Ground state solution for a class of supercritical nonlocal equations with variable exponent

In this paper, we obtain the existence of positive critical point with least energy for a class of functionals involving nonlocal and supercritical variable exponent nonlinearities by applying the variational method and approximation techniques. We apply our results to the supercritical Schrödinger–Poisson type systems and supercritical Kirchhoff type equations with variable exponent, respectively.


Introduction and main results
We divide this section into two parts. In the first part, we present a critical point theory of abstract functional inspired by the article of Marcos do Ó, Ruf and Ubilla [21]. The second part is devoted to introduce its applications to a class of Schrödinger-Poisson type systems and a class of Kirchhoff type equations.

Abstract critical point theory
In the pioneering article [8], Brézis and Nirenberg considered the existence of solution to the following nonlinear elliptic equation in Ω, where Ω is a bounded domain in R 3 . If f (x, u) = 0 and Ω is star shaped, a well-known nonexistence result of Pohozaev [26] asserts that (1.1) has no solution. But the lower-order terms perturbation can reverse this situation. Brézis and Nirenberg [8] proved the existence of solutions to (1.1) under the assumptions on the lower-order perturbation term f (x, u). On X. Feng the other hand, the topology and the shape of the domain can affect the existence of solution for (1.1) with f (x, u) = 0. For example, Coron [12] used a variational approach to prove that (1.1) is solvable if Ω exhibits a small hole. Rey [27] established existence of multiple solutions if Ω exhibits several small holes. As Ω is an annulus, Kazdan and Warner [17] observed that there exists a solution to (1.1) without any constraint by critical exponent. It is worth noticing that there are also a few papers concerning on the supercritical equations except adding lower-order perturbation terms or changing the topology of region Ω. The papers in [10,21] considered the following nonlinear supercritical elliptic problem −∆u = |u| 4+|x| α u, in B, where B ⊂ R 3 is the unit ball and 0 < α < 1. By using the mountain pass lemma and approximation techniques, a radial positive solution for (1.2) is obtained by Marcos doÓ, Ruf and Ubilla in [21]. Cao, Li and Liu [10] considered the existence of infinitely many nodal solutions to (1.2) by looking for a minimizer of a constrained minimization problem in a special space.
Let H be the subspace of H 1 0 (B) consisting of radially symmetric functions. From [21], we know that (1.2) possesses a variational structure, its solutions can be found as critical points of the functional The solutions to this kind of supercritical elliptic equations involving nonlocal nonlinearities can be found to look for the critical points of a suitable perturbation of I 0 , where λ ∈ R and R ∈ C(H, R). In order to obtain the nontrivial critical point of J, we need to consider the approximation functional I : H → R associated to J given by In this paper, we are interested in researching the least energy critical point of J, the following assumptions are needed: (i) R ∈ C 1 (H, R + ) with R + = [0, +∞); (ii) there exist C, q > 0 such that for t > 0, R(tu) = t q R(u), R(u) ≤ C u q , ∀ u ∈ H; (iii) qR(u) = R (u)u , u ∈ H; (iv) if {u n } is a (PS) c sequence of J for some c > 0 and u n u weakly in H as n → ∞, then J (u) = 0.
Inspired by above papers, the main purpose of this paper is to consider the existence of ground state for the functional J. Our main result reads as follows.

