Fractional (p, q)-Schrödinger Equations with Critical and Supercritical Growth

In this paper, we complete the study started in Ambrosio and Rădulescu (J Math Pures Appl (9) 142:101–145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrödinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional (p, q)-Laplacian problems: (-Δ)psu+(-Δ)qsu+V(εx)(up-1+uq-1)=f(u)+uqs∗-1inRN,u∈Ws,p(RN)∩Ws,q(RN),u>0inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p}u+(-\Delta )^{s}_{q}u + V(\varepsilon x) (u^{p-1} + u^{q-1})= f(u)+u^{q^{*}_{s}-1} \, \text{ in } \mathbb {R}^{N}, \\ u\in W^{s, p}(\mathbb {R}^{N})\cap W^{s,q}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$\end{document}where ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is a small parameter, s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0, 1)$$\end{document}, 1<p<q<Ns\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q<\frac{N}{s}$$\end{document}, qs∗=NqN-sq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{*}_{s}=\frac{Nq}{N-sq}$$\end{document} is the fractional critical Sobolev exponent, (-Δ)rs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{s}_{r}$$\end{document}, with r∈{p,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\in \{p, q\}$$\end{document}, is the fractional r-Laplacian operator, V:RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:\mathbb {R}^{N}\rightarrow \mathbb {R}$$\end{document} is a positive continuous potential such that inf∂ΛV>infΛV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf _{\partial \Lambda }V>\inf _{\Lambda } V$$\end{document} for some bounded open set Λ⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \mathbb {R}^{N}$$\end{document}, and f:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\mathbb {R}\rightarrow \mathbb {R}$$\end{document} is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik–Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when f(t)=tγ-1+μtτ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(t)=t^{\gamma -1}+\mu t^{\tau -1}$$\end{document}, with 1<p<q<γ<qs∗<τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< p<q<\gamma<q^{*}_{s}<\tau $$\end{document} and μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0$$\end{document} sufficiently small, by combining a truncation argument with a Moser-type iteration.


