Inﬁnitely many solutions for quasilinear Schrödinger equation with concave-convex nonlinearities

In this work, we study the existence of inﬁnitely many solutions to the following quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity:


Introduction and main result
In this paper, we are interested in the existence of infinitely many solutions to a class of quasilinear Schrödinger equations with a parameter α and a concave-convex nonlinearity where p u = div(|∇u| p-2 ∇u)(1 < p < N) and α > 1 2 is a parameter.For the case p = 2, α = 1, solutions of (1.1) are standing waves of the following Schrödinger equation: where z : R × R N → C and W : R N → R is a given potential, h 1 , g : R + → R are real functions.
It is well known that the standing wave solutions of the form z(t, x) = exp(-iωt)u(x) satisfy (1.2) with g(s) = s if and only if the function u(x) solves the equation of elliptic type u + V (x)uu 2 u = h(u), x ∈ R N , (1.3) where V (x) = W (x)ω, ω ∈ R and h(u) ≡ h 1 (|u| 2 )u.Quasilinear Schrödinger equations of form (1.2) appear naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of nonlinear term g.The case g(s) = s was used for the superfluid film equation in plasma physics by Kurihura in [11] (see also [12]).In the case g(s) = (1 + s) 1/2 , Eq. (1.2) models the self-channeling of a high power ultra short laser in matter, see [7].Equation (1.2) also appears in plasma physics and fluid mechanics [20], in mechanics [9], and in condensed matter theory [18].More information on this subject can be found in [15] and the references therein.
For p = 2, several methods can be used to solve (1.1), e.g., the existence of positive ground state solution was proved in [17,19] by using a constrained minimization argument; Eq. (1.1) was transformed to a semilinear one in [4-6, 10, 15] by a change of variables (dual approach); Nehari method was used to get the existence results of ground state solutions in [16,22].Especially, in [13,[15][16][17]25], the existence of the ground state solutions for the following problem with a parameter α(> 1  2 ): - was studied with subcritical nonlinearities g(x, u).
For (1.4), we find in the literature several types of potentials V (x) to obtain a solution.Wu in [25] studied Eq. (1.4) considering the subcritical case and a potential V (x), which is unbounded in R N and satisfies the following assumption: (A 1 ) The potential V (x) ∈ C(R N ) and 0 < V 0 := inf x∈R N V (x), and for each M > 0, meas({x ∈ R N : V (x) ≤ M}) < ∞.In [15][16][17], Liu et al. proved the existence of a positive solution to problem (1.4) with V (x) ∈ C(R N ), inf x∈R N V (x) > 0 and the following conditions: Similar assumptions also appeared in Severo [24], Ruiz and Siciliano [22], Fang and Szulkin [8].By the variational principle in a suitable Orlicz space, do Ó and Severo in [3] established the existence of positive standing wave solutions for (1.4) with a concaveconvex nonlinearity and the following condition: Recently, Aires and Souto [1] considered (1.4) with α = 1 and the vanishing potential V (x) at infinity.
Clearly, it is well known that assumption (A 1 ) or (A 2 ) guarantees that the embedding Similarly, the application of (A 3 ) in [2,15,24] shows that the solution is nontrivial.
It is worth pointing out that the aforementioned authors always assumed that the potential V (x) has some special characteristic.As far as we know, there are few papers that deal with a general bounded potential case for (1.1).Motivated by papers [1,25], in the present paper we consider problem (1.1) with positive and more general bounded potential V (x) by a dual approach and establish the existence of infinitely many high energy solutions under a concave-convex nonlinearity and different type weight functions h 1 (x), h 2 (x).It is easy to verify that for a general continuous and bounded function V (x), assumptions (A 1 ) -(A 6 ) fail to hold.We shall use mountain pass theorem under the Cerami condition to study Eq.(1.1).
Throughout this paper, we always assume the potential V (x) ∈ C(R N ) and the weight function h 2 (x) ≥ 0, ≡ 0 in R N .Furthermore, we let C, C 1 , C 2 , . . .be positive generic constants that can change from line to line.
The main result in this paper is as follows.
This paper is organized as follows.In Sect.2, with a convenient change of variable, we set up the variational framework for (1.1).In Sect.3, we verify that the energy functional associated with (1.1) satisfies the Cerami condition.In Sect.4, the geometric conditions of the mountain pass theorem are verified, and the proof of Theorem 1.1 is given.

