} \\ (6.9) & + C_{1} | g |_{2 N / [N (2 - θ) + θ]} {| γ (v v_{T}^{β}) |}_{2^{#} (2 / θ)}^{2}] . \end{array}$

Since $v v_{T}^{β} \to v_{+}^{1 + β}$ , it follows from (6.9) that $\begin{array}{l} {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2} & ⩽ 2 C_{2^{#}}^{2} C_{β} [(| V |_{\infty} + C C_{1} (2 + M)) {| γ {(v_{+})}^{1 + β} |}_{2}^{2} \\ + C_{1} | g |_{2 N / [N (2 - θ) + θ]} {| γ {(v_{+})}^{1 + β} |}_{2^{#} (2 / θ)}^{2}], \end{array}$ and we are done. (Observe, however, that M depends on β.) □
Proposition 6.2.
For all $p \in [2, \infty)$ we have $γ (v) \in L^{p} (R^{N})$ .
Proof.
Since $\frac{2 N}{N - 1} \frac{2}{θ} ⩽ 2$ never occurs, we have $2 < \frac{2^{#} 2}{θ} = \frac{2 N}{N - 1} \frac{2}{θ} < 2^{#}$ .

According to the Proposition 6.1, we have $\begin{array}{l} (6.10) & {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2} & ⩽ [D_{1} {| γ {(v_{+})}^{1 + β} |}_{2}^{2} + E_{1} {| γ {(v_{+})}^{1 + β} |}_{2^{#} (2 / θ)}^{2}], \end{array}$ where $D_{1}$ and $E_{1}$ are positive constants.

Choosing $β_{1} + 1 : = (θ / 2) > 1$ , it follows from (2.4) that $\begin{matrix} {| γ {(v_{+})}^{1 + β} |}_{2^{#} (2 / θ)}^{2} = {| γ (v_{+}) |}_{\frac{2 N}{N - 1}}^{θ} < \infty, \end{matrix}$ from what follows that the right-hand side of (6.10) is finite. We conclude that $| γ (v_{+}) | \in L^{\frac{2 N}{N - 1} \frac{θ}{2}} (R^{N}) < \infty$ . Now, we choose $β_{2}$ so that $β_{2} + 1 = {(θ / 2)}^{2}$ and conclude that $\begin{matrix} | γ (v_{+}) | \in L^{\frac{2 N}{N - 1} \frac{θ^{2}}{2^{2}}} (R^{N}) . \end{matrix}$

After k iterations we obtain that $\begin{matrix} | γ (v_{+}) | \in L^{\frac{2 N}{N - 1} \frac{θ^{k}}{2^{k}}} (R^{N}), \end{matrix}$ from what follows that $γ (v_{+}) \in L^{p} (R^{N})$ for all $p \in [2, \infty)$ . Since the same arguments are valid for $v_{-}$ , we have $γ (v) \in L^{p} (R^{N})$ for all $p \in [2, \infty)$ . □

By simply adapting the proof given in [12], we present, for the convenience of the reader, the proof of our next result:
Proposition 6.3.
Let $v \in H^{1} (R_{+}^{N + 1})$ be a weak solution of ( 1.3 ). Then $γ (v) \in L^{p} (R^{N})$ for all $p \in [2, \infty]$ and $v \in L^{\infty} (R_{+}^{N + 1})$ .
Proof.
We recall equation (6.5): $\begin{array}{l} {‖ v v_{T}^{β} ‖}^{2} ⩽ C_{β} [- \int_{R^{N}} V γ {(v v_{T}^{β})}^{2} + \int_{R^{N}} (W * F (γ (v))) f (γ (v)) γ (v) γ {(v_{T})}^{2 β}], \end{array}$ where $C_{β} = max {m^{- 2}, (1 + β^{2})}$ .

