The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem where R+N:={x∈RN:xN>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$\end{document} for some N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 1$$\end{document} and p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p > 1$$\end{document}, c>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c > 0$$\end{document} are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (Pc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c$$\end{document}). We prove that the existence and multiplicity of bounded positive solutions to (Pc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c$$\end{document}) depend in a striking way on the value of c>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c > 0$$\end{document} and also on the dimension N. We find an explicit number cp∈(1,e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_p}\in (1,\sqrt{e})$$\end{document}, depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 2$$\end{document}, we prove that, for 0<c<cp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< c < {c_p}$$\end{document}, problem (Pc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c$$\end{document}) admits infinitely many bounded positive solutions, whereas, for c>cp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c > {c_p}$$\end{document}, there are no bounded positive solutions to (Pc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_c$$\end{document}).


Introduction
Due to its relevance within several models arising in physics and biology, the nonlinear stationary Schrödinger equation received extensive attention in the last four decades. In particular, let us mention that the study of solitary wave solutions for the (focusing) NLS i∂ t ϕ + ∆ϕ + |ϕ| p−1 ϕ = 0, (t, x) ∈ R × R N , is reduced to problem (1.1) via a time-harmonic ansatz. For classic existence and multiplicity results, we refer the reader e.g. to the seminal papers [2,3,6,8,9,27,28] and the monographs [29,31]. We also recall the fundamental works [16,21] where the radial symmetry and uniqueness, up to translations, of positive solutions to (1.1) satisfying the decay condition (1.2) v(x) → 0, as |x| → ∞, are proved in the case 1 < p < 2 * − 1. In particular, these results imply the uniqueness, up to translations, of positive finite energy solutions u ∈ H 1 (R N ). In contrast, (1.1) admits an abundance of sign-changing finite energy solutions satisfying (1.2), see e.g. [2,3,24,25] and the references therein. Moreover, more recent geometric constructions of different solution shapes highlight the rich structure of the set of positive solutions which do not satisfy the decay assumption (1.2), see e.g. [1,11,23,25] and the references therein. Whereas it seems impossible to provide an exhaustive list of references for the full space problem (1.1), much less is known regarding the half-space Dirichlet problem on ∂R N + , where R N + := {x ∈ R N : x N > 0} for some N ≥ 1 and c ≥ 0 is a constant. In the case c = 0, general nonexistence results are available for (1.3). More precisely, the non-existence of finite energy solutions u ∈ H 1 0 (R N + ) to (1.3) in the case c = 0 follows from [14,Theorem I.1], while [4,Corollary 1.3] yields, in particular, the non-existence of positive solutions to (1.3) with c = 0 and the the decay property (1.4) lim The aim of the present paper is to analyse the existence, non-existence and multiplicity of bounded positive solutions v to the problem (1.3)- (1.4) in the case c > 0, for which we are not aware of any previous result in general dimensions N . As we shall see below, the multiplicity of positive solutions depends on a striking way 1 on the value c > 0 and, somewhat surprisingly, also on the dimension N . Let us stress that we cannot expect the existence of finite energy solutions u ∈ H 1 (R N + ) to (1.3) in the case N ≥ 2. Actually, we cannot expect solutions to (1.3) belonging to L p (R N + ) for any 1 ≤ p < ∞. The one-dimensional decay condition (1.4) therefore seems natural.
Not surprisingly, problem (1.3) is completely understood in the case N = 1. This is due to the fact that the one-dimensional equation (1.5) − w ′′ + w = w p admits a first integral, and from this one can easily deduce that, for 1 < p < ∞, (1.5) admits, up to sign and translation, a unique global non trivial solution satisfying w(t) → 0 as t → ±∞. See e.g. [8,Theorem 5] and [31,Theorem 3.16]. By direct computations, one can verify that this solution is precisely given by with c p := p + 1 2 1 p−1 = w 0 (0) = sup t∈R w 0 (t).
As we shall see below, the value c p will be of key importance also for the higher dimensional version of (1.3).
The following complete characterization of the one dimensional case is an immediate consequence of these facts.
where, here and in the following, we write At first glance, it seems natural to establish such a uniqueness result also with the help of a sliding argument as mentioned in (b) above, but additional difficulties appear in the case c = c p , and non-uniqueness remains a possibility for now.
