Electronic Journal of Qualitative Theory of Differential Equations2017, No. 29, 1–18; doi: 10.14232/ejqtde.2017.1.29 http://www.math.u-szeged.hu/ejqtde/

Existence of infinitely many radial and non-radialsolutions for quasilinear Schrödinger equations with

general nonlinearity

Jianhua Chen1, Xianhua TangB 1, Jian Zhang2 and Huxiao Luo1

1School of Mathematics and Statistics, Central South UniversityChangsha, 410083, Hunan, P. R. China

2School of Mathematics and Statistics, Hunan University of CommerceChangsha, 410205, Hunan, P.R. China

Received 31 December 2016, appeared 3 May 2017

Communicated by László Simon

Abstract. In this paper, we prove the existence of multiple solutions for the followingquasilinear 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 solutionsfor N ≥ 2, many non-radial solutions for N = 4 and N ≥ 6, and a non radial solutionfor N = 5. Our results generalize and extend some existing results.

Keywords: quasilinear Schrödinger equation, radial solutions, non-radial solutions,symmetric mountain pass theorem.

2010 Mathematics Subject Classification: 35J60, 35J20.

1 Introduction and preliminaries

This article deals mainly with the following quasilinear Schrödinger equation

− ∆u− u∆(|u|2) + V(|x|)u = f (|x|, u), x ∈ RN , (1.1)

where N ≥ 2, V : [0, ∞)→ R and f : [0, ∞)×R→ R.It is well known that Schrödinger equation has already found a great deal of interest in

recently years because not only it is very important for other fields to study the Schrödingerequation but also it provides a good model for developing mathematical methods. By virtueof variational methods, Schrödinger equation has been widely studied for multiplicity of non-trivial solutions over the past several years. See, e.g., [4,6,10,12,13,24,29,34] and the referencesand quoted in them. However, quasilinear Schrödinger equation is taken as a generalisation

BCorresponding author. Email: tangxh@mail.csu.edu.cn

2 J. Chen, X. Tang, J. Zhang and H. Luo

of the Schrödinger equation. Some authors studied the multiplicity of solutions for quasi-linear problem. See, e.g., [1, 20, 23, 31] and the references and quoted in them. In the mostof the aforementioned references, there are rarely papers to study the radial and non-radialsolutions for quasiliner and semilinear Schrödinger equation which has the properties of ra-dial 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 solutionsfor semilinear Schrödinger equation when sN = [N−1

2 ] + (−1)N , N ≥ 4, where the elements indifferent sequences cannot be compared from symmetrical point of view. The idea comes fromthe solution of the Rubik cube, and it has been extended to Heisenberg groups by Kristály andBalogh [15]. Based on this fact, recently, Yang et al. [30] first studied infinitely many radialand non-radial solutions for the problem (1.1) under the following assumptions on V and f :

(V) V ∈ C([0, ∞), R) ∩ L∞([0, ∞), R) and 0 < V0 := infr≥0 V(r) ≤ V(r) for all r ≥ 0.

( f1) f ∈ C([0, ∞)×R, R), and there exist c > 0 and 4 < p < 22∗ such that

| f (r, u)| ≤ c(|u|+ |u|p−1) for any r ≥ 0 and u ∈ R,

where 2∗ = 2NN−2 if N ≥ 3 and 2∗ = ∞ if N = 2.

( f2) f (r, u) = o(|u|) as |u| → 0 uniformly in r.

( f3) There exists R > 0 such that

C0 = infx∈RN ,|u|≥R

F(|x|, u) > 0,

where F(r, u) =∫ u

0 f (r, s)ds.

( f4) There exists α > 4 such that

αF(r, u) ≤ u f (r, u) for any r ≥ 0 and u ∈ R.

( f5) f (r,−u) = − f (r, u) for any r ≥ 0 and u ∈ R.

Moreover, the authors gave the following theorems in [30]. (Note that ` is defined in (2.1) inthe rest paper.)

