Existence of infinitely many radial and non-radial solutions for quasilinear Schrödinger equations with general nonlinearity

In this paper, we prove the existence of multiple solutions for the following quasilinear Schrödinger equation −∆u− u∆(|u|2) + V(|x|)u = f (|x|, u), x ∈ RN . Under some generalized assumptions on f , we obtain infinitely many radial solutions for N ≥ 2, many non-radial solutions for N = 4 and N ≥ 6, and a non radial solution for N = 5. Our results generalize and extend some existing results.


Introduction and preliminaries
This article deals mainly with the following quasilinear Schrödinger equation of the Schrödinger equation.Some authors studied the multiplicity of solutions for quasilinear problem.See, e.g., [1,20,23,31] and the references and quoted in them.In the most of the aforementioned references, there are rarely papers to study the radial and non-radial solutions for quasiliner and semilinear Schrödinger equation which has the properties of radial symmetry except for the papers [3,5,7,8,18,19,27,28] and the references.Especially, in [14], Kristály et al. proved the existence of sequences of non-radial, sign-changing solutions for semilinear Schrödinger equation when s N = [ N−1 2 ] + (−1) N , N ≥ 4, where the elements in different sequences cannot be compared from symmetrical point of view.The idea comes from the solution of the Rubik cube, and it has been extended to Heisenberg groups by Kristály and Balogh [15].Based on this fact, recently, Yang et al. [30] first studied infinitely many radial and non-radial solutions for the problem (1.1) under the following assumptions on V and f : ( f 3 ) There exists R > 0 such that where F(r, u) = u 0 f (r, s)ds.
In 2013, Tang [29] gave some much weaker conditions and studied the existence of infinitely many solutions for Schrödinger equation via symmetric mountain pass theorem with sign-changing potential.Using Tang's conditions, some authors studied the existence of infinitely many solutions for different equations.See, e.g., [9,16,25,32,33,35,36] and the references quoted in them.These results generalized and extended some existing results.Especially, Zhang et al. [37] proved many radial and non-radial solutions for a fractional Schrödinger equation by using Tang's conditions and methods which are more weaker than (AR)-condition and super-quadratic conditions.
Inspired by the above references, we consider problem (1.1) with the following more general super-quartic conditions, and establish the existence of infinitely many radial and nonradial solutions by symmetric mountain pass theorem in [2,26].To state our results, we give the following much weaker conditions: Next, we are ready to state the main results of this paper.(Note that is defined later in (2.1).) Theorem 1.4.Suppose that N ≥ 2, (V ), ( f 1 ), ( f 2 ), ( f 3 ), ( f 4 ) and ( f 5 ) hold.Then problem (1.1) has a sequence of radial solutions {u n } such that (u n ) → ∞ as n → ∞.
Remark 1.7.On the one hand, note that the condition (V ) is weaker than (V).In (V), V ∈ L ∞ ([0, +∞)), it is very important for to prove the boundedness of (C) c -sequence {v n }.But in (V ), there is no need to assume that V ∈ L ∞ ([0, +∞)), and we give a different approach to prove the boundedness of (C) c -sequence {v n }, which is different from Yang's methods (see [30]).On the other hand, note that condition ( f 4 ) is somewhat weaker than the condition ( f 4 ).
As for the specific examples, we can see the reference [29].

Variational framework and some lemmas
Before stating this section, we first recall the following important notions.As usual, for 1 ≤ s < +∞, let with the norm .
Let S be the best Sobolev constant Our working spaces is defined by and the norm u H = (u, u) H .To this end, we define the functional by and define the derivative of J at u in the direction of φ ∈ C ∞ 0 (R N ) as follows: In order to prove Theorem 1.4, we denote by E the space of radial functions of H, namely, For the proof of Theorem 1.5, following [5], choose an integer 2 ≤ m ≤ N/2 with 2m = N − 1, and write the elements of on H and define by lu(x) = u(l −1 x).
Let ς ∈ O(N) be involution given by ς(x 1 , x 2 , x 3 ) = (x 2 , x 1 , x 3 ).The action of G := {id, ς} on Note that 0 is the only radially symmetric function in E for this case.
In both cases, E is a closed subspace of H, and the embedding We know that J is not well defined in general in E. To overcome this difficulty, we apply an argument developed by Liu et al. [17] and Colin and Jeanjean [11].We make the change of variables by v = g −1 (u), where g is defined by Let us recall some properties of the change of variables g : R → R which are proved in [11,17,21] as follows.
Lemma 2.1.The function g(t) and its derivative satisfy the following properties: (1) g is uniquely defined, C ∞ and invertible; (2) |g (t)| ≤ 1 for all t ∈ R; (3) |g(t)| ≤ |t| for all t ∈ R; (4) g(t)/t → 1 as t → 0; (5) g(t)/ √ t → 2 1 4 as t → +∞; (6) g(t)/2 ≤ tg (t) ≤ g(t) for all t > 0; there exists a positive constant C such that (10) for each α > 0, there exists a positive constant C(α) such that Hence, by making the change of variables, from J(u) we obtain the following functional which is well defined on the space E. Similar to the proof of [30,37], it is easy to see that ∈ C 1 (E, R), and for any ω ∈ E.Moreover, the critical points of are the weak solutions of the following equation We also know that if v is a critical point of the functional , then u = g(v) is a critical point of the functional J, i.e. u = g(v) is a solution of problem (1.1).
To prove our results, we need the principle of symmetric criticality theorem (see [22, Theorem 1.28]) as follows.