The following result provides some information on the shape of the solutions we construct.
with a nonnegative function u ∈ H 1 0 (R N + ) \ {0}. Here and in the following, 2 * denotes the critical Sobolev exponent, i.e. 2 * = 2N N −2 for N ≥ 3 and 2 * = 2 * −1 = ∞ for N = 1, 2. By the remarks above and since all exponents p < ∞ are subcritical in the case N = 2, Theorem 1.3 implies Theorem 1.2 (i) in the case N = 2 and therefore for all N ≥ 2. It also allows us to distinguish different solution orbits under translations and rotations in R N −1 . In the following, we call two solutions geometrically distinct if they do not belong to the same orbit of solutions under translations and rotations in R N −1 .
with a nonnegative u ∈ H 1 0 (R N + )\{0}. Clearly, these ansatzs give rise to solutions that are geometrically different. It is natural to guess that the change of the solution set when passing from c > c p to c < c p is a bifurcation phenomenon. More precisely, one may guess that the solutions constructed in Theorem 1.3 have the property that u = u c → 0 ∈ H 1 0 (R N + ) as c ր c p for the functions u in the ansatz (1.8).
This remains an open question, and the answer could even depend on the value of p. We note that standard results from bifurcation theory do not apply here since the linearized problem on ∂R N + , at the parameter value c = c p has purely essential spectrum due to its invariance with respect to translations in directions parallel to the boundary ∂R N + = R N −1 . Bifurcation from the essential spectrum has been observed succesfully in other contexts (see e.g. the survey paper [30] and the references therein), but there is still no general functional analytic framework which provides sufficient abstract conditions. We now give some ideas of the proof of Theorem 1.3. For this we fix c ∈ (0, c p ) and define the functions where t c,p is given in (1.7). We recall that z c and z c are the unique positive solutions to (1.5) such that z c (0) = z c (0) = c. Moreover, we define u c : R N + → R and u c : and we directly notice that u c and u c are both solutions to (1.3)-(1.4). Furthermore, it follows that Proving Theorem 1.3 now amounts to find a nonnegative solution u ∈ H 1 0 (R N + ) \ {0} to the non-autonomous Schrödinger type equation because in this case v = u c + u is of the form (1.8), solves (1.3) and it is easy to see that also satisfies (1.4). Since we are interested in finding non-negative solutions to (1.11), we truncate the nonlinearity and define with f given in (1.12). We then consider the auxiliary problem (1.14) − ∆u + u = g(x, u), u ∈ H 1 0 (R N + ).
Considering u − ∈ H 1 0 (R N + ) as test function in (1.14), one can easily check that every solution to (1.14) is nonnegative and so, that every solution to (1.14) is a non-negative solution to (1.11). It might be worth pointing out that the one-dimensional function u := u c − u c ∈ C 2 (R N + ) is a positive solution to the equation in (1.14) and also satisfies u = 0 on ∂R N + . However, u H 1 0 (R N + ) since it only depends on the x N variable. Hence, u is not a solution to (1.14).
We shall look for a non-trivial solution to (1.14) as a critical point of the associated functional More precisely, we are going to prove the existence of a non-trivial critical point of mountain pass type. This requires new and subtle estimates. The key difficulties in the variational approach are the non-standard shape of the nonlinearity g in (1.14) and the lack of compactness due to the unboundedness of R N + . To overcome these difficulties, we need new estimates within the analysis of Cerami sequences and for comparing the mountain pass energy value for E with the corresponding one of the limit energy functional In particular, we shall use the asymptotic decay properties of the unique positive radial solution to (1.1) in order to build suitable test functions. We now comment on the proofs of the non-existence part (ii) of Theorem 1.2. We argue by contradiction and use a suitable modification of the so-called sliding method introduced by H. Berestycki   . The proof of Proposition 1.5 follows by a rather standard blow up argument based on the doubling lemma by P. Poláčik, P. Quittner and P. Souplet in [26]. For the convenience of the reader, we include the proof in Section 5 below.
Organization of the paper. In Section 2, we collect estimates related to the nonlinearity g in (1.13) and the functional E associated with (1.14). With the help of these estimates, we establish the mountain pass geometry of E in Section 3, and we show that Cerami sequences at nontrivial energy levels are bounded and admit nontrivial weak limits after suitable translation. In Section 4, we then prove a key energy estimate which shows that, in dimensions N ≥ 2, the mountain pass energy of the functional E is strictly smaller than the corresponding one for the limit energy functional E ∞ given in (1.17). With the help of this energy estimate, we then complete the proof of Theorem 1.3 in Section 5. Finally, we give the proof of Theorem 1.2 (ii) in Section 6.