Theorem 1.1 ([30]). Assume that N ≥ 2, (V), ( f1)–( f5) hold. Then problem (1.1) has a sequence ofradial solutions un such that `(un)→ ∞ as n→ ∞.

Theorem 1.2 ([30]). Assume that N = 4 or N ≥ 6, (V), ( f1)–( f5) hold. Then problem (1.1) has asequence of non-radial solutions un such that `(un)→ ∞ as n→ ∞.

Theorem 1.3. [30] Assume that (V) and ( f1)–( f4) hold. If N = 5 and

( f6) for all z = (x, y) ∈ R×R4 and for all g ∈ O(R4)

f (|(x + 1, y)|, u) = f (|(x, g(y))|, u) and V(|(x + 1, y)|) = V(|(x, g(y))|),

where O(R4) is the orthogonal transform group in R4. Then problem (1.1) has a nontrivial non-radialsolution.

Existence of infinitely many radial and non-radial solutions 3

In 2013, Tang [29] gave some much weaker conditions and studied the existence of in-finitely many solutions for Schrödinger equation via symmetric mountain pass theorem withsign-changing potential. Using Tang’s conditions, some authors studied the existence ofinfinitely many solutions for different equations. See, e.g., [9, 16, 25, 32, 33, 35, 36] and thereferences 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 fractionalSchrödinger 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 gen-eral super-quartic conditions, and establish the existence of infinitely many radial and non-radial solutions by symmetric mountain pass theorem in [2, 26]. To state our results, we givethe following much weaker conditions:

(V ′) V ∈ C([0, ∞)) is bounded from below by a positive constant V0;

( f ′3) lim|u|→∞

|F(r,u)||u|4 = ∞, uniformly in r ∈ [0,+∞) and there exists r0 ≥ 0 such that

F(r, u) ≥ 0, ∀ u ∈ R, |u| ≥ r0;

( f ′4) F (r, u) := 14 u f (r, u)− F(r, u) ≥ 0, and there exist c0 > 0 and κ > max1, 2N

N+2 such that

|F(r, u)|κ ≤ c0|u|2κF (r, u), ∀ u ∈ R, |u| ≥ r0.

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 ′), ( f1), ( f2), ( f ′3), ( f ′4) and ( f5) hold. Then problem (1.1) hasa sequence of radial solutions un such that `(un)→ ∞ as n→ ∞.

Theorem 1.5. Suppose that N = 4 or N ≥ 6, (V ′), ( f1), ( f2), ( f ′3), ( f ′4) and ( f5) hold. Then problem(1.1) has a sequence of non-radial solutions un such that `(un)→ ∞ as n→ ∞.

Theorem 1.6. Suppose that N = 5, (V ′), ( f1), ( f2), ( f ′3), ( f ′4) and ( f6) hold. Then problem (1.1) hasa nontrivial non-radial solution.

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 vn. Butin (V ′), there is no need to assume that V ∈ L∞([0,+∞)), and we give a different approachto prove the boundedness of (C)c-sequence vn, which is different from Yang’s methods (see[30]). On the other hand, note that condition ( f ′4) is somewhat weaker than the condition ( f4).As for the specific examples, we can see the reference [29].

Remark 1.8. By conditions ( f ′3) and ( f ′4), we can get

F (r, u) ≥ 1c0

(|F(r, u)||u|2

)κ

→ ∞

uniformly in r as |u| → ∞.

4 J. Chen, X. Tang, J. Zhang and H. Luo

2 Variational framework and some lemmas

Before stating this section, we first recall the following important notions.As usual, for 1 ≤ s < +∞, let

‖u‖s =

(∫RN|u|s) 1

s

, u ∈ Ls(RN).

LetH1(RN) =

u ∈ L2(RN) : ∇u ∈ L2(RN)

with the norm

‖u‖H1 =

(∫RN

(|∇u|2 + u2)dx) 1

2

.