Introduction
In this paper, we investigate the multiplicity and concentration phenomenon for the following class of fractional ( p, q)-Laplacian problems: where ε > 0 is a small parameter, s ∈ (0, 1), 1 < p < q < N s , q * s = Nq N −sq is the fractional critical Sobolev exponent, V : R N → R and f : R → R are continuous functions. The leading operator (− ) s r , with r ∈ {p, q}, is the fractional r -Laplacian operator which may be defined, up to a normalization constant, by setting for any u ∈ C ∞ c (R N ). When s = 1, equation in (1.1) is closely connected to the study of stationary solutions of general reaction diffusion systems of the form (1.2) Equation (1.2) has a wide range of applications in physical and related sciences, e.g. in biophysics, plasma physics, and chemical reaction design; see [13]. In such contexts, the function u in (1.2) describes a concentration, div[D(u)∇u] corresponds to the diffusion with diffusion coefficient D(u), and c(x, u) is related to sources and loss processes. Typically, in chemical and biological applications, the reaction term c(x, u) is a polynomial of u with variable coefficients. We refer to [1,20,24,29,31] for some engaging existence and multiplicity results for ( p, q)-Laplacian problems in bounded or unbounded domains. In the last years, the study of nonlocal elliptic problems driven by fractional operators has gained a tremendous popularity both for their interesting theoretical structure and in view of concrete applications, such as, obstacle problem, optimization, finance, crystal dislocations, phase transitions, conservation laws, ultra-relativistic limits of quantum mechanics, quasi geostrophic flows and so on; see [18] for more details.
When s ∈ (0, 1) and p = q = 2, after rescaling, equation (1.1) boils down to the following nonlinear fractional Schrödinger equation with γ ∈ {0, 1}, for which several existence, multiplicity and concentration results have been established via different variational and topological methods. Since we cannot review the huge bibliography on this topic, we refer to the monograph [3] and the references therein.
In the case s ∈ (0, 1) and p = q = 2 in (1.1), we derive the following fractional p-Laplacian equation We point out that when p = 2, we have to tackle not only the usual nonlocal character of (− ) s p , but also the difficulties given by the corresponding nonlinear behavior. In particular, standard arguments used in the linear case p = 2 are not so easy to adapt in the case p = 2 due to the non-Hilbertian structure of W s, p (R N ) with p = 2. For these reasons, there has been a source of interest around nonlocal and fractional problems driven by the fractional p-Laplacian operator; see for instance [6,16,27,30] and the references therein.
On the other hand, in recent years, some existence, multiplicity and regularity results for fractional ( p, q)-Laplacian problems appeared in the literature; see [4,7,8,10,15,22,23]. Indeed, such problems involve the sum of two nonlocal nonlinear operators with different scaling properties and so some nontrivial additional technical difficulties arise. For what concerns problems like (1.1), in [8] the authors obtained a multiplicity result for a fractional subcritical ( p, q)-Laplacian problem by combining a penalization approach and the Ljusternik-Schnirelmann category theory. We also mention [7] where a multiplicity result was established by requiring that the potential V satisfies the global condition proposed by Rabinowitz [33]: Particularly motivated by [1,7,8], in the first part of this paper we are interested in the multiplicity and concentration behavior as ε → 0 of positive solutions to the fractional critical problem (1.1). Next we introduce the assumptions on the potential V and the nonlinearity f . Throughout the paper we will assume that V ∈ C(R N , R) ∩ L ∞ (R N ) satisfies the following conditions: there exists an open bounded set ⊂ R N such that and f ∈ C(R, R) fulfills the following hypotheses: there exist σ 1 , σ 2 ∈ (q, q * s ) and λ > 1 such that ( f 4 ) the map t ∈ (0, ∞) → f (t) t q−1 is increasing. Since we look for positive solutions to (1.1), we assume that f (t) = 0 for t ≤ 0. The first main result of this work can be stated as follows. Theorem 1.1 Assume that (V 1 )- (V 2 ) and ( f 1 )-( f 4 ) hold. Then there exists λ * > 0 such that, for any λ > λ * and for any δ > 0 such that there exists ε δ,λ > 0 such that, for any ε ∈ (0, ε δ,λ ), problem (1.1) has at least cat M δ (M) positive solutions. Moreover, if u ε denotes one of these solutions and x ε ∈ R N is a global maximum point of u ε , then The proof of Theorem 1.1 will be carried out by using variational and topological arguments. First, we modify in a suitable way the nonlinearity outside of the set and we handle an auxiliary problem. The main feature of the corresponding modified energy functional J ε is that it satisfies all the geometric assumptions of the mountainpass theorem [2]. Differently from the subcritical cases examined in [7,8], a more accurate analysis will be needed to recover compactness. To overcome the lack of compactness caused by the critical exponent, we don't construct cut-off functions of the extremal functions for the best constant in the Sobolev inequality, as for the equations (1.3) and (1.4) studied in [3,6], to control the mountain-pass level. Indeed, the nonhomogeneity of the fractional ( p, q)-Laplacian operator does not permit to develop this argument. For this reason, we employ a different strategy by considering the solution of an appropriate fractional ( p, q)-Laplacian problem in a bounded domain with nonlocal Dirichlet condition; see Lemma 3.3. After that, by invoking the variant of concentration-compactness principle of Lions [25,26] established in [4], we will prove that the energy functional J ε verifies a local Palais-Smale condition; see Lemma 3.5.
To accomplish multiple solutions for the modified problem, we use the technique due to Benci and Cerami [9] based on precise comparisons between the category of some sublevel sets of J ε and the category of the set M. Note that f is only continuous, so standard Nehari manifold arguments for C 1 functionals are not applicable. The nondifferentiability of the Nehari manifold associated with J ε will be overcame through the generalized Nehari manifold method by Szulkin and Weth [34]. Finally, we prove that the solutions u ε 's of the modified problem are also solutions of (1.1) for ε > 0 small, by exploiting a Moser iteration argument [28] and the Hölder regularity result in [8].
In the second part of this paper, we treat the following supercritical fractional problem: where ε, μ > 0 and 1 < p < q < γ < q * s < τ. Our multiplicity result for the supercritical case reads as follows.
Then there exists μ 0 > 0 such that, for any μ ∈ (0, μ 0 ) and for any δ > 0 satisfying there exists ε δ,μ > 0 such that, for any ε ∈ (0, ε δ,μ ), problem (1.5) has at least cat M δ (M) positive solutions. Moreover, if u ε denotes one of these solutions and x ε ∈ R N is a global maximum point of u ε , then The main difficulty in the study of (1.5) consists in the fact that τ > q * s is supercritical, and we cannot directly apply variational techniques because the corresponding energy functional is not well-defined on the space W s, p (R N ) ∩ W s,q (R N ). In order to circumvent this obstacle, we perform some arguments inspired by [12,21,32] which can be summarized as follows. We first truncate in a suitable way the nonlinearity on the right hand side of (1.5), so we deal with a new problem but with subcritical growth. In light of Theorem 1.1 in [8], we know that a multiplicity result for this truncated problem is available. Then we prove a priori bound (independent of μ) for these solutions, and by means of an appropriate Moser type iteration [28] we show that, for μ > 0 sufficiently small, the solutions of the truncated problem also solve the original one.
We stress that our theorems complement and improve the main results in [1,7,8], in the sense that we are considering multiplicity results for critical and supercritical problems involving continuous nonlinearities and imposing local conditions on the potential V .
The paper is organized as follows. In Sect. 2 we collect some notations and technical results. In Sect. 3 we introduce the modified problem. In Sect. 4 we focus our attention on the limiting problem associated with (1.1). In Sect. 5 we obtain a multiplicity result for the modified problem. Section 6 is devoted to the proof of Theorem 1.1. In the last section we investigate the supercritical problem (1.5).