Notation. For 1 ≤ p < ∞, we let · L p (R N + ) denote the standard norm on the usual Lebesgue space L p (R N + ). The Sobolev space H 1 0 (R N + ) is endowed with the standard norm Also, for a function v, we define v + := max{v, 0} and v − := max{−v, 0} and we write x = (x ′ , x N ) for x ∈ R N + with x ′ ∈ R N −1 . We denote by ′ → ′ , respectively by ′ ⇀ ′ , the strong convergence, respectively the weak convergence in corresponding space and denote by B R (x) the open ball in R N of center x and radius R > 0. Also, we shall denote by C i > 0 different constants which may vary from line to line but are not essential to the analysis of the problem. Finally, at various places, we have to distinguish the cases p ≤ 2 and p > 2. For this it is convenient to introduce the special constant Acknowledgements. Part of this work was done while the first author was visiting the Goethe-Universität Frankfurt. He wishes to thank his hosts for the warm hospitality and the financial support.

Preliminaries
In this section we collect some estimates related to the transformed nonlinearity g defined in (1.13), its primitive G and the functional E defined in (1.15). For this we fix, throughout Sections 2-Section 5, c ∈ (0, c p ), p ∈ (1, 2 * ), and we let u c be given in (1.9). We recall that we have the uniform estimate We start with an elementary inequality for nonnegative real numbers which will be used in the energy estimates in Section 4 below.
2) holds in symmetric form with κ q = q, see e.g. [19,Theorem 1]. If q ∈ (2, 3), it is easy to see that one has to choose κ q < q.
Proof of Lemma 2.1. We first note that, since q − 1 > 1, we have, by convexity of the function τ → (1 + τ) q−1 , for s ≥ t > 0. Now, to prove the claim, it suffices to consider a, b > 0, since the inequality holds trivially if a = 0 or b = 0. Moreover, it suffices to prove that the inequality holds for b ≥ a > 0 with some κ q ∈ (0, q], since then it also follows for arbitrary a, b > 0. For fixed a > 0, we consider the function Then we have ℓ(0) = 0 and Since, by Young's inequality, Hence, (2.2) holds for b ≥ a > 0 with κ q = q min{ q−2 q−1 , κ q,1 } ∈ (0, q). The proof is finished.
Next we provide basic but important estimates for the nonlinearity g defined in (1.13) and its primitive G.
Then we have Remark 2.2. The constants C i,p and D i,p , i = 1, 2, are not optimal. However, this choice simplifies the presentation. Moreover, they do not play an important role in our proofs below.
We now distinguish two cases. If p ∈ (1, 2], we have and therefore, if p ∈ (1, 2], by the convexity of the function a → (a + τ) p−1 and therefore, using also (2.1), Note also that, since p > 2, Consequently, if p > 2, On the other hand, since p > 1, we have, by the mean value theorem, . This also implies that H(x, s) ≥ 0 for s ≥ 0. It thus remains to prove (2.6) for s ≥ 0. For this we first note that It therefore remains to show that By (2.7) and integration by parts we have If p > 2, arguing as (2.9), we have and therefore (2.12) yields (2.14) for s > 0. Now (2.11) follows by combining (2.13) and (2.14). The proof is finished.
(a) From the growth estimates given in Lemma 2.2 (i) and the fact that g is continuous, it follows in a standard way that the functional E is well-defined on H 1 0 (R N + ) and of class C 1 . (b) Part (ii) of Lemma 2.2 will be useful in the analysis of Cerami sequences of the functional E, see Section 3 below.
Next, we consider then the quadratic form q c : H 1 0 (R N + ) → R given by . As we show in the following lemma, q c is positive definite on H 1 0 (R N + ). Proposition 2.3. We have (2.17) q c := inf Indeed, if (2.18) holds, then for δ ∈ (0, 1) we have To show (2.18), we first consider the case N = 1. Arguing by contradiction, we assume that ). Then, there exists a sequence (u n ) n such that u n L 2 (R + ) = 1 for all n ∈ N and q c (u n ) → λ as n → ∞. Hence, (u n ) n is a bounded sequence in H 1 0 (R + ), and thus u n ⇀ u * weakly in H 1 0 (R + ) after passing to a subsequence. Moreover, with v n : and therefore It thus follows that v n → 0 in L 2 (R + ) and hence u n → u * in L 2 (R + ), which yields that u * L 2 = 1. Moreover, by weak lower semicontinuity of q c and the definition of λ, it follows that q c (u * ) = λ, so u * is a constrained minimizer for q c . A standard argument (based on replacing u * by |u * |) shows that u * ∈ H 1 0 (R + ) is a positive or negative solution of −u ′′ * + V c (t)u * = λu * in R + , u * (0) = 0.