Let S be the best Sobolev constant

S‖u‖22∗ ≤

∫RN|∇u|2dx

for any u ∈ H1(RN).Our working spaces is defined by

H :=

u ∈ H1(RN) :∫

RNV(|x|)u2dx < ∞

with the inner product

(u, v)H =∫

RN(∇u∇v + V(|x|)uv)dx

and the norm‖u‖H = (u, u)

12H.

To this end, we define the functional by

J(u) =12

∫RN

[(1 + 2u2)|∇u|2 + V(|x|)u2]dx−∫

RNF(|x|, u)dx

and define the derivative of J at u in the direction of φ ∈ C∞0 (RN) as follows:

〈J(u), φ〉 =∫

RN[(1 + 2u2)∇u∇φ + |∇u|2uφ + V(|x|)uφ]dx−

∫RN

f (|x|, u)φdx.

In order to prove Theorem 1.4, we denote by E the space of radial functions of H, namely,

E := u ∈ H : u(x) = u(|x|) .

For the proof of Theorem 1.5, following [5], choose an integer 2 ≤ m ≤ N/2 with 2m 6=N− 1, and write the elements of RN = Rm ×Rm ×RN−2m as x = (x1, x2, x3) with x1, x2 ∈ Rm

and x3 ∈ RN−2m. Now, consider the action of

Gm := O(m)×O(m)×O(N − 2m)

on H and define bylu(x) = u(l−1x).

Existence of infinitely many radial and non-radial solutions 5

Let ς ∈ O(N) be involution given by ς(x1, x2, x3) = (x2, x1, x3). The action of G := id, ς on

Fix(Gm) := u ∈ H : lu = u, ∀ l ∈ Gm

is defined by

(lu)(x) =

u(x), if l = id,

−u(l−1x), if l = ς.

LetE := Fix(G) = u ∈ Fix(Gm) : hu = u, ∀ h ∈ G .

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 E → Ls(RN) are continuous

for s ∈ [2, 2∗] and the embeddings E → Ls(RN) are compact for s ∈ (2, 2∗) (see [26, Lemma 2]).It follows from the embedding E → Ls(RN) for s ∈ [2, 2∗] that

‖u‖s ≤ γs‖u‖E = γs‖u‖H, ∀ u ∈ E, s ∈ [2, 2∗].

We know that J is not well defined in general in E. To overcome this difficulty, we applyan argument developed by Liu et al. [17] and Colin and Jeanjean [11]. We make the change ofvariables by v = g−1(u), where g is defined by

g′(t) =1

(1 + 2g2(t))12

on [0, ∞) and g(t) = −g(−t) on (−∞, 0].

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→ 214 as t→ +∞;

(6) g(t)/2 ≤ tg′(t) ≤ g(t) for all t > 0;

(7) g2(t)/2 ≤ tg(t)g′(t) ≤ g2(t) for all t ∈ R;

(8) |g(t)| ≤ 21/4|t|1/2 for all t ∈ R;

(9) there exists a positive constant C such that

|g(t)| ≥

C|t|, if |t| ≤ 1,

C|t| 12 , if |t| ≥ 1;

(10) for each α > 0, there exists a positive constant C(α) such that

|g(αt)|2 ≤ C(α)|g(t)|2;

(11) |g(t)||g′(t)| ≤ 1√2.

6 J. Chen, X. Tang, J. Zhang and H. Luo

Hence, by making the change of variables, from J(u) we obtain the following functional

`(v) =12

∫RN

[|∇v|2 + V(|x|)g2(v)]dx−∫

RNF(|x|, g(v))dx, (2.1)

which is well defined on the space E. Similar to the proof of [30, 37], it is easy to see that` ∈ C1(E, R), and

〈`′(v), ω〉 =∫

RN[∇v∇ω + V(|x|)g(v)g′(v)ω]dx−

∫RN

f (|x|, g(v))g′(v)ωdx, (2.2)

for any ω ∈ E. Moreover, the critical points of ` are the weak solutions of the followingequation

−∆v =1√

1 + 2|g(v)|2( f (|x|, g(v))−V(|x|)g(v)) in RN .

