Ground states for scalar field equations with anisotropic nonlocal nonlinearities

We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.


Introduction and main results
In the present paper we seek ground states, namely least energy solutions, for the following nonlocal anisotropic scalar field equation: Here, N ≥ 2, V ∈ L ∞ (R N ) is a weight function satisfying In other terms, (1.1) has a ground state if and only if λ 1 is attained. The constant exponent case p(x) ≡ p ∞ ∈ (2, 2 * ) of equation (1.1) has been studied extensively for more than three decades (see Bahri and Lions [1] for a detailed account). Ground states are quite well understood in this case. Set for all u ∈ H 1 (R N ) M ∞ := u ∈ H 1 (R N ) : I ∞ (u) = 1 .
The infimum is not attained in general. The asymptotic functional attains its infimum (1.5) λ ∞ 1 := inf u∈M ∞ J ∞ (u) > 0 at a positive radial function w ∞ 1 (see Berestycki and Lions [3] and Byeon et al. [4]). Moreover, such minimizer is unique up to translations (see Kwong [7]). By (1.2) and the translation invariance of J ∞ , one can easily see that λ 1 ≤ λ ∞ 1 , and λ 1 is attained if this inequality is strict (see Lions [8,9]). In this paper we give sufficient conditions on the weight V and the exponent p for the existence of a ground state of (1.1). Precisely, we shall prove the following result: then λ 1 is attained at a positive minimizer w 1 ∈ C 1 (R N ).
In particular, there exists a positive ground state if p ∞ = p − and λ 1 < λ ∞ 1 (hence our result is consistent with the constant exponent case). Hypothesis (1.6) is global, but it can be assured by making convenient local assumptions on p, for instance when p(x) slowly approaches p ∞ from below as |x| → ∞: be a mapping such that ψ(r) → ∞ as r → ∞ and the function e −ψ(|·|) ∈ H 1 (R N ), and let R > 0. Then there exists a > 0 such that, if , |x| ≥ R, then λ 1 is attained.
For example, given R > 0, we can find a > 0 such that, if then λ 1 is attained. We note that the only assumption on V in Theorem 1.2 is (1.2). Finally, we address the problem of symmetry of minimizers. Apparently, the best we can achieve under the assumption of radial symmetry of the data V and p is axial symmetry of all ground states: (1.4), and both are radially symmetric in R N . Moreover, assume that (1.6) holds. Then, for every minimizer w there exist a line L though 0 and a function w : L × R + → R such that where P L : R N → L denotes the projection onto L.
As in the constant exponent case, the main difficulty is the lack of compactness inherent in this problem, which originates from the invariance of R N under the action of the noncompact group of translations, and manifests itself in the noncompactness of the embedding of H 1 (R N ) into the variable exponent Lebesgue space L p(·) (R N ). This in turn implies that the manifold M is not weakly closed in H 1 (R N ) and that J| M does not satisfy the Palais-Smale compactness condition (shortly (PS) c ) at all energy levels c ∈ R. We will use the concentration compactness principle of Lions (see [8][9][10]), expressed as a suitable profile decomposition for (PS) sequences of J| M , to overcome these difficulties. Developing this argument, we will also prove an extension to the variable exponent case of the compactness lemma of Benci and Cerami [2, Lemma 3.1].
The paper has the following structure: in Section 2 we introduce the mathematical background and establish some technical lemmas; in Section 3 we prove that (PS) c holds for all c below a threshold level; and in Section 4 we deliver the proofs of our main results.

Preliminaries
We consider the space H 1 (R N ), endowed with the norm defined by which is equivalent to the standard norm. Clearly we have J ∈ C 1 (H 1 (R N )) with We recall some basic features from the theory of variable exponent Lebesgue space, referring the reader to the book of Diening et al. [5] for a detailed account on this subject.
This is a reflexive Banach space under the following modified Luxemburg norm, introduced by Franzina and Lindqvist [6]: The following relation will be widely used in our study: It can be proved noting that, for all u ∈ L p(·) , Analogously, for all q > 1 we endow the constant exponent Lebesgue space L q (R N ) with the norm

By (1.3) and [5, Theorem 3.3.11], the embedding
by the Sobolev embedding theorem, also H 1 (R N ) ֒→ L p(·) (R N ) is continuous. As a consequence, the functional I : H 1 (R N ) → R is well defined and continuous. Moreover, reasoning as in [6, Lemma A.1], we see that In particular, as 1 is a regular value of I, M turns out to be a C 1 Hilbert manifold. By the Lagrange multiplier rule, u ∈ M is a critical point of J| M if and only if there exists µ ∈ R such that that is (recalling that I(u) = 1), if and only if u is a (weak) solution of (1.1) with λ = µ/2. Moreover, testing (1.1) with u yields J(u) = λ.
We set σ(u) := max{ u , and we prove the following properties that will be used later: and integrating over R N gives (note that u the last inequality following from (2.2).
the last inequality following from (i).
The autonomous case V (x) ≡ V ∞ , p(x) ≡ p ∞ represents a limit case for (1.1). We set Morevoer, we have the following asymptotic laws: Proof. We prove (i). For all k ∈ N, the change of variable z = x − y k gives Since p(· + y k ) → p ∞ by (1.4) and |u(z)| p(z+y k ) ≤ |u(z)| p − + |u(z)| p + , the last integral converges to ρ ∞ (u) by the dominated convergence theorem.
We prove (ii). As in the proof of (i), for all γ > 0 If u k p(·) → u p ∞ , then there exists ε 0 > 0 such that, on a renumbered subsequence, either u k p(·) ≤ u p ∞ − ε 0 or u k p(·) ≥ u p ∞ + ε 0 . In the former case, u p ∞ ≥ ε 0 and, taking ε 0 smaller if necessary, we may assume that this inequality is strict. Then Passing to the limit as k → ∞, (2.6) implies a contradiction. The latter case leads to a similar contradiction.
Finally we prove (iii). We have for all k ∈ N , the last integral converges to J ∞ (u) by the dominated convergence theorem.