Applications to two nonlocal problems
As a first application, we consider the existence of nontrivial solution to the supercritical Schrödinger-Poisson type systems with variable exponent where B ⊂ R 3 is the unit ball and 0 < α < 1. The Schrödinger-Poisson system as a model describing the interaction of a charge particle with an electromagnetic field arises in many mathematical physics context (we refer to [7] for more details on the physical aspects). There are a few references which investigated the well-known Schrödinger-Poisson system with nonlocal critical growth in a bounded domain (see e.g. [3][4][5]). Azzollini, d'Avenia [3] considered the following problem involving the nonlocal critical growth They proved the existence of positive solution depending on the value of λ and (1.4) has no solution for λ ≤ 0 via Pohozaev's identity. Later, Azzollini, d'Avenia and Vaira [5] improved the results in [3]. They proved existence and nonexistence results of positive solutions for (1.4) when λ is in proper region. By applying the variational arguments and the cut-off function technique, Azzollini, d'Avenia and Luisi [4] studied the following generalized Schrödinger- in Ω, where Ω ⊂ R 3 is a bounded domain with smooth boundary ∂Ω, In the case where f is critical growth, they obtained the existence and nonexistence results.
In the recent years, there have been a lot of researches dealing with the Schrödinger-Poisson systems (1.5) When f (x, u) = |u| p−1 u with p ∈ (1, 5), Ruiz and Siciliano [29] considered the existence, nonexistence and multiplicity results by using variational methods. Alves and Souto [2] studied system (1.5) when f has a subcritical growth. They obtained the existence of least energy nodal solution by using variational methods. Ba and He [6] proved the existence of ground state solution for system (1.5) with a general 4-superlinear nonlinearity f by the aid of the Nehari manifold. Pisani and Siciliano [25] proved the existence of infinitely many solutions of (1.5) by means of variational methods. In [1], Almuaalemi, Chen and Khoutir obtained the existence of nontrivial solutions for (1.5) when f has a critical growth via variational methods. Motivated by above papers, by applying Theorems 1.1 and 1.2, we obtain the existence of positive ground state solution for system (1.3) with both nonlinearity supercritical growth and nonlocal critical growth. From the technical point of view, there are two difficulties to prove our result. Firstly, the supercritical nonlinearity in the system sets an obstacle since the bounded (PS) sequence could not converge. Secondly, due to the system has two critical terms, it is difficult to estimate the critical level of mountain pass. In order to overcome these difficulties, by employing the ideas of [21], we first estimate the critical level of the mountain pass for the functional corresponding to (1.3) via approximation techniques and then show that the level is below the non-compactness level of the functional. Finally, the existence of positive ground state solution is obtained by applying the Nehari manifold method and regularity theory. Hence, we have the following result: Remark 1.8. By the Pohozaev's identity used in [3], we can deduce that (1.3) has no nontrivial solution if |x| α = 0. Hence, our result is interesting phenomena due to the nonlinearity |u| 4+|x| α u has supercritical growth everywhere in B except in the origin and critical growth in the origin.
Next, as the second application, we consider the following Kirchhoff type equations: where b > 0, 0 < α < 1. This kind of equation is related to the stationary analogue of the equation presented by Kirchhoff in [18]. The equation extends the classical d'Alembert's wave equation by considering the effects of the changes in the length of the strings during the vibrations. The solvability of the Kirchhoff type equations has been well studied in a general dimension by many authors after Lions [20] introduced an abstract framework to this problem. By using new analytical skills and non-Nehari manifold method, Tang and Cheng [31] obtained the ground state sign-changing solutions for a class of Kirchhoff type problems in bounded domains. In [11], Chen, Zhang and Tang considered the existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity based on variational and some new analytical techniques. There are also many papers devoted to the existence and multiplicity of solutions for the following critical Kirchhoff type equations with subcritical disturbance where a, b are positive constants. By using concentration-compactness principle and variational method, Naimen in [22] obtained the existence and multiplicity of (1.7) with f (x, u) = λu. Xie, Wu and Tang [34] derived the existence and multiplicity of solutions to (1.7) via variational method by discussing the sign of a and b and adding different conditions on f . By controlling concentrating Palais-Smale sequences, Naimen and Shibata [23] proved the existence of two positive solutions for (1.7) with f (x, u) = u q , 1 ≤ q < 5.
In particular, there are some papers considered the equations with critical and supercritical growth by adding the smallness of the coefficient in front of critical and supercritical which is used to overcome the difficulty provoked by supercritical growth. By combining an appropriate method of truncation function with Moser's iteration technique, Corrêa and Figueiredo [13,14] considered the existence of positive solution for a class of p-Kirchhoff type equations and Kirchhoff type equations with supercritical growth, respectively. Motivated by the above fact, we study the existence of positive ground state solution for (1.6) with variable exponential perturbation by using the similar method introduced by Marcos do Ó, Ruf and Ubilla in [21]. The result reads as follows.   [22], if |x| α = 0, (1.6) has no nontrivial solution by Pohozaev's identity. Hence, our result is interesting phenomena for this kind of Kirchhoff type equations due to the nonlinearity |u| 4+|x| α u has supercritical growth everywhere in B except the origin and critical growth in the origin. Remark 1.11. Throughout the paper we denote by C > 0 various positive constants which may vary from line to line and are not essential to the problem.
The paper is organized as follows: in Section 2, some notations and preliminary results are presented. We obtain the existence of nontrivial critical point to the functional J in Section 3. By using Nehari manifold method, the least energy critical point of the functional J is derived X. Feng in Section 4. Sections 5 and 6 are devoted to show that the Theorems 1.1 and 1.2 can be applied to the nonlinear Schrödinger-Poisson type systems and the Kirchhoff type equations, respectively.