It follows that $W * F (γ (v)) \in L^{\infty} (R^{N})$ , since $γ (v) \in L^{p} (R^{N})$ for all $p ⩾ 2$ , by Proposition 6.2. We also know that $| f (t) | ⩽ C_{1} (| t | + | t |^{θ - 1})$ and V is bounded. Therefore, if $C = max {| V |_{\infty}, C_{1} | W * F (γ) |_{\infty}}$ , we have $\begin{array}{l} {‖ v v_{T}^{β} ‖}^{2} & ⩽ C_{β} C [\int_{R^{N}} γ {(v v_{T}^{β})}^{2} + \int_{R^{N}} (| γ (v) | + {| γ (v) |}^{θ - 1}) γ (v) γ {(v_{T})}^{2 β}] \\ ⩽ C_{β} [2 C \int_{R^{N}} γ {(v v_{T}^{β})}^{2} + C \int_{R^{N}} {| γ (v) |}^{θ - 2} γ {(v v_{T}^{β})}^{2}] . \end{array}$ Since $| γ (v) |^{p - 2} = | γ (v) |^{p - 2} χ_{{| γ (v) ⩽ 1}} + | γ (v) |^{p - 2} χ_{{| γ (v) > 1}}$ , the fact that $\begin{matrix} {| γ (v) |}^{p - 2} χ_{{| γ (v) > 1}} = : g_{3} \in L^{2 N} (R^{N}) \end{matrix}$ allows us to conclude that $\begin{array}{l} 2 C γ {(v v_{T}^{β})}^{2} + C {| γ (v) |}^{θ - 2} γ {(v v_{T}^{β})}^{2} ⩽ (C_{3} + g_{3}) γ {(v v_{T}^{β})}^{2} \end{array}$ for a positive constant $C_{3}$ and a positive function $g_{3} \in L^{2 N} (R^{N})$ that depends neither on T nor on β.

Therefore, $\begin{array}{l} {‖ v v_{T}^{β} ‖}^{2} & ⩽ \int_{R^{N}} (C_{3} + g_{3}) γ {(v v_{T}^{β})}^{2} \end{array}$ and $\begin{array}{l} {‖ v_{+}^{β + 1} ‖}^{2} & ⩽ C_{β} \int_{R^{N}} (C_{3} + g_{3}) γ {(v_{+}^{β + 1})}^{2} . \end{array}$ Since $\begin{array}{l} \int_{R^{N}} g_{3} γ {(v_{+}^{β + 1})}^{2} & ⩽ | g_{3} |_{2 N} {| γ {(v_{+})}^{1 + β} |}_{2} {| γ {(v_{+})}^{1 + β} |}_{2^{#}} \\ ⩽ | g_{3} |_{2 N} (λ {| γ {(v_{+})}^{1 + β} |}_{2}^{2} + \frac{1}{λ} {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2}), \end{array}$ we conclude that $\begin{array}{l} {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2} & ⩽ C_{2^{#}}^{2} {‖ v_{+}^{β + 1} ‖}^{2} \\ ⩽ C_{2^{#}}^{2} C_{β} (C_{3} + λ | g_{3} |_{2 N}) {| γ {(v_{+})}^{1 + β} |}_{2}^{2} + \frac{C_{2^{#}}^{2} C_{β} | g_{3} |_{2 N}}{λ} {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2} \end{array}$ and, by taking $λ > 0$ so that $\begin{matrix} \frac{C_{2^{#}}^{2} C_{β} | g_{3} |_{2 N}}{λ} < \frac{1}{2}, \end{matrix}$ we obtain $\begin{array}{l} {| γ {(v_{+})}^{1 + β} |}_{2^{#}}^{2} & ⩽ C_{β} (2 C_{2^{#}}^{2} C_{3} + 2 C_{2^{#}}^{2} λ | g_{3} |_{2 N}) {| γ {(v_{+})}^{1 + β} |}_{2}^{2} \\ (6.11) & ⩽ C_{4} C_{β} {| γ {(v_{+})}^{1 + β} |}_{2}^{2} . \end{array}$ Since $\begin{matrix} C_{4} C_{β} ⩽ C_{4} (m^{- 2} + 1 + β) ⩽ M^{2} e^{2 \sqrt{1 + β}} \end{matrix}$ for a positive constant M, it follows from (6.11) that $\begin{array}{l} {| γ (v_{+}) |}_{2^{#} (1 + β)} & ⩽ M^{1 / (1 + β)} e^{1 / \sqrt{1 + β}} {| γ (v_{+}) |}_{2 (1 + β)} . \end{array}$ We now apply an iteration argument, taking $2 (1 + β_{n + 1}) = 2^{#} (1 + β_{n})$ and starting with $β_{0} = 0$ . This produces $\begin{matrix} {| γ (v_{+}) |}_{2^{#} (1 + β_{n})} ⩽ M^{1 / (1 + β_{n})} e^{1 / \sqrt{1 + β_{n}}} {| γ (v_{+}) |}_{2 (1 + β_{n})} . \end{matrix}$ Because $(1 + β_{n}) = {(\frac{2^{#}}{2})}^{n} = {(\frac{N}{N - 1})}^{n}$ , we have $\begin{matrix} \sum_{i = 0}^{\infty} \frac{1}{1 + β_{n}} < \infty and \sum_{i = 0}^{\infty} \frac{1}{\sqrt{1 + β_{n}}} < \infty . \end{matrix}$