Lemma 2.2 ([22]
). Assume that the action of the topological group G on the Hilbert space X is isometric.
Proof.Let {v n } be a (C) c -sequence, then we have Hence, by (6) in Lemma 2.1, there is a constant C 1 > 0 such that Next, we prove that there exists a constant Suppose to the contrary that On the one hand, setting g(v n ) := g(v n ) S n , by (2) in Lemma 2.1, then g(v n ) E ≤ 1. Passing to a subsequence, we may assume that g
Proof.By Lemma 2.3, it can conclude that {v n } is bounded in E. Going if necessary to a subsequence, we can assume that v n v in E. By the embedding, we have Firstly, we prove that there exists C 13 > 0 such that Indeed, we may assume v n = v (otherwise the conclusion is trivial).Set We argue by contradiction and assume that then g(t)g (t) is strictly increasing and for each C 14 > 0 there is δ > 0 such that By a similar fashion as (2.19) and (2.20), we can get a contradiction.This implies that (2.21) holds.

Proof of Theorem 1.4 and Theorem 1.5
To prove our results, we state the following symmetric mountain pass theorem.
Lemma 3.3.Suppose that (V ), ( f 1 ), ( f 2 ), ( f 3 ) and ( f 4 ) are satisfied.Then for any finite dimensional subspace E ⊂ E, there exists constant R = R( E) > 0 such that Proof.Arguing indirectly, assume that for some sequence Since E is finite dimensional, passing to a subsequence, then we assume that and so ω E = 1, which implies that ω = 0. Let Λ = {x ∈ R N : ω(x) = 0}, then meas(Λ) > 0. Since Therefore, by (2) in Lemma 2.1, we have On the other hand, set g(v n ) = g(v n ) S n , then g(v n ) E ≤ 1. Passing to a subsequence, we may assume that g(v n ) σ in E, g(v n ) → σ in L s (R N ) for all 2 < s < 2 * , g(v n ) → σ a.e. on R N , and so σ E ≤ 1.Hence, we can conclude a contradiction by a similar fashion as (2.15) and (2.17).This completes the proof.Corollary 3.4.Suppose that (V ), ( f 1 ), ( f 2 ), ( f 3 ) and ( f 4 ) are satisfied.Then for any E ⊂ E, there exists R = R( E) > 0, such that ).This completes the proof.
Proof of Theorem 1.5.Using a similar way as Theorem 1.4, we can complete the proof of Theorem 1.5.

Proof of Theorem 1.6
In this section, we want to prove Theorem 1.6.Before proving our results, we need the following mountain pass theorem without compactness (see [22], Theorem 1.15) Lemma 4.1 ([22]).Let X be an Hilbert space, ∈ C 1 (X, R), e ∈ X, and r > 0 such that e > r and inf v =r (v) > (0) ≥ (e).Then there exists a sequence and The following lemma, has been proved in [30], which is very useful for the proof of Theorem 1.6.
For the proof of Theorem 1.6, following [19], let G be a subgroup of [22]).For simplicity of notation, we denote G := O(R 2 ) × O(R 2 ).We consider the action of G on H 1 (R 5 ), defined by and ς ∈ O(N) be the involution in R 5 = R × R 2 × R 2 given by ς(x 1 , x 2 , x 3 ) = (x 2 , x 1 , x 3 ).We define an action of the group G 1 := {id, ς} on H 1 (R 5 ) by Set ).It is clear that u = 0 is only radial function in E, which is a Hilbert space with the inner product of H 1 (R 5 ).
The following compactness result is due to [19].
Making the change of variable x = x + j n , one has Ω×R 4 v n (x + j n , y)dx dy ≥ > 0.

Ω×R 4 w
n (x , y)dx dy ≥ > 0. (4.3)Note that {v n } is also a (C) c sequence of .Hencew n w in E, w n → w in L s loc (R 5), where 2 < s < e. on R 5 .
Proof of Theorem 1.4.Let X = E, Y = Y m and Z = Z m .By Lemmas 2.3, 2.4, 3.2 and Corollary 3.4, all conditions of Lemma 3.1 are satisfied.Thus, problem (1.1) possesses has a sequence of radial solutions {v n } such that