We also know that if v is a critical point of the functional `, then u = g(v) is a critical point ofthe 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, The-orem 1.28]) as follows.

Lemma 2.2 ([22]). Assume that the action of the topological group G on the Hilbert space X isisometric. If Φ ∈ C1(X, R) is invariant and if u is a critical point of Φ restricted to Fix(G), then u isa critical point of Φ.

Therefore, from the above lemma, if v is a critical point of Φ := `|E, then v is a criticalpoint of `, i.e. u = g(v) is a solution of (1.1).

A sequence vn ⊂ E is said to be a (C)c-sequence if `(v)→ c and ‖`′(v)‖(1 + ‖vn‖)→ 0.` is said to satisfy the (C)c-condition if any (C)c-sequence has a convergent subsequence.

Lemma 2.3. Suppose that (V ′), ( f1), ( f2), ( f ′3) and ( f ′4) are satisfied. Then any (C)c-sequence vn of` is bounded.

Proof. Let vn be a (C)c-sequence, then we have

`(vn)→ c and 〈`′(vn), vn〉 → 0. (2.3)

Hence, by (6) in Lemma 2.1, there is a constant C1 > 0 such that

C1 ≥ `(vn)−12〈`′(vn), vn〉 ≥

∫RNF (|x|, g(vn))dx. (2.4)

Firstly, let S2n =

∫RN

(|∇vn|2 + V(|x|)g2(vn)

)dx. Next, we prove that there exists a constant

C2 > 0 such thatS2

n =∫

RN

(|∇vn|2 + V(|x|)g2(vn)

)dx ≤ C2.

Suppose to the contrary that

S2n =

∫RN

(|∇vn|2 + V(|x|)g2(vn)

)dx → ∞, as n→ ∞.

On the one hand, setting g(vn) := g(vn)Sn

, by (2) in Lemma 2.1, then ‖g(vn)‖E ≤ 1. Passingto a subsequence, we may assume that g(vn) σ in E, g(vn)→ σ in Ls(RN), 2 < s < 2∗, andg(vn)→ σ a.e. on RN .

Existence of infinitely many radial and non-radial solutions 7

By (2.1) and (2.3), we can get

limn→∞

∫RN

|F(|x|, g(vn))|S2

ndx =

12

. (2.5)

On the other hand, let ψn = g(vn)g′(vn)

, by (6) in Lemma 2.1, then there is a constant C3 > 0such that ‖ψn‖ ≤ C3‖vn‖E. Moreover, by (2.4), we know that there exists a constant C4 > 0such that

C4 ≥ `(vn)−14〈`′(vn), ψn〉

=14

∫RN

(g′(vn))2|∇vn|2dx +

14

∫RN

V(|x|)g2(vn)dx

+∫

RN

(14

f (|x|, g(vn))g(vn)− F(|x|, g(vn))

)dx

=14

∫RN

(g′(vn))2|∇vn|2dx +

14

∫RN

V(|x|)g2(vn)dx +∫

RNF (|x|, g(vn))dx,

which implies that ∫RNF (|x|, g(vn))dx ≤ C4. (2.6)

Let

η(r) := infF (x, g(vn)) | x ∈ RN with |g(vn)| ≥ r

,

for r > 0. By Remark 1.8, η(r)→ ∞ as r → ∞. For 0 ≤ a < b, let

Ωn(a, b) =

x ∈ RN : a ≤ |g(vn)| < b

and

Cba := inf

F (|x|, g(v))|g(v)|2 : x ∈ RN and v ∈ R with a ≤ |g(v)| < b

.

Since F (|x|, u) > 0 if u 6= 0, we have Cba > 0 and

F (|x|, g(vn)) ≥ Cba |g(vn)|2.