Preliminary Results
Let p ∈ [1, ∞] and A ⊂ R N be a measurable set. We will use the notation | · | L p (A) to denote the L p (A)-norm, and | · | p when A = R N . Let s ∈ (0, 1), p ∈ (1, ∞) and N > sp. We define D s, p (R N ) as the closure of C ∞ c (R N ) with respect to The following embeddings are well-known. For the reader's convenience, we also recall the following vanishing lemma.
Let s ∈ (0, 1), p, q ∈ (1, ∞) and consider the space endowed with the norm Note that W is a separable reflexive Banach space (since W s,r (R N ), with r ∈ (1, ∞), is a separable reflexive Banach space). For any ε > 0, we introduce the space Due to the presence of the critical exponent in (1.1), the following variant of the concentration-compactness lemma of Lions [25,26] proved in [4] will be crucial. In what follows, we use the notation Let us assume that in the sense of measure, where μ and ν are two non-negative bounded measures on R N . Then, there exist an at most a countable set I , a family of distinct points

Variational Framework and Modified Problem
As in [8], we use a del Pino-Felmer penalization type approach [17] to deal with problem (1.1). Take and a > 0 such that We definef and where χ A denotes the characteristic function of A ⊂ R N . By ( f 1 )-( f 4 ), we deduce that g : R N × R → R is a Carathéodory function satisfying the following conditions: . Let us introduce the following auxiliary problem: Notice that if u ε is a solution to (3.1) such that u ε (x) ≤ a for all x ∈ c ε , where ε = {x ∈ R N : εx ∈ }, then u ε is also a solution to (1.1). Then we consider the functional J ε : X ε → R associated with (3.1), that is Obviously, J ε ∈ C 1 (X ε , R) and it holds for any u, ϕ ∈ X ε . We define the Nehari manifold N ε associated with J ε , i.e.
and we set Let X + ε be the open set given by The next lemma ensures that J ε possesses a mountain pass geometry [2].