Without loss of generality, we may assume that u * is positive, which implies that u ′ * (0) > 0. We also recall that Consequently, we have Here we have used the result in the case N = 1 and the fact that u(x ′ , ·) ∈ C ∞ c (R + ) ⊂ H 1 0 (R + ) for every x ′ ∈ R N −1 . We thus have proved (2.18) for general N ≥ 1, and the proof is complete.
Remark 2.4. From Proposition 2.3 it follows that (q c (·)) 1/2 is an equivalent norm to · in H 1 0 (R N + ). Having at hand Proposition 2.3, we prove a lower estimate on the functional E given in (1.15) that will be useful at several points below.
with q c given in (2.17) and C 1,p , C 2,p given in Lemma 2.2 (i).

Mountain-pass geometry and boundedness of the Cerami sequences
This section is devoted to show that the functional E has a Mountain-pass geometry and that, for any d ∈ R, the Cerami sequences for E and level d are bounded. We keep using the notation of the introduction and of Section 2, which depends on the fixed quantities c ∈ (0, c p ) and p ∈ (1, 2 * ). We begin by proving that the functional E has indeed a Mountain-pass geometry.
Lemma 3.1. The functional E has the following properties.
Claim (iii) follows taking t sufficiently large and thus the proof is complete.
We now prove the boundedness of Cerami sequences of the functional E.
Proposition 3.2. Cerami sequences for E at any level d ∈ R are bounded.
The proof of Proposition 3.2 is inspired by [20,Section 3]. However, since our problem is not invariant under translations in R N and our nonlinearity g has a non-standard shape, several difficulties appear.

Proof of Proposition 3.2.
Let d ∈ R be an arbitrary but fixed constant and let (u n ) n ⊂ H 1 0 (R N + ) be a Cerami sequence for E at level d ∈ R. First of all, observe that n → 0, as n → ∞. In particular, we deduce that (u − n ) n is bounded. It then remains to prove that (u + n ) n is bounded. We assume by contradiction that u n → ∞ and we set v n := u n / u n for all n ∈ N. Since (v n ) n and (u − n ) n are bounded, up to a subsequence if necessary, we have with v ≥ 0. We now consider separately two different cases: , v n → 0 in L q (R N + ) for all 2 < q < 2 * , and so, by uniqueness of the limit we have v ≡ 0. We define then the sequence (z n ) n ⊂ H 1 0 (R N + ) by z n := t n u n with t n ∈ [0, 1] satisfying E(z n ) = max (if, for n ∈ N, t n is not unique, we choose the smallest value) and we split the proof in the vanishing case (Case 1) into three steps.
Step where q c > 0 is the constant given by Proposition 2.3. First, observe that (3.4) k n ⇀ 0 in H 1 0 (R N + ), k n → 0 in L q (R N + ), for 2 < q < 2 * , and k n → 0 a.e. in R N + . Then, by Corollary 2.4 and (3.4), we obtain that (1). Taking M bigger if necessary, we have that, for all n ∈ N large enough, On the other hand, observe that, for n ∈ N large enough, 4M 1]. Hence, we have that which is a contradiction. Thus, the Step 1.1 follows.
Step 2.1: E ′ (z n ), z n = 0 for all n ∈ N large enough.

By
Step 1.1 we know that E(z n ) → ∞ as n → ∞. On the other hand, E(0) = 0 and E(u n ) → d as n → ∞. Hence, for n ∈ N large enough, t n ∈ (0, 1) and so, by the definition of z n , the Step 2.1 follows.
Observe that, by Step 2.1, for all n ∈ N large enough, where H is given in Lemma 2.2 (ii). By Step 1.1, we have that On the other hand, since (u n ) n is a Cerami sequence, Then, using the definition of z n and the fact that H(x, s) is non-decreasing in s by Lemma 2.2 (ii), we obtain that which clearly contradicts (3.6). Hence, the proof of the result in the vanishing case (Case 1) is concluded.