Variational setting of the equation
Let E = W 1,p (R N ) be the Sobolev spaces with the norm By hypothesis (H 1 ), it is equivalent to the standard norm in E. It is well known that there is a constant S > 0 such that From the approximation argument, we see that (2.2) holds on E.
We observe that the natural energy functional associated with Eq. (1.1) is given by where It should be pointed out that the functional I is not well defined in general in E. To overcome this difficulty, we employ an argument developed by Colin and Jeanjean [6] for the case p = 2 and Severo [24] for 1 < p ≤ N .We make the change of variables u = f (v) or v = f -1 (u), where f is defined by and by f (t) = -f (-t) on (-∞, 0].Then we have the following.

Lemma 2.1 The function f (t) satisfies the following properties:
(f 1 ) f is uniquely defined, odd, increasing, and invertible in R; There exists a ∈ (0, (2α) 1/2αp ] such that f (t) Proof The proof of properties (f 1 ) -(f 8 ) can be found in [24](for the case 1 < p ≤ N and α = 1) and in [25] (for the case p = 2 and 1 2 < α ≤ 1).For the case 1 < p < N and α > 1 2 , the proof of (f 1 ) -(f 8 ) is similar and omitted.Here we prove (f 9 ).Note that Then For the second integral in (2.8), we take s = t + ξ and h(s) Similarly, we have f (nt) ≤ nf (t) for t ≥ 0 and n ∈ N. Since f (t) is odd and increasing in R, we obtain (2.6).
So, after the change of variables, we can write I(u) as which is well defined on E under assumptions (H 0 ) -(H 4 ).
As in [24], we observe then v is a weak solution of the equation and u = f (v) is a weak solution of (1.1).By using Theorem 1 in [23], we can conclude that v is locally bounded in R N .So, we consider the existence of solutions to (2.12) in E.

The boundedness of the Cerami sequences
To obtain the existence of solutions to problem (2.12), we need to prove that the functional J defined by (2.10) satisfies the Cerami condition.We first recall that a sequence The functional J satisfies the Cerami condition if any Cerami sequence possesses a convergent subsequence in Proof Without loss of generality, we assume f (v n (x)) .Then, using (f 2 ) and (f 5 ) in Lemma 2.1, we have This estimate and the assumption m ∈ (1, p) prove that { ∇v n p } is bounded.Moreover, where In the following, we show that there exists a constant C 0 > 0 such that We argue by contradiction and assume that, up to a subsequence, v n ∈ E such that which shows Moreover, since we conclude Similar to the idea of [25], we assert that for each ε > 0 there exists Otherwise, there are ε 0 > 0 and subsequence {v n k } ⊂ {v n } such that | n k | ≥ ε 0 , where (3.12) By (f 8 ), one sees as k → ∞.This is a contradiction.Hence the assertion is true.
) and (f 9 ), we get for some C 2 > 0. Thus, as n → ∞, On the other hand, from the integral absolute continuity, it follows that there is δ > 0 such that whenever ⊂ R N and | | < δ, For this δ > 0, we have Letting n → ∞, one sees from (3.11) and (3.17) that 1 ≤ 1 2 .It is impossible.So, (3.6) is true and {v n } is bounded in E.
Since the sequence {v n } given by Lemma 3.1 is a bounded sequence in E, there exist a constant M > 0 and v ∈ E, and a subsequence of {v n }, still denoted by {v n }, such that and for any r > 0, where On the other hand, we see from Hölder's inequality and (2.2) that as r → ∞.By Fatou's lemma, we obtain Then, the application of (3.21)- (3.23) gives that (3.19).Similarly, noticing that (f 6 ) and and Proof If (H 3 ) is satisfied, we use a similar argument in the proof of Lemma 3.2 to get limits (3.24) and (3.25).We now assume (H 4 ).Choose t ∈ (0, 1) such that q = 2α(pt + (1t)p * ). Then and Moreover, it follows from (3.18) that for all r > 0, Then the application of (3.26)-(3.28)yields (3.24).Similarly, from (f 6 ), it follows that and