A compactness result
In this section we prove that J| M satisfies (PS) c whenever c ∈ R lies below a certain threshold level.
The main technical tool that we will use for handling the convergence matters is the following profile decomposition of Solimini [12] for bounded sequences in H 1 (R N ).
Proposition 3.1. Let (u k ) be a bounded sequence in H 1 (R N ), and assume that there is a constant δ > 0 such that, if u k (· + y k ) ⇀ w = 0 on a renumbered subsequence for some sequence (y k ) in R N with |y k | → ∞, then w ≥ δ. Then there exist m ∈ N, w (1) , . . . w (n) ∈ H 1 (R N ), and sequences (y k = 0 for all k ∈ N, w (n) = 0 for all 2 ≤ n ≤ m, such that, on a renumbered subsequence, Since y Next we show that the sublevel sets of J are bounded. Set Lemma 3.2. For all a ∈ R, J a is bounded.
We prove (ii). As above, we only need to show that for all v ∈ C ∞ 0 (R N ) We have supp v ⊂ Ω for some bounded domain Ω ⊂ R N . Testing (3.2) with v(· − y k ) and making the change of variable z = x − y k gives for all k ∈ N where u k = u k (· + y k ). Since u k ⇀ w and V (· + y k ) → V ∞ uniformly on Ω, we have Besides, µ k /(2ρ(u k )) → c/ρ 0 . Finally, exploiting again (1.3), (2.3) as in the proof of (i) we have, on a renumbered subsequence, which proves (ii).
The main result of this section is the following extension to the variable exponent case of the compactness lemma of Benci and Cerami [2, Lemma 3.1].
By (ii) and Lemma 2.2 (ii) we have Since w k p(·) → 1 and ρ(w k ) → ρ 0 , then by Lemma 2.1 (ii) we can find a constant C > 0 such that On the other hand, for all sufficiently large k, the (compact) supports of w (n) (· − y (n) k ) are pairwise disjoint by (ii) and hence by (ii) and Lemma 2.2 (i). So By Lemmas 2.1 (ii) and (2.5) we have by Lemmas 2.1 and 2.2. Since ε > 0 is arbitrary, (viii) follows.
We conclude by proving (x). Set v k = u k − w k and u k = u k − w (1) for all k ∈ N. Note that both ( u k ) and (v k ) are bounded in H 1 (R N ). By (3.2), (iv) and (v) we have . Testing with v k and using Lemma 2.1 (i) and (2.4) gives Since u k is bounded, so are u k , σ(u k ), and v k . If v k → 0, then there exists ε 0 > 0 such that, on a renumbered subsequence, v k ≥ ε 0 . By (1.2), there exists R > 0 such that Then, from the equation above, Hölder inequality, Proposition 3.1 (iv) and (3.1) we get Assume now that (1.6) holds. Since J satisfies the Palais-Smale condition at the level λ 1 by Theorem 3.5, it has a minimizer w 1 by a standard argument. Then |w 1 | is a minimizer too and hence we may assume that w 1 ≥ 0. Since w 1 = 0, then w 1 > 0 by the strong maximum principle. Observe that a solution u ∈ H 1 (R N ) of (1.1) satisfies −∆u = g(x, u) and g(x, u) u ≤ C + C|u| p(x)−2 ≤ C + C|u| It follows from (4.1), (2.1) and (4.2) that (1.6) holds if a > 0 is sufficiently large.

4.3.
Proof of Corollary 1.3. If N ≥ 3, V and p are radially symmetric in R N , we can get some symmetry properties of minimizers by applying the results of Mariş [11]. We can equivalently define For any hyperplane Π through 0, splitting R N in two half-spaces Π + and Π − , and all u ∈ H 1 (R N ) we define functions u Π + , u Π − : R N → R by setting where P (·) is the orthogonal projection from R N to an affine submanifold (·). Clearly u Π ± ∈ H 1 (R N ), so hypothesis A1 of [11] is satisfied. Since u is of class C 1 (R N ), so hypothesis A2 holds as well. By [11,Theorem 1] we learn that, for every minimizer w ofJ, there exists a line L through 0 such that w(x) =w(P L (x), |x − P L (x)|) for all x ∈ R N , for a convenient functionw : L × R + → R.