Thus, $\begin{matrix} {| γ (v_{+}) |}_{\infty} = lim_{n \to \infty} {| γ (v_{+}) |}_{2^{#} (1 + β_{n})} < \infty, \end{matrix}$ from what follows $| γ (v_{+}) |_{p} < \infty$ for all $p \in [2, \infty]$ . The same argument applies to $γ (v_{-})$ , proving that $γ (v) \in L^{p} (R^{N})$ for all $p \in [2, \infty]$ .

By taking $λ = 1$ and $| γ {(v_{+})}^{1 + β} |_{p} < C_{5}$ for all p, we obtain for any $β > 0$ , $\begin{array}{l} (6.12) & {‖ v_{+}^{β + 1} ‖}^{2} & ⩽ C_{β} (C_{3} + | g_{3} |_{2 N}) C_{5}^{2} + C_{β} | g_{3} |_{2 N} C_{5}^{2} . \end{array}$

But $‖ v_{+} ‖_{2^{} (1 + β)}^{1 + β} = ‖ v_{+}^{1 + β} ‖_{2^{}} ⩽ C_{2^{}} ‖ v_{+}^{1 + β} ‖$ and for a positive constant $\tilde{c}$ results from (6.12) that $\begin{matrix} ‖ v_{+} ‖_{2^{} (1 + β)}^{2 (1 + β)} ⩽ \tilde{c} C_{β} C_{5}^{2 (1 + β)} . \end{matrix}$ Thus, $\begin{matrix} ‖ v_{+} ‖_{2^{} (1 + β)} ⩽ {\tilde{c}}^{1 / 2 (1 + β)} C_{β}^{1 / 2 (1 + β)} C_{5} \end{matrix}$ and the right-hand side of the last inequality is uniformly bounded for all $β > 0$ . We are done. □

We now state [12, Proposition 3.9]: Proposition 6.4.
Suppose that* $v \in H^{1} (R_{+}^{N + 1}) \cap L^{\infty} (R_{+}^{N + 1})$ is a weak solution of $\begin{matrix} (6.13) & \{\begin{matrix} - Δ v + m^{2} v = 0, & in R_{+}^{N + 1}, \\ - \frac{\partial v}{\partial x} (0, y) = h (y) & for all y \in R^{N}, \end{matrix} \end{matrix}$ where $h \in L^{p} (R^{N})$ for all $p \in [2, \infty]$ .

Then $v \in C^{α} ([0, \infty) \times R^{N}) \cap W^{1, q} ((0, R) \times R^{N})$ for all $q \in [2, \infty)$ and $R > 0$ .

In addition, if $h \in C^{α} (R^{N})$ , then $v \in C^{1, α} ([0, \infty) \times R^{N}) \cap C^{2} (R_{+}^{N + 1})$ is a classical solution of (6.13).
Proof of Theorem 2.
In the proof of Proposition 6.4 (see [12, Proposition 3.9]), defining $\begin{matrix} ρ (x, y) = \int_{0}^{x} v (t, y) d t, \end{matrix}$ taking the odd extension of h and ρ to the whole $R^{N + 1}$ (which we still denote simply by h and ρ), in [12] is obtained that ρ satisfies the equation $\begin{matrix} (6.14) & - Δ ρ + m^{2} ρ = h in R^{N + 1} \end{matrix}$ and $ρ \in C^{1, α} (R^{N + 1})$ for all $α \in (0, 1)$ by applying Sobolev’s embedding. Therefore, $v (x, y) = \frac{\partial ρ}{\partial x} (x, y) \in C^{α} (R^{N})$ .