Lemma 3.1
The functional J ε has the following properties: There exists e ∈ X ε such that e X ε > ρ and J ε (e) < 0. Therefore, Taking u X ε = ρ ∈ (0, 1) and using 1 < p < q, we have u V ε , p < 1 and thus u and invoking Theorem 2.1, we deduce that Then we can find α > 0 such that J ε (u) ≥ α for all u ∈ X ε such that u X ε = ρ.
Thus, for all u ∈ X + ε and t > 0, we have The next two results are very useful since they allow us to overcome the nondifferentiability of N ε and the incompleteness of S + ε . Lemma 3.2 The following properties hold: (ii) There exists τ > 0, independent of u, such that t u ≥ τ for any u ∈ S + ε . Moreover, for each compact set Proof (i) From the proof of Lemma 3.1, we see that h u (0) = 0, h u (t) > 0 for t > 0 small enough and h u (t) < 0 for t > 0 sufficiently large. Then there exists a global maximum point t u > 0 for h u in [0, ∞) such that h u (t u ) = 0 and t u u ∈ N ε . We claim that t u > 0 is the unique number such that h u (t u ) = 0. Let t 1 , t 2 > 0 be such that h u (t 1 ) = h u (t 2 ) = 0, or equivalently Hence, Accordingly, From (g 4 ) and q > p, we obtain that t 1 = t 2 .
(ii) Let u ∈ S + ε . By (i), we can find t u > 0 such that h u (t u ) = 0, that is Fix ξ > 0. From (g 2 ), ( f 1 ), ( f 2 ) and Theorem 2.1, we have that Taking ξ > 0 sufficiently small and recalling that 1 = u X ε ≥ u V ε ,q , we get , and using the fact ε be a compact set, and assume by contradiction that there exists a sequence {u n } n∈N ⊂ K such that t n = t u n → ∞. Then there exists u ∈ K such that Taking v n = t u n u n ∈ N ε in the above inequality, we find which is impossible because K > 1 and u = 0. Therefore, m −1 ε (u) = u u Xε ∈ S + ε is well-defined and continuous. Since, we infer that m ε is a bijection. To prove thatm ε : and letting n → ∞ we obtain Thus,m ε and m ε are continuous maps.
(iv) Let {u n } n∈N ⊂ S + ε be a sequence such that dist(u n , ∂S + ε ) → 0. Then, for each v ∈ ∂S + ε and n ∈ N, we have u + n ≤ |u n − v| a.e. in ε . Therefore, by (V 1 ), (V 2 ) and Theorem 2.1, we can see that for each r ∈ [p, q * s ] there exists C r > 0 such that Consequently, Now, we note that K > q p > 1, and From the definition of m ε (u n ) and employing (3.6), (3.7), we obtain

Now we define the mapŝ
by settingψ ε (u) = J ε (m ε (u)) and ψ ε =ψ ε | S + ε . From Lemma 3.2 and arguing as in the proofs of Proposition 9 and Corollary 10 in [34], we may obtain the following result.

Proposition 3.1 The following properties hold:
the corresponding critical values coincide and

Remark 3.2 As in [34]
, we have the following variational characterization of the infimum of J ε over N ε : then we can argue as in [17,33,35] to verify that c ε = c ε .
Next we establish a very useful upper bound for the minimax level c ε . To accomplish this, we require that the parameter λ > 0 appearing in ( f 2 ) is sufficiently large.
Proof For simplicity, we take ε = 1. Define V ∞ = max V . Consider the following problem Define the Nehari manifold and c ∞ = inf N ∞ J ∞ . Let us prove that there exists a non-negative function We first show that, for all u ∈ W s, p The uniqueness of a such t u is a consequence of the fact that the map [11,18]), we have that, for some C > 0 depending only on N , s, q, σ 1 and | |, and using the fact that σ 1 > q we get the desired estimate. In a similar way, and then we can find k 1 > 0 such that [u] s, p ≥ k 1 > 0 for all u ∈ N ∞ . We can also see that, for some Let us now {u n } n∈N be a minimizing sequence for J ∞ in N ∞ . Note that the above discussion guarantees that {u n } n∈N is bounded in W s, p for all r ∈ [1, q * s ) and u n → u a.e. in R N . Observe that |supp(u + ) ∩ | > 0. Indeed, by (3.9), we also see that Taking Since t ∈ (0, 1), p < q, and J ∞ (w σ 1 ), w σ 1 = 0, we find Using (W 2) and the fact that Taking λ > λ * in the hypothesis ( f 2 ) we get the thesis.
The main feature of the modified functional is that it satisfies a compactness condition. We start by proving the boundedness of Palais-Smale sequences.
Proof In light of (g 3 ) and ϑ > q > p, we see that whereC > 0 since K > 1 and ϑ > q. Now, assume by contradiction that u n X ε → ∞. We distinguish the following cases: (1) u n V ε , p → ∞ and u n V ε ,q → ∞; (2) u n V ε , p → ∞ and u n V ε ,q is bounded; (3) u n V ε ,q → ∞ and u n V ε , p is bounded.
In case (1), for n large, we have u n q− p V ε ,q ≥ 1, that is u n q V ε ,q ≥ u n p V ε ,q . Therefore, which gives a contradiction. In case (2), we have and thus Since p > 1 and letting n → ∞, we obtain 0 <C ≤ 0 which is impossible. The last case is similar to the case (2), so we skip the details. In conclusion, {u n } n∈N is bounded in X ε .