Case 2 (Non-vanishing):
We split the proof into two steps.
Step 1.2: There exists M > 0 such that y n N := dist(y n , ∂R N + ) ≤ M for all n ∈ N. We assume by contradiction that y n N → +∞ as n → +∞. Then, for all n ∈ N, we introduce w n := v n (· + y n ) and observe that (3.7) w n ⇀ w in H 1 (R N ), w n → w in L q loc (R N ) for 1 ≤ q < 2 * , and w n → w a.e. in R N , for some w ∈ H 1 (R N ) with w 0 (by (3.3)) and w ≥ 0. Now, observe that, since (u n ) n is a Cerami sequence, Lemma 2.2 (ii) implies that where C > 0 is a constant independent of n. Here we also used Sobolev embeddings and the fact that u n → ∞ as n → ∞. Since p > 1, we thus conclude by Fatou's Lemma that Hence w = w + ≡ 0, which clearly is a contradiction. Thus, Step 1.2 follows.

By
Step 1.2 we know there exists M > 0 such that y n N ≤ M for all n ∈ N. We then define, for all n ∈ N, w n := v n (· + ξ n ), where ξ n = (y n 1 , . . . , y n N −1 , 0). Again by (3.3), we have (3.8) w n ⇀ w in H 1 0 (R N + ), w n → w in L q loc (R N + ) for 1 ≤ q < 2 * , and w n → w a.e. in R N + , for some w ∈ H 1 0 (R N + ) with w 0 and w ≥ 0. For n ∈ N, let ϕ n := w(· − ξ n ) ∈ H 1 0 (R N + ). Since (u n ) n is a Cerami sequence with u n → ∞ as n → ∞, we have On the other hand, since p > 1, we have that Proof. We assume by contradiction that, for all R > 0, Then, by Lions' [22, Lemma I.1], we have that u n → 0 in L q (R N + ) for all 2 < q < 2 * . Now, since (u n ) n is a Cerami sequence, using Lemma 2.2 (i), we get . Hence, since u n → 0 in L q (R N + ) for all 2 < q < 2 * , we deduce that u n → 0. Since E is continuous, this implies that E(u n ) → 0 as n → ∞, contradicting our assumption that d 0. The proof is finished.

Energy estimates
We keep using the notation of the introduction and of Section 2, which depends on the fixed quantities c ∈ (0, c p ) and p ∈ (1, 2 * ). Moreover, we will assume N ≥ 2 throughout this section, which will be of key importance in order to derive the energy estimates we need. The mountain pass value associated to (1.14) is given by (1)) < 0 . We note that b > 0 by Lemma 3.1. We also note that the functional E (given in (1.15)) can be written as Now, we introduce the auxiliary (limit) problem and its associated energy E ∞ : H 1 (R N ) → R given by Also, we define According to [8,Theorem 1], [6, Théorème 1] and [16,Theorem 2], there exists a ground-state solution ψ ∈ C 2 (R N ) to (4.3) which is positive, radially symmetric, and such that for some C GS > 0 depending only on N and p. Moreover, ψ is strictly decreasing in the radial variable.
Let us also emphasize that The aim of this section is to show, based on the assumption N ≥ 2, that This strict inequality will be crucial to prove the existence result to (1.14) contained in Section 5. To this end, let us recall that u c (x) ∼ e −x N as x N → ∞. More precisely, it follows from (1.6) and the definition of u c that Moreover, for r > 0, we introduce the function where e N = (0, . . . , 0, 1) is the N -th coordinate vector and ε r > 0 is uniquely defined by (4.7) and the property that ψ > ε r in B r (0) and ψ ≤ ε r in R N \ B r (0). We note that, as a consequence of (4.6), we have We also note that ψ r ∈ H 1 0 (R N + ) for every r ≥ 0. The rest of the section is devoted to prove the following result from which (4.9) immediately follows.
Proposition 4.1. There exists R > 0 and k > 0 with the following properties: We split the proof of this proposition into several lemmas.