In our case $\begin{matrix} h (y) = - V (y) v (0, y) + (W * F (v (0, y))) f (v (0, y)) . \end{matrix}$

We now rewrite equation (6.14) as $\begin{matrix} - Δ ρ + V (y) \frac{\partial ρ}{\partial x} (0, y) + m^{2} ρ = (W * F (\frac{\partial ρ}{\partial x} (0, y))) f (\frac{\partial ρ}{\partial x} (0, y)) . \end{matrix}$ Since $f \in C^{1}$ and $\frac{\partial ρ}{\partial x} (x, y)$ is bounded, the right-hand side of the last equality belongs to $C^{α} (R^{N + 1})$ . Thus, classical elliptic boundary regularity yields $\begin{matrix} ρ \in C^{2} (R^{N + 1}) \Rightarrow v \in C^{1, α} (R_{+}^{N + 1}) . \end{matrix}$ Hence, by applying classical interior elliptic regularity directly to v, we deduce that $v \in C^{1, α} (R_{+}^{N + 1}) \cap C^{2} (R_{+}^{N + 1})$ is a classical solution of problem (1.3). □

7. Proof of Theorem 3

We now adjust [12, Theorem 3.14] to our needs. The original statement guarantees that $v \in C^{\infty} ([0, \infty) \times R^{N})$ , a result that depends on the function h (of Proposition 6.4) considered in that paper. For the convenience of the reader, we present the proof of the next result:

Theorem 6.

Suppose that $v \in H^{1} (R_{+}^{N + 1})$ is a critical point of the energy functional I, then $\begin{matrix} | v (x, y) | e^{λ x} \to 0 \end{matrix}$ as $x + | y | \to \infty$ , for any $λ < m$ .

Proof.

Let us consider a solution v of the problem $\begin{matrix} \{\begin{matrix} - Δ v + m^{2} v = 0 in R_{+}^{N + 1} \\ v (0, y) = v_{0} (y) \in L^{2} (R^{N}), y \in R^{N} = \partial R_{+}^{N + 1} . \end{matrix} \end{matrix}$

By applying the Fourier transform with respect to variable $y \in R^{N}$ we obtain $\begin{matrix} F v (x, k) = e^{- \sqrt{| 2 π k |^{2} + m^{2}} x} F v_{0} (k), \end{matrix}$ from what follows $\begin{matrix} sup_{y \in R^{N}} | v (x, y) | ⩽ C | v_{0} |_{2} e^{- m x} . \end{matrix}$ Since Proposition 6.4 shows that $v \in W^{1, q} ((0, R) \times R^{N})$ for all $q \in [2, \infty)$ and $R > 0$ , we conclude that $| v (x, y) | \to 0$ when $| y | \to \infty$ for any x and $| v (x, y) | e^{λ x} \to 0$ as $x + | y | \to \infty$ for any $λ < m$ . □

We now adapt the proof of [12, Theorem 5.1]. In that paper is assumed that $W (y) \to 0$ as $| y | \to \infty$ , a condition that is not necessary.

Proof of Theorem 3.

We denote $\begin{matrix} K (y) = W * F (\frac{\partial w}{\partial x} (0, y)) . \end{matrix}$ It follows easily that K is bounded.

By Theorem 1 we have $w (x, y) ⩾ 0$ . Applying Harnack’s inequality we conclude that w is strictly positive.

Following [12], for any $R > 0$ we denote $\begin{array}{l} B_{R}^{+} = {(x, y) \in R_{+}^{N + 1} : \sqrt{x^{2} + | y |^{2}} < R} \\ Ω_{R}^{+} = {(x, y) \in R_{+}^{N + 1} : \sqrt{x^{2} + | y |^{2}} > R} \\ Γ_{R}^{+} = {(0, y) \in R_{+}^{N + 1} : | y | > R} \end{array}$ and define $\begin{matrix} f_{R} (x, y) = C_{R} e^{- α x} e^{- (m - α) \sqrt{x^{2} + | y |^{2}}}, \end{matrix}$ where the positive constants $C_{R}$ and $α \in (V_{0}, m)$ will be chosen later on. A simple computation shows that $\begin{matrix} Δ f_{R} = (α^{2} + {(m - α)}^{2} + \frac{2 α (m - α) x}{\sqrt{x^{2} + | y |^{2}}} - \frac{N (m - α)}{\sqrt{x^{2} + | y |^{2}}}) f_{R} . \end{matrix}$ Thus, for R large enough, we have $\begin{matrix} \{\begin{matrix} - Δ f_{R} + m^{2} f_{R} ⩾ 0 in Ω_{R}^{+} \\ - \frac{\partial f_{R}}{\partial x} = \frac{\partial f_{R}}{\partial η} = α f_{R} on Γ_{R}^{+} . \end{matrix} \end{matrix}$