Lemma 3.5 J ε satisfies the (P S) c condition at any level c
Proof Let {u n } n∈N ⊂ X ε be a (P S) c sequence for J ε . From Lemma 3.4, going to a subsequence if necessary, we may assume that u n u in X ε and u n → u in L r loc (R N ) for all r ∈ [1, q * ). It is standard to verify that the weak limit u is a critical point of J ε . Indeed, taking into account that for all φ ∈ C ∞ c (R N ) and that J ε (u n ), φ = o n (1), we see that J ε (u), φ = 0 for any φ ∈ C ∞ c (R N ). By the density of C ∞ c (R N ) in X ε , we obtain that u is a critical point of J ε . In particular, J ε (u), u = 0. Now, we claim that for any η > 0 there exists R = R(η) > 0 such that (3.11) Take R > 0 such that ε ⊂ B R 2 (0). By the definition of ψ R and (g 3 )-(ii), we have that (3.12) Now, from the Hölder inequality and the boundedness of {u n } n∈N in X ε , we derive, for t ∈ {p, q}, Using the definition of ψ R , polar coordinates and the boundedness of {u n } n∈N in X ε , we can see that (3.14) In light of (3.14) and (3.13), we find Combining (3.12), (3.13) and (3.15), we get from which we deduce that (3.10) holds. Next we show that (3.10) is useful to infer that u n → u in L r (R N ) for any r ∈ [p, q * s ). Fixed η > 0 we can find R = R(η) > 0 such that (3.10) is valid. Recalling the compact embedding X ε L p loc (R N ) in Theorem 2.1, we deduce that The arbitrariness of η implies the strong convergence in L p -norm. By interpolation, we can see that u n → u in L r (R N ) for any r ∈ [p, q * s ). In order to establish the strong convergence in X ε , we prove that (3.17) Note that the Sobolev inequality in Theorem 2.1, 0 ≤ ψ R ≤ 1, (3.16) and (3.14) yield Clearly, the strong convergence in L r (R N ) for all r ∈ [p, q * s ) gives Then, using the growth assumption on g, (3.18) and (3.19), for all η > 0 there exists (3.20) On the other hand, choosing R > 0 large enough, we may assume that It follows from the definition of g that Since B R (0) ∩ c ε is bounded, we can use the above estimate, ( f 1 ), ( f 2 ), Theorem 2.1 and the dominated convergence theorem to infer that (3.27) It suffices to prove that {x i } i∈I ∩ ε = ∅. Assume, by contradiction, that x i ∈ ε for some i ∈ I . For ρ > 0, define and |∇ζ | ∞ ≤ 2. We suppose that ρ is chosen in such way that the support of ζ ρ is contained in ε . Since {ζ ρ u n } n∈N is bounded, J ε (u n ), u n ζ ρ = o n (1) and thus Since f has subcritical growth and ζ ρ has compact support, we have Now, we verify that, for t ∈ {p, q}, The Hölder inequality yields  (1).
From this and (3.27), we obtain that Proof Let {u n } n∈N ⊂ S + ε be a (P S) c sequence for ψ ε . Therefore, Thus, by Lemma 3.5, we deduce that J ε satisfies the (P S) c condition in X ε and thus there exists u ∈ S + ε such that, up to a subsequence, Using Lemma 3.2-(iii), we obtain that u n → u in S + ε .
We conclude this section by proving an existence result for (3.1).
Proof In light of Lemma 3.1, Remark 3.2 and Lemma 3.5, we can apply the mountain pass theorem [2] to deduce that, for all ε > 0, there exists a nontrivial critical point Arguing as in the proof of Lemma 4.1 in [8] (when p < 2, we use Theorem 2.2 in [22] instead of Corollary 2.1 in [8]), we have that u ε ∈ L r (R N ) ∩ C 0,α (R N ) for all r ∈ [p, ∞] and u ε (x) → 0 as |x| → ∞. From the strong maximum principle [5], we derive that u ε > 0 in R N .