Preliminary
In this Section, we will give some notations and lemmas which will be used throughout this paper. Let B ⊂ R 3 denote the unit ball, Then for each p ∈ C + (B), the variable exponent function space L p(x) (B) is defined as follows

Lemma 2.2 ([21]
). Let q(x) = 6 + β|x| α , x ∈ B and α, β > 0. The following embedding is continuous: It is easy to check by (i), Lemma 2.2 and Hölder type inequality that J is well defined on H and J ∈ C 1 (H, R), and In the following we define the best embedding constant S by Then define u ε = χ(x)U ε (x), the following estimates can be deduced via standard arguments as ε → 0 + (see [33]),

The nontrivial critical point
In this section, we first show that the functional J possesses the mountain pass structure under the assumption λ < 0, q > 6 or λ > 0, 0 < q < 6, respectively. And hence J has a (PS) c sequence {u n } with some c > 0. Then we prove that {u n } is bounded and is also a (PS) c sequence of I, which is a key in the existence of nontrivial critical point.
Proof. (a) For ρ 1 > 0, let We deduce, from the Sobolev inequality and Lemma 2.1, that for u ∈ ∂Σ ρ 1 and C > 0, Hence, by letting ρ 1 > 0 small enough, it is easy to see that there is η 1 > 0 such that (a) holds.

X. Feng
This together with (2.2) implies that for t ≥ 1 and ε > 0 small enough, By taking e 1 =h(1), then (b) is valid. The proof is completed.
(a) There exist It follows from the Sobolev inequality and Lemma 2.1 that for u ∈ ∂Σ ρ 2 and C > 0, Hence, by letting ρ 2 > 0 small enough, it is easy to see that there is η 2 > 0 such that (a) holds. (b) By using (2.2) and (3.1) again, we have for t ≥ 1 and ε > 0 small enough, By taking e 2 =ĥ(1), we proof (b). The proof is completed.
From Lemmas 3.1 and 3.2, we know that the functional J possesses the mountain pass geometry. Then there is a (PS) c sequence {u n } ⊂ H for J with the property that where c is given by and Proof. For n large enough, it is easy to deduce from (iii) that which implies that {u n } is bounded in H. The proof is completed.
Proof of Theorem 1.1. By using Lemmas 3.1 and 3.2 respectively, there exists a sequence {u n } ⊂ H satisfying J(u n ) → c, J (u n ) → 0 as n → ∞, where c is given in (3.2). By Lemma 3.3, {u n } is a bounded sequence in H. Passing to a subsequence if necessary, we may assume that there exists u ∈ H such that u n u in H, and u n (x) → u(x), a.e. x ∈ B.
If u = 0, then u is a nontrivial critical point of the functional J follows from the assumption (iv). In what follows, we will deal with the case of u = 0 and show that this is impossible. In In the following, we will show that {u n } is also a (PS) c sequence of I. Hence, it is sufficient to prove (a) J(u n ) = I(u n ) + o(1);