We now define $\begin{matrix} ρ (x, y) = f_{R} (x, y) - w (x, y) . \end{matrix}$ We clearly have $- Δ ρ (x, y) + m^{2} ρ (x, y) ⩾ 0$ in $Ω_{R}^{+}$ . Choosing $\begin{matrix} C_{R} = e^{m R} max_{\partial B_{R}^{+}} v, \end{matrix}$ we also have $ρ (x, y) ⩾ 0$ on $\partial B_{R}^{+}$ and $ρ (x, y) \to 0$ when $x + | y | \to \infty$ .

We claim that $ρ (x, y) ⩾ 0$ in ${\overline{Ω}}_{R}^{+}$ . Supposing the contrary, let us assume that ${inf}_{{\overline{Ω}}_{R}^{+}} ρ (x, y) < 0$ . By the strong maximum principle, there exist $(0, y_{0}) \in Γ_{R}^{+}$ such that $ρ (0, y_{0}) = {inf}_{{\overline{Ω}}_{R}^{+}} ρ (x, y) < ρ (x, y)$ for all $(x, y) \in Ω_{R}^{+}$ . Defining $\begin{matrix} z (x, y) = ρ (x, y) e^{λ x} \end{matrix}$ for some $λ \in (V_{0}, m)$ , a straightforward calculation shows that $\begin{matrix} - Δ ρ + m^{2} ρ = e^{- λ x} (Δ z + 2 λ \partial_{x} z + (m^{2} - λ^{2}) z) . \end{matrix}$ Since $- Δ ρ + m^{2} ρ ⩾ 0$ , we conclude that $Δ z + 2 λ \partial_{x} z + (m^{2} - λ^{2}) z ⩾ 0$ .

Another application of the strong maximum principle yields $\begin{matrix} z (0, y_{0}) = inf_{Γ_{R}^{+}} z = inf_{Γ_{R}^{+}} ρ = ρ (0, y_{0}) < 0 . \end{matrix}$

An application of Hopf’s lemma produces $\frac{\partial z}{\partial η} (0, y_{0}) < 0$ , that is, $\begin{matrix} (7.1) & - \frac{\partial z}{\partial x} (0, y_{0}) < 0 . \end{matrix}$ Since $\frac{\partial z}{\partial x} = \frac{\partial ρ}{\partial x} e^{λ x} + λ ρ e^{λ x}$ , we conclude that $\begin{matrix} \frac{\partial z}{\partial x} (0, y_{0}) = \frac{\partial ρ}{\partial x} (0, y_{0}) + λ ρ (0, y_{0}) \end{matrix}$ and so $\begin{array}{l} - \frac{\partial z}{\partial x} (0, y_{0}) & = - \frac{\partial f_{R}}{\partial x} (0, y_{0}) + \frac{\partial w}{\partial x} (0, y_{0}) - λ f_{R} (0, y_{0}) + λ w (0, y_{0}) \\ = (α - λ) f_{R} (0, y_{0}) + V (y_{0}) w (0, y_{0}) - K (y_{0}) f (w (0, y_{0})) \\ + λ w (0, y_{0}) \\ = (α - λ) f_{R} (0, y_{0}) + (V (y_{0}) + V_{0}) w (0, y_{0}) - K (y_{0}) f (w (0, y_{0})) \\ + (λ - V_{0}) w (0, y_{0}) . \end{array}$ Now, choosing $α = λ$ , since $λ > V_{0}$ (so that the last term in the above inequality is non-negative), the positiveness of $(V (y_{0}) + V_{0}) w (0, y_{0})$ and hypothesis $(f_{1})$ guarantees that $- \frac{\partial z}{\partial x} (0, y_{0}) > 0$ , thus reaching a contradiction with (7.1). □

Footnotes

Acknowledgements

P. Belchior was partially supported by CAPES/Brazil. O.H. Miyagaki received research grants from CNPq/Brazil 307061/2018-3 and INCTMAT/CNPQ/Brazil. G.A. Pereira received research grants by PNPD/CAPES/Brazil.

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