The Autonomous Problem
Since we are interested in giving a multiplicity result for the auxiliary problem (3.1), we consider the limiting problem associated with (1.1), namely and It is standard to check that and for any u, ϕ ∈ Y V 0 . We also consider the Nehari manifold M V 0 associated with E V 0 , that is Arguing as in Sect. 3, we can deduce the following results.
(ii) There exists τ > 0, independent of u, such that t u ≥ τ for any u ∈ S + V 0 . Moreover, for each compact set Let us consider the mapŝ

Proposition 4.1
The following properties hold:

the corresponding critical values coincide and
Moreover, arguing as in the proof of Lemma 3.3, we see that 0 The next lemma allows us to assume that the weak limit of a (P S) d V 0 sequence of E V 0 is nontrivial.
In particular, using ( f 1 ) and ( f 2 ), we see that Up to a subsequence, there exists ≥ 0 such that Assume by contradiction that > 0. Since , u n = 0 and q > p, we deduce that which implies that By Theorem 2.1, we see that and letting n → ∞ we obtain and this contradicts Remark 4.1.

Remark 4.2
As it has been mentioned earlier, if u is the weak limit of a (P S) d V 0 sequence for E V 0 , then we can assume u = 0. Otherwise, if u n 0 and u n 0 in Y V 0 , it follows from Lemma 4.2 that there are {y n } n∈N ⊂ R N and R, β > 0 such that Set v n (x) = u n (x + y n ). Then, using the invariance of R N by translation, we see that In the next result we obtain a positive ground state solution for the autonomous problem (4.1).

Theorem 4.1 Problem (4.1) admits a positive ground state solution.
Proof Invoking a variant of the mountain-pass theorem without the Palais-Smale condition (see [35]), there exists a (P S) d V 0 sequence {u n } n∈N ⊂ Y V 0 for E V 0 . Arguing as in the proof of Lemma 3.5, we can verify that {u n } n∈N is bounded in Y V 0 and so, passing to a subsequence if necessary, we may assume that Standard arguments (see proof of Lemma 3.5) show that E V 0 (u) = 0. From Remark 4.2, we may assume that u = 0. On the other hand, by Fatou's lemma and ( f 3 ), we deduce that which gives u − = 0, that is u ≥ 0 in R N . Therefore, u ≥ 0 and u ≡ 0 in R N . Arguing as in the proof of Lemma 4.1 in [8], we see that u ∈ L r (R N ) ∩ C 0,α (R N ) for all r ∈ [p, ∞] and |u(x)| → 0 as |x| → ∞. By the strong maximum principle [5], we arrive at u > 0 in R N .
The next lemma is a useful compactness result for the autonomous problem (4.1).

Lemma 4.3 Let {u
and Let us define G : S Note that • G is bounded below, by Proposition 4.1-(d).
Hence, applying the Ekeland variational principle [19] (1). Now the remainder of the proof follows from Proposition 4.1, Theorem 4.1, and arguing as in the proof of Corollary 3.1.
Finally we prove a useful relation between the minimax levels c ε and d V 0 .

Multiplicity of Solutions to (3.1)
This section is devoted to provide some technical results needed to implement the barycenter machinery below. Let δ > 0 be such that and w ∈ Y V 0 be a positive ground state solution to the autonomous problem (4.1) (whose existence is ensured by where t ε > 0 satisfies max t≥0 J ε (t ε,y ) = J ε (t ε ε,y ).
By construction, ε (y) has compact support for any y ∈ M.