Proof. Let t, r > 0. Directly observe that, by the definition of ψ r , On the other hand, since ψ is a solution to (4.3) and (ψ − ε r ) + ∈ H 1 (R N ), using (ψ − ε r ) + as test function in (4.3), we obtain that Substituting the above identity into (4.13) and using the mean value theorem, we find that Using (4.8) and (4.11), we deduce that Hence (4.12) holds with C 1 = pC GS p+1 R N ψ p dx.
Lemma 4.3. There exists R ′ ≥ 1 and C 2 > 0 with (4.14) Proof. Let t > 0 be arbitrary but fixed. Using Lemma 2.1 with q = p + 1, κ := κ q > 0, the identity (4.2), the lower bound in (4.10) and the mean value theorem, we deduce that, for all r ≥ 1, Since N ≥ 2, we may choose R ′ ≥ 1 sufficiently large to guarantee that and therefore Hence the claim follows. Proof. Let k > 0. For r ≥ R ′ ≥ 1 we have, by Lemma 4.3 and since the map r → ε r is strictly decreasing by (4.7), Since N ≥ 2, we may fix R ≥ R ′ with the property that kC 1 r − N −1 2 ≤ C 2 2 for r ≥ R, which implies that Since also E(0) = 0 < b ∞ , we thus obtain that E(tψ r ) < b ∞ for t ∈ [0, k], r ≥ R. Moreover, by Lemma 4.4 we have E(kψ r ) < 0 for all r ≥ R since R ≥ R ′ . The proof is finished.

The existence result
We keep using the notation of the introduction and of Section 2, which depends on the fixed quantities c ∈ (0, c p ) and p ∈ (1, 2 * ). Moreover, we will assume N ≥ 2 throughout this section, which will allow us to prove the existence of a non-trivial solution to (1.14). This will conclude the proof of Theorem 1.3.
Proof. Since the functional E has a mountain pass geometry (see Lemma 3.1), there exists a Cerami sequence for E at the corresponding mountain pass level b defined in (4.1) (see e.g. [10] or [13, Theorem 6, Section 1, Chapter IV]), i.e. there exists (u n ) n ⊂ H 1 0 (R N + ) such that By Proposition 3.2 we know that (u n ) n is bounded in H 1 0 (R N + ). Moreover, Let (y n ) n ⊂ R N + be the sequence of points obtained in Lemma 3.3 applied to (u n ) n , i.e., we have We split the argument into two steps.
Step 1: There exists M > 0 such that y n N = dist(y n , ∂R N + ) ≤ M for all n ∈ N.
We assume by contradiction that Then, let us define, for all n ∈ N, w n := u n (· + y n ). By Lemma 3.3 and (5.1), it follows that (5.4) w n ⇀ w in H 1 (R N ), w n → w in L q loc (R N ) for 1 ≤ q < 2 * , and w n → w a.e. in R N , for some w ∈ H 1 (R N ) with w ≥ 0, w 0. We also observe that H(x + y n , w + n (x))dx, as n → ∞, with the function H defined in Lemma 2.2 (ii). Next, we note that by Lemma 2.2 (ii) and (4.10). Thus, (5.5) and Fatou's Lemma imply that Next we claim that w ∈ K ∞ , i.e., w is a nontrivial solution of (4.3). To see this, we fix an arbitrary ϕ ∈ C ∞ c (R N ), and we show that Since (5.3) holds, we have that supp(ϕ) ⊂ {x N ≥ −y n N } for n ∈ N sufficiently large. Hence, for n ∈ N large enough, we have that Hence (5.7) follows, and therefore w ∈ K ∞ . Together with (4.8) and (5.6) it then follows that b ≥ b ∞ , but this contradicts (4.9). Hence, (5.3) cannot happen and Step 1 follows.
Let us define, for all n ∈ N, v n := u n (· + ξ n ) with ξ n := (y n 1 , . . . , y n N −1 , 0) and observe that, after passing to a subsequence Hence, if v 0, we will have that v is a non-trivial solution to (1.14). Since v n → v in L q loc (R N + ) and y n N ≤ M for all n ∈ N, the lower integral bound (5.2) implies that v 0, and the result follows.
We now prove Proposition 1.5. Let us first state a technical lemma due to Poláčik, Quittner and Souplet that will be key to prove this result.