Hence, from (2.6) and the above inequality, we can get

C4 ≥∫

RNF (|x|, g(vn))dx

≥∫

Ωn(0,r)F (|x|, g(vn))dx +

∫Ωn(r,+∞)

F (|x|, g(vn))dx

≥∫

Ωn(0,r)F (|x|, g(vn))dx + η(r)meas(Ωn(r,+∞)),

which shows that meas(Ωn(r,+∞)) → 0 as r → ∞ uniformly in n. Thus, for any s ∈ [2, 22∗),

8 J. Chen, X. Tang, J. Zhang and H. Luo

by (11) in Lemma 2.1, Hölder’s inequality and Sobolev’s embedding, we get∫Ωn(r,+∞)

gs(vn)dx ≤(∫

Ωn(r,+∞)g22∗(vn)dx

) s22∗

(meas(Ωn(r,+∞)))22∗−s

22∗

=1

Ssn

(∫Ωn(r,+∞)

g22∗(vn)dx) s

22∗

(meas(Ωn(r,+∞)))22∗−s

22∗

≤ C4

Ssn

(∫Ωn(r,+∞)

|∇g2(vn)|2dx) s

4

(meas(Ωn(r,+∞)))22∗−s

22∗

≤ C5

Ssn

(∫Ωn(r,+∞)

|∇vn|2dx) s

4

(meas(Ωn(r,+∞)))22∗−s

22∗

≤ C5

Ss2n

(meas(Ωn(r,+∞)))22∗−s

22∗ → 0

(2.7)

as r → ∞ uniformly in n.If σ = 0, then g(vn) → 0 in Ls(RN) for all s ∈ (2, 2∗), and g(vn) → 0 a.e. in RN . By virtue

of ( f2), we can find some number r1 > 0 such that r0 > r1 and

| f (|x|, u)| < ε|u|, for |u| ≤ r1,

where r0 is given in ( f ′3). Then

∫Ωn(0,r1)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx ≤∫

Ωn(0,r1)

(ε2 |g(vn)|2

|g(vn)|2

)|g(vn)|2dx

≤ ε

2‖g(vn)‖2

2.

(2.8)

It follows from (2.4) that

C1 ≥∫

Ωn(0,r1)F (|x|, g(vn))dx +

∫Ωn(r1,r0)

F (|x|, g(vn))dx +∫

Ωn(r0,+∞)F (|x|, g(vn))dx.

≥∫

Ωn(0,r1)F (|x|, g(vn))dx + Cr0

r1

∫Ωn(r1,r0)

|g(vn)|2dx +∫

Ωn(r0,+∞)F (|x|, g(vn))dx,

thus we have ∫Ωn(r1,r0)

|g(vn)|2dx ≤ C1

Cr0r1

. (2.9)

By ( f1) and ( f2), for any ε > 0, there exists a Cε > 0 such that

| f (r, u)| ≤(

ε|u|+ Cε|u|p−1)

and |F(r, u)| ≤(

ε

2|u|2 + Cε

p|u|p

), (2.10)

and then∫Ωn(r1,r0)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx ≤∫

Ωn(r1,r0)

(ε2 |g(vn)|2 + Cε

p |g(vn)|p

|g(vn)|2

)|g(vn)|2dx

≤(

ε

2+

Cε

prp−2

0

) ∫Ωn(r1,r0)

|g(vn)|2dx.

(2.11)

By using (2.9), we have∫Ωn(r1,r0)

|g(vn)|2dx ≤ 1S2

n

∫Ωn(r1,r0)

|g(vn)|2dx ≤ 1S2

n

C1

Cr0r1

. (2.12)

Existence of infinitely many radial and non-radial solutions 9

Therefore, it follows from (2.11) and (2.12) that

∫Ωn(r1,r0)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx ≤(

ε

2+

Cε

prp−2

0

)1

S2n

C1

Cr0r1

→ 0, as n→ ∞. (2.13)

Let κ′ = κ/(κ− 1). Since κ > max1, 2NN+2, we obtain 2κ′ ∈ (2, 22∗). Hence from ( f ′4), (2.4)

and (2.7), one has

∫Ωn(r0,∞)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx

≤[∫

Ωn(r0,∞)

(|F(|x|, g(vn))||g(vn)|2

)κ

dx] 1

κ[∫

Ωn(r0,∞)|g(vn)|2κ′dx

] 1κ′

≤ c1κ0

[∫Ωn(r0,∞)

F (|x|, g(vn))dx] 1

κ(∫

Ωn(r0,∞)|g(vn)|2κ′dx

) 1κ′

≤ [C1c0]1κ

(∫Ωn(r0,∞)

|g(vn)|2κ′dx) 1

κ′

→ 0.