X. Feng
We first claim that (a) is valid, indeed we only need to estimate For any ε > 0, there exist δ > 0 and n 1 ∈ N such that for any n ≥ n 1 , we have, by (3.4), where ω is the surface area of the unit sphere in R 3 . Similarly, for above ε > 0, there exist δ 1 > 0 small enough and n 2 ∈ N such that for any n ≥ n 2 , it follows from (3.3) and (3.4) that (3.7) Hence, combining (3.5), (3.6) and (3.7), we have for above ε > 0, there exists n 0 = max{n 1 , n 2 }, such that for any n ≥ n 0 , which implies that (a) is true.
Secondly, we will devoted to verify that (b) is correct. In fact, by Lemma 3.4, for 0 < η < 1 small enough and v ∈ H, η 0 |u n | 5 |v|(|u n | r α − 1)r 2 Hence, for any ε > 0, there exists η = η(ε) > 0 sufficiently small such that On the other hand, it follows that for above ε > 0, there exists n 1 ∈ N such that for n > n 1 , Similarly, we have for above ε > 0, there exists n 2 ∈ N such that for n > n 2 , Combining (3.8), (3.9) and (3.10), we obtain for ε > 0, there exists n 0 = max{n 1 , n 2 } such that for n > n 0 , which ensures that (b) is valid. Thereby, it is obvious that {u n } is also a (PS) c sequence for the functional I. Recall that I satisfies (PS) c condition, we have that u n → u = 0 strongly in H, which is a contradiction to I(u n ) → c > 0. The proof is completed.

The least energy critical point
In this section, we will use the Nehari manifold method to show the existence of nontrivial nonnegative ground state of the functional J. In order to obtain the ground state, we need the Nehari manifold associated with J given by Lemma 4.1. Assume that λ < 0, q > 2 or λ > 0, 2 < q < 6 and the assumptions (i)-(ii) hold. Then, for each u ∈ H \ {0}, there exists a unique t(u) > 0 such that t(u)u ∈ N . Moreover, J(t(u)u) = max t≥0 J(tu).
Proof. For u ∈ N , it follows from (i) and (ii) that which implies that there exists a positive constant C such that u ≥ C. On the other hand, we have Hence, J is bounded below. The proof is completed.
The following lemma can be also obtained by Implicit Function Theorem or by the Lusternik Theorem. We give the other proof by applying the Lagrange multiplier method.

Lemma 4.4.
Assume that λ < 0, q > 6 or λ > 0, 2 < q < 6 and the assumptions (i)-(iii) hold. If c * is attained at some u ∈ N , then u is a critical point of J in H.
Proof. Let G(u) = J (u), u , then G ∈ C 1 (H, R). By Lemma 4.1, N = ∅. We claim that 0 / ∈ ∂N . In fact, for any u ∈ H with u small. Note that for any u ∈ N Hence, G (u) = 0 for any u ∈ N . Then the implicit function theorem implies that N is a C 1 manifold. Recall that u is minimizer of J on u ∈ N . Then by the Lagrange multiplier method, there exists λ ∈ R such that J (u) = λG (u).
This shows that J(u) = c * . It is easy to see that J(|u|) = J(u) = c * . Thus, Lemma 4.4 implies that |u| is a ground state of J. The proof is completed.