Lemma 5.1
The functional ε verifies the following limit: Proof Assume by contradiction that there exist δ 0 > 0, {y n } n∈N ⊂ M and ε n → 0 such that (we recall that w is continuous and positive in R N ). If t ε n → ∞, using the fact that q > p and that the dominated convergence theorem yields ε n ,y n V εn ,r → w s,r ∈ (0, ∞) for all r ∈ {p, q}, (5.6) we find On the other hand, by ( f 3 ), we get lim n→∞ f (t ε n w(ẑ)) (t ε n w(ẑ)) q−1 = ∞. (5.8) In light of (5.5), (5.7), (5.8) and that q * s > q, we achieve a contradiction. Consequently, {t ε n } n∈N is bounded and, up to a subsequence, we may suppose that t ε n → t 0 for some t 0 ≥ 0. From (5.4), (5.6), ( f 1 )-( f 2 ), we can see that t 0 > 0. Now we claim that t 0 = 1. Letting n → ∞ in (5.4), and using (5.6) and the dominated convergence theorem, we have that Since w ∈ M V 0 , it holds Then we get which together with ( f 4 ) implies that t 0 = 1. Therefore, letting n → ∞ in (5.3), we deduce that which contradicts (5.2).
The next compactness result plays an important role to verify that the solutions of the modified problem are also solutions of the original one.
Proof Arguing as in the proof of Lemma 3.4, it is easy to check that {u n } n∈N is bounded in Y V 0 . Clearly, u n X εn 0 since d V 0 > 0. Consequently, proceeding as in the proof of Lemma 4.2 and Remark 4.2, we obtain a sequence {ỹ n } n∈N ⊂ R N and constants R, β > 0 such that Put v n (x) = u n (x +ỹ n ). Then, {v n } n∈N is bounded in Y V 0 , and, going to a subsequence if necessary, we may assume that v n v ≡ 0 in Y V 0 . Take t n > 0 such thatṽ n = t n v n ∈ M V 0 and set y n = ε nỹn . From u n ∈ N ε n and (g 2 ), we get which implies that Moreover, {ṽ n } n∈N is bounded in Y V 0 and thus, up to a subsequence,ṽ n ṽ in Y V 0 .
Using standard arguments, we may assume that t n → t 0 > 0. From the uniqueness of the weak limit, we haveṽ = t 0 v ≡ 0. By Lemma 4.3, we getṽ n →ṽ in Y V 0 , and so v n → v in Y V 0 . Moreover, Next we show that {y n } n∈N admits a bounded subsequence. Otherwise, assume that there is a subsequence of {y n } n∈N , still denoted by itself, such that |y n | → ∞. Choose R > 0 such that ⊂ B R (0). Then, for n large enough, |y n | > 2R, and for each x ∈ B R/ε n (0) we have |ε n x + y n | ≥ |y n | − |ε n x| > R.
Using v n → v in Y V 0 , the definition of g, and the dominated convergence theorem, Since v n → v ≡ 0 and K > 1, we reach a contradiction. Hence, {y n } n∈N is bounded in R N and, up to a subsequence, we can assume that y n → y 0 . If y 0 / ∈ , we can proceed as above to get v n → 0 in Y V 0 . Then, y ∈ . Now, suppose by contradiction that V (y 0 ) > V 0 . Fromṽ n →ṽ in Y V 0 , Fatou's lemma and the invariance of R N by translations, we have which gives a contradiction. Therefore, V (y 0 ) = V 0 and y 0 ∈ . The assumption (V 2 ) ensures that y 0 / ∈ ∂ and thus y 0 ∈ M. Let us define where π(ε) = sup y∈M |J ε ( ε (y)) − d V 0 |. By Lemma 5.1, we know that π(ε) → 0 as ε → 0. By the definition of π(ε), we have that, for all y ∈ M and ε > 0, ε (y) ∈ N ε and thus N ε = ∅. Now we prove an interesting relation between N ε and the barycenter map.
From E V 0 (tu n ) ≤ J ε n (tu n ) for all t ≥ 0, and {u n } n∈N ⊂ N ε n ⊂ N ε n , we obtain which leads to J ε n (u n ) → d V 0 . By invoking Lemma 5.2, there exists {ỹ n } ⊂ R N such that y n = ε nỹn ∈ M δ for n large enough. Hence, Since u n (· +ỹ n ) strongly converges in Y V 0 and ε n z + y n → y ∈ M δ for all z ∈ R N , we can see that β ε n (u n ) = y n + o n (1). The proof of the lemma is now complete.