The following proof is inspired by [ is the unique even non-trivial positive solution to (1.5). Throughout this section, we will use the following notation. We define v 0 : R N → R as Also, recall that for a bounded positive solution to (1.3)-(1.4), we mean a function v ∈ C 2 (R N Proof. Let us fix an arbitrary c > c p . We assume by contradiction that there exists a bounded positive solution v to (1.3)-(1.4) and we define, for all t ∈ R, v t := v 0 (· + te N ) where v 0 is given in (6.1) and e N = (0, . . . , 0, 1) is the N -th coordinate vector. We split the proof into three steps.
Step 1: First of all, fixed an arbitrary Hence, there exists t 0 > 0 such that, for all t ≥ t 0 , We fix t 0 > 0 such that (6.2) holds and we are going to prove the Step 1 for this t 0 . To that end, we fix an arbitrary t ≥ t 0 > 0. First, we are going to prove that v ≥ v t in R N + . Since c > c p ≥ max x∈R v t (x), we have that We assume by contradiction that Then, using the mean value theorem and (6.2), we deduce that, for all x ∈ {x ∈ R N + : v(x) < v t (x)}, Hence, in each connected component D of {x ∈ R N + : v(x) < v t (x)} we have that with c t satisfying (6.7). Then, applying the weak maximum principle [5, Lemma 2.1], we obtain that v ≥ v t in D which contradicts the fact that D ⊂ {x ∈ R N + : v(x) < v t (x)}. Hence, we conclude that {x ∈ R N + : v(x) < v t (x)} = ∅ and so, that v ≥ v t in R N + . Having this at hand and substituting in (6.3), we deduce that (6.9) on ∂R N + , and so, the Step 1 follows from the strong maximum principle and the fact that t ≥ t 0 is arbitrary.
Step 2: v > v t in R N + for all t ∈ R. Note that, if we prove that v ≥ v t in R N + for all t ∈ R, then the claim follows from the Strong Maximum principle. Also, by the Step 1, we know that Hence, we can define (6.10) We argue by contradiction and suppose that t ⋆ > −∞. First note that, by continuity, v ≥ v t ⋆ in R N + . Also, t ⋆ > −∞ implies the existence of M > 0 such that We now consider separately two cases.
Next, we are going to prove that, for all t ∈ [t ⋆ − η 0 , t ⋆ ], it follows v ≥ v t in Σ M . To that end, we fix an arbitrary t ∈ [t ⋆ − η 0 , t ⋆ ]. Arguing as in Step 1, we have that where (6.14) c t (x) := We assume by contradiction that (6.15) x ∈ Σ M : v(x) < v t (x) ∅.
Then, using the mean value theorem and (6.11), we deduce that, for all x ∈ {x ∈ Σ M : v(x) < v t (x)}, Hence, in each connected component D of {x ∈ Σ M : v(x) < v t (x)} we have that with c t satisfying (6.16). Then, applying the weak maximum principle [5, Lemma 2.1], we obtain that v ≥ v t in D which contradicts the fact that D ⊂ {x ∈ Σ M : v(x) < v t (x)}. Hence, we conclude that {x ∈ Σ M : v(x) < v t (x)} = ∅ and so, that v ≥ v t in Σ M . Taking into account (6.12), we infer that, for all η ∈ [0, η 0 ], v ≥ v t ⋆ −η in R N + . This is in contradiction with the definition of t ⋆ . Hence, Case 1 cannot happen.

Case 2:
inf In this case there exists a sequence of points (x n ) n ⊂ R N −1 × [0, M] such that Up to a subsequence, it follows that x n N → x N for some x N ∈ [0, M]. We define then v n (x) = v x ′ + (x n ) ′ , x N , for all n ∈ N, and, for all n ∈ N, we have v n ≥ v t ⋆ in R N + and Moreover, for all n ∈ N, it follows that and so, by the Strong Maximum principle, we have that v n − v t ⋆ > 0, in R N + , for all n ∈ N.
Now, arguing as in [15, Proof of Theorem 2.1, Step 1], we deduce that the sequence (v n ) n admits a subsequence, still denoted by (v n ) n , converging to a function v in C 2 loc (R N + ). This function v still solves and satisfies v ≥ v t ⋆ , in R N + , and Note that (6.19) and Hence, by the Strong Maximum principle, it follows that v > v t ⋆ in R N + which gives a contradiction with (6.19). Case 2 cannot happen either and hence the Step 2 follows.
Observe that v > v t in R N + for all t ∈ R implies that v ≥ v 0 (0) = c p in R N + . This gives a contradiction with (1.4) and so the proof is complete.