The Schrödinger-Poisson type system
This section is devoted to apply the Theorems 1.1 and 1.2 to a class of Schrödinger-Poisson type system. We first estimate the critical level of mountain pass of the functionalJ associated to (1.3) and then show that the critical level of mountain pass is below the non-compactness level ofJ. Secondly, we are devoted to verify that the (PS) sequence of the functionalJ is also the one of the approximation functional associated toJ by using approximation techniques. Finally, by using the regularity theory, the positive ground state solution of (1.3) is obtained. We establish the following lemmas, which guarantee that the conditions in the Theorems 1.1 and 1.2 are valid. We observe that by [3], for given u ∈ H, there exists a unique solution φ = φ u ∈ H satisfying −∆φ u = |u| 5 in B, u = 0 on ∂B in a weak sense and it has the following properties. (ii) φ tu = t 5 φ u for all t > 0; where S is defined in (2.1); (iv) if u n u in H, then, up to a subsequence, φ u n φ u in H.

Moreover, (1.3) is variational and its solutions are the critical points of the functional defined in H byJ
It is easy to check by Lemmas 2.2 and 5.1 thatJ is well defined on H andJ ∈ C 1 (H, R), and Then Proof. For t ≥ 0, we have Substituting it into f 1 (t), the result is obtained. The proof is completed.
then we have, as ε → 0 + , Then thanks to (2.2) we derive that, for ε > 0 sufficiently small, This together with Lemma 5.2 and the estimate (2.2) implies that for ε > 0 sufficiently small. The proof is completed.

X. Feng
From Lemma 3.1, we know that the functionalJ possesses the mountain pass geometry. Then there is a (PS) c 1 sequence {u n } ⊂ H forJ with the property that where c 1 is given by In the following we give an estimate of the upper bound of the critical level c 1 by using above two lemmas. Proof. It follows from (3.1) that, for ε small enough, Thus, there exists R 1 > 0 sufficiently large which is independent of ε, such that ϕ(R 1 ) = 0 andJ(R 1 u ε ) ≤ 0 for ε small enough. Hence, we can find 0 < t ε < R 1 satisfying Hence we deduce from (2.2) that where A ε = O(ε α | log ε|) + O(ε 3/2 ) is given in [21]. For convenience, we set A = S

) can be rewritten as
It is easy to see that for ε small, Thereby, for ε small enough, there holds In what follows, we will estimate the term It follows from (5.4) again that Combining (5.5)-(5.8) and using Lemma 5.3, we derivẽ By choosing ε > 0 small enough, we derive by (5.9), The proof is finished.
Lemma 5.5. If {u n } is a (PS) c 1 sequence ofJ, then there exists u ∈ H such that, up to a subsequence, u n u andJ (u) = 0.
Proof. From Lemma 3.3 we see that {u n } is bounded in H. Then, up to a subsequence, we can assume that {u n } converges to u weakly in H and u n → u a.e. in B. By taking ϕ ∈ C ∞ 0 (B), we find It follows from Lemma 5.1 that φ u n φ u in H, which implies φ u n φ u in L 6 (B). Then Since u n → u a.e. in B and B |φ u n (|u n | 3 u n − |u| 3 u)| For any measurable subset Q ⊂ B, we have where p(x) = 6 + |x| α . Hence, Vitali's theorem (see [28]) implies B |u n | 4+|x| α u n ϕ → B |u| 4+|x| α uϕ, as n → ∞. Therefore, by density, we derive thatJ (u) = 0. The proof is completed.
In order to obtain the nontrivial solution of (1.3), we need define the approximation func-tionalĨ : H → R associated toJ given bỹ Lemma 5.6. The functionalĨ satisfies the (PS) c 1 condition with c 1 ∈ (0, Λ).
Similarly to Lemma 3.3, it is easy to see that {u n } is bounded in H. Going if necessary to a subsequence, we can find u ∈ H such that u n u in H. By the same argument used in Lemma 5.5, we deduce thatĨ (u) = 0, hencẽ Now, let v n = u n − u, it is obvious to see that From Brézis-Lieb Lemma in [9,19], we have These three equalities imply that and similarly We will show that v n → 0. Otherwise, there exists a subsequence still denoted by {v n } such that v n 2 → l > 0. For convenience, let a n = B φ v n |v n | 5 and b n = B |v n | 6 . Without loss of generality, we may assume a n → a 1 and b n → b 1 , as n → ∞. Notice that then as n → ∞ passing to the limit, we conclude that Taking ε 2 = √ 5−1 2 , and combining with (5.15) leads to from which we get by (5.13), (5.14) and (5.15) that On the other hand, (5.1) and (5.15) yield Therefore we get l 2 ≥ −1+ √ 5 2 S 3 . This together with (5.16) implies that c 1 ≥ Λ, which will come to a contradiction. Therefore v n → 0 strongly in H, or equivalently, u n → u in H as n → ∞. The proof is completed.
Proof of Theorem 1.7. The Lemmas 5.4, 5.5 and Theorem 1.2 imply that (1.3) admits a nonnegative nontrivial ground state solution u ∈ H, which satisfies the following equation in weak sense −∆u = φ u |u| 3 u + u 5+|x| α in B.