Lemma 3.4
Assume that all hypotheses in Theorem 1.1 hold.Let {v n } be a Cerami sequence and satisfy (3.18).Then the following statements hold: (i).For each ε > 0, there exists r 0 ≥ 1 such that r ≥ r 0 , and (ii).The weak limit v ∈ E is a critical point for functional J.
In the following, we prove (3.35).We first note that (f 6 ) and (3.38) show This shows that there exists a constant r 0 ≥ 1 such that and consequently and then for every ε > 0. Therefore, limit (3.35) is true.The proof of part (i) is completed.(ii).From (3.18), one sees that as n → ∞ As in the proof of (i), we can derive as n → ∞ By the dense C ∞ 0 (R N ) in E, we have J (v)ϕ = 0, ∀ϕ ∈ E. In particular, J (v)v = 0. Hence, v is a critical point of J in E. This completes the proof of Lemma 3.4.

Lemma 3.5
Assume that all hypotheses in Theorem 1.1 hold.Let {v n } be a Cerami sequence and satisfy (3.18).Then v n → v in E, that is, the functional J satisfies the Cerami condition in E. The application of Brezis-Lieb lemma in [14] yields As in the proof of (3.6), we see that Clearly, it follows from (3.57) and (3.58) that, to conclude v n → v in E, it remains to prove Indeed, by Fatou's lemma, for any r > 0, we have (3.60) On the other hand, from (3.34), one sees Altogether, we get (3.59) and v n → v in E. This completes the proof of Lemma 3.5.

Proof of Theorem 1.1
We need the following mountain pass theorem to prove our result.
Lemma 4.1 ([21], Theorem 9.12).Let E be an infinite dimensional real Banach space, J ∈ C 1 (E, R) be even and satisfy the Cerami condition, and J(0 Then J possesses an unbounded sequence of critical values. Proof of Theorem 1.1 Clearly, the functional J defined by (2.10) is even in E. By Lemmas 3.1-3.5 in Sect.3, the functional J satisfies the Cerami condition.Next, we prove that J satisfies (J 1 ) and (J 2 ).From (f 5 ) and Hölder's inequality, we deduce that with some constant C 1 > 0. Similarly, if (H 3 ) is true, then one sees that If (H 4 ) holds, one has Moreover, it follows from (f 3 ), (f 5 ) and Hölder's inequality that with some C 2 > 0 and h 0 = h 2 L ∞ (B c 1 ) , q 0 = pt + p * (1t), t = 2αp * -q 2αp * -p .Clearly, q 0 > q 2α .Then (4.3) and (4.4) show that there is a constant C 3 > 0 such that As in the proof of (3.6), we can derive Then, from (4.1),(4.2),and (4.5), we conclude that where Choose z 1 ∈ (0, 1) such that We now verify (J 2 ).For any finite dimensional subspace E 0 ⊂ E, we assert that there exists a constant R 0 > ρ such that J < 0 on E 0 \ B R 0 .Otherwise, there is a sequence {v n } ⊂ E 0 such that v n E → ∞ and J(v n ) ≥ 0. Hence, On the other hand, from (f 7 ), we derive |v n | q 2α -p ω p n dx → ∞ as n → ∞ (4.14) Therefore, But it is easy to see that We have a contradiction from (4.12), (4.15), and (4.16).So, | | = 0 and ω(x) = 0 a.e. on R N .By the equivalency of all norms in E 0 , there exists a constant β > 0 such that R N |h 2 ||v| q dx 1/q ≥ β v E , ∀v ∈ E 0 , and |h 2 ||v n | q dx ≥ β q v n q E , ∀n ∈ N. (4.17) Hence, It is impossible.This shows that there is a constant R 0 > 0 such that J < 0 on E 0 \ B R 0 .Therefore, the existence of infinitely many solutions {v n } for problem (2.12) follows from Lemma 4.1, and so u n = f (v n ) is a solution of Problem (1.1) for n = 1, 2, . . . .We finish the proof of Theorem 1.1.