(6.1) and thus u n (x) < a for any x ∈ B c R (ỹ n ) , n ∈ N.
On the other hand, there exists ν ∈ N such that, for any n ≥ ν, Hence, u n (x) < a for any x ∈ c ε n and n ≥ ν, which contradicts (6.3). Letε δ > 0 be given by Theorem 5.1 and set ε δ = min{ε δ ,ε δ }. Take ε ∈ (0, ε δ ). Applying Theorem 5.1, we obtain at least cat M δ (M) positive solutions to (3.1). If u ε is one of these solutions, we have that u ε ∈ N ε , and we can use (6.2) and the definition of g to deduce that g(εx, . This means that u ε is also a solution of (1.1). Consequently, (1.1) admits at least cat M δ (M) positive solutions. Now we consider ε n → 0 and take a sequence {u n } n∈N ⊂ X ε n of solutions to (1.1) as above. Let us investigate the behavior of the maximum points of u n . By the definition of g and (g 1 ), there exists σ ∈ (0, a) such that As before, we can take R > 0 such that Up to a subsequence, we may also assume that Otherwise, if (6.6) fails, we have |u n | ∞ < σ . Then, in view of J ε n (u n ), u n = 0 and (6.4), we get which gives a contradiction. Therefore, (6.6) is true. In light of (6.5) and (6.6), we deduce that if p n is a global maximum point of u n , then p n =ỹ n + q n for some q n ∈ B R (0). Since ε nỹn → y 0 ∈ M and |q n | < R for all n ∈ N, we see that ε n p n → y 0 , and using the continuity of V we achieve The proof of Theorem 1.1 is now complete.
Now, we know that, for any ε ∈ (0,ε(δ, μ)), problem (7.2) possesses at least cat M δ (M) positive solutions. Let u ε,μ be one of these solutions. We shall assume thatε(δ, μ) is small in such a way that the thesis of Lemma 7.1 is valid. Our goal is to show that u ε,μ is a solution of the original problem (1.5) whenever μ is sufficiently small. More precisely, we will prove that there exists K 0 > 0 such that for any K ≥ K 0 , there exists μ 0 = μ 0 (K ) > 0 such that |u ε,μ | ∞ ≤ K for all μ ∈ [0, μ 0 ]. (7.6) To see this, we use a Moser iteration argument [28]. For simplicity, we set u = u ε,μ . For any L > 0, we define u L = min{u, L} ≥ 0, where β > 1 will be chosen later, and let w L = uu On the other hand, arguing as at the beginning of the proof of Lemma 3.18 in [6] (see formula (85) there), we can see that, for t ∈ {p, q}, u, z L s,t ≥ 1 β t S t |w L | t Putting together (7.7), (7.1) and (V 1 ), we deduce that (7.9) where C μ,K = 1 + μK τ −γ . In light of (7.8) and (7.9), and applying the Hölder inequality, we have that where C 1 = S −1 q > 0 and ∈ (q, q * s ).

Final Comments
We conclude the paper with some comments which we believe useful for future developments. First we observe that the variational techniques used along the paper are rather flexible and can be adapted to study problems like (1.1) and involving the more are considered, as for instance the g-Laplacian case, that is, in the variational case, minima of functionals of the type where, for every fixed x, it holds p < G (x, t)t G(x, t) < q for every t > 0. (8.1) In our context, this would correspond to take G(x, t) = t p + t q . In the same way in the literature are considered functionals of the type where G 1 and G 2 are two different functions satisfying (8.1). The fractional analog of this setting is given by functionals of the type that would be the natural driving energy to combine with the lower order terms making the problem non-homogeneous. In this situation, we need to modify the conditions on the nonlinearity f as in [1]. We will address these questions in a future paper.
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