The Kirchhoff type equation
In this section, we obtain the existence of positive ground state solution of (1.6) by using Theorem 1.2 with λ = 1, q = 4. Similarly to Section 4, we first estimate the level of mountain critical of the functionalĴ corresponding to (1.6) and show that the critical level is below the non-compactness level ofĴ by using approximation techniques. Then we are devoted to verify that the (PS) sequence of the functionalĴ is also the one of the approximation functional associated toĴ. Finally, by the regularity theory of the elliptic equation, the positive ground state solution of (1.6) is obtained. In order to find the weak solutions to (1.6) and it is natural to consider the energy functional on H: Then we have from Lemma 2.2 thatĴ is well defined on H and is of C 1 , and It is standard to verify that the weak solutions of (1.6) correspond to the critical points of the functionalĴ. Then Proof. For t ≥ 0, we have Let α 2 + β 2 t 2 − γ 2 t 4 = 0, we write at Substituting it into f 2 (t), the result is valid. The proof is completed.
This together with the fact thatĴ (u n ) → 0 ensures that By taking v = u in (6.6), there holds Ĵ (u), u < 0. Similarly to the proof of Lemma 3.1, we have Ĵ (tu), tu > 0 for small t > 0. Thus, there exists a t u ∈ (0, 1) such thatĴ(t u u) = max t≥0Ĵ (tu) and Ĵ (t u u), t u u = 0. Then, we deduce by the weak lower semicontinuity of the norm and Fatou's lemma that which is impossible. Thus, B |∇u| 2 = A 2 andĴ (u) = 0. The proof is completed.
By repeating the arguments used in Lemma 3.3, it is easy to show that {u n } is bounded in H. Then passing to a subsequence, we can find u ∈ H such that u n u in H. Now, let v n = u n − u, we claim that v n → 0. In fact, we use an argument of contradiction and suppose that there exists a subsequence still denoted by {v n } such that v n →l > 0. It is easy to verify that u n 2 = v n 2 + u 2 + o(1) (6.7) and u n 4 = v n 4 + u 4 + 2 v n 2 u 2 + o(1). (6.11) On the other hand, combining (6.7), (6.8) with (6.9) leads tô (6.12) Similarly, by using (6.10) again, we deduce (6.13) Then, taking the limit on the both sides in (6.13) as n → ∞, we findl 2 + bl 4 + bl 2 u 2 ≤ S −3l6 , which implies thatl 2 ≥ S 3 b + S S 4 b 2 + 4(1 + b u 2 )S 2 . (6.14) It follows from (6.12) and (6.13) that This together with (6.14) ensures that which contradicts to (6.11). Therefore v n → 0 strongly in H, or equivalently, u n → u in H as n → ∞. The proof is completed.

X. Feng
Proof of Theorem 1.9. By Lemmas 6.4, 6.5, we know that the assumptions in Theorem 1.2 are valid. Hence, (1.6) possesses a nonnegative nontrivial ground state solution u ∈ H, which satisfies the following equation in weak sense Let us defineĝ (u(x)) = u 5+|x| α 1 + b B |∇u| 2 , x ∈ B.