(2.14)

Thus it follows from (2.8), (2.13) and (2.14) that

∫R3

|F(|x|, g(vn))|S2

ndx =

∫Ωn(0,r1)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx

+∫

Ωn(r1,r0)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx

+∫

Ωn(r0,∞)

|F(|x|, g(vn))||g(vn)|2

|g(vn)|2dx

→ 0, as n→ ∞,

(2.15)

which contradicts (2.5).

Now, we consider the case σ 6= 0. Set A :=

x ∈ RN : σ(x) 6= 0

. Thus meas(A) > 0. Fora.e. x ∈ A, we have lim

n→∞|g(vn(x))| = ∞. Hence A ⊂ Ωn(r0, ∞) for large n ∈ N, where r0 is

given in ( f ′3). By ( f ′3), we can get

limn→∞

F(|x|, g(vn))

|g(vn)|4= +∞.

It follows from Fatou’s Lemma that

limn→∞

∫A

F(|x|, g(vn))

|g(vn)|4dx = +∞. (2.16)

10 J. Chen, X. Tang, J. Zhang and H. Luo

Hence, from (2.3), (2.10) and (2.16), we can get

0 = limn→∞

c + o(1)S2

n= lim

n→∞

`(vn)

S2n

= limn→∞

1S2

n

(12

∫R3

[|∇vn|2 + V(|x|)g2(vn)

]dx−

∫RN

F(|x|, g(vn))dx)

= limn→∞

[12−∫

Ωn(0,r0)

F(|x|, g(vn))

g2(vn)|g(vn)|2dx−

∫Ωn(r0,∞)

F(|x|, g(vn))

g2(vn)|g(vn)|2dx

]≤ 1

2+ (ε + Cεr

p−20 )|g(vn)|22 − lim inf

n→∞

∫A

F(|x|, g(vn))

g4(vn)|g(vn)g(vn)|2dx

≤ 12+ (ε + Cεr

p−20 )γ2

2‖g(vn)‖2E − lim inf

n→∞

∫A

F(|x|, g(vn))

g4(vn)|g(vn)g(vn)|2dx

≤ 12+ (ε + Cεr

p−20 )γ2

2 − lim infn→∞

∫A

F(|x|, g(vn))

g4(vn)|g(vn)g(vn)|2dx

→ −∞,

(2.17)

which is a contradiction. Thus there exists C2 > 0 such that

S2n =

∫RN

(|∇vn|2 + V(|x|)g2(vn)

)dx ≤ C2.

Next, we prove vn is bounded in E, i.e. we only need to prove that there exists C7 > 0such that

S2n =

∫RN

(|∇vn|2 + V(|x|)g2(vn)

)dx ≥ C7‖vn‖2

E. (2.18)

Now, we may assume that vn 6= 0 (otherwise, the conclusion is trivial). If this conclusion isnot true, passing to a subsequence, we have S2

n‖vn‖2

E→ 0. Let ωn = vn

‖vn‖Eand hn = g2(vn)

‖vn‖2E

. Then

limn→∞

∫RN

(|∇ωn|2 + V(|x|)hn(x)

)dx = 0.

Thus ∫RN|∇ωn|2dx → 0,

∫RN

V(|x|)hn(x)dx → 0 and∫

RNV(|x|)ω2

n(x)dx → 1.

Similar to the idea of [23], we assert that for each ε > 0, there exists C8 > 0 independent of nsuch that meas(Θn) < ε, where Θn := x ∈ RN : |vn(x)| ≥ C8. Otherwise, there is an ε0 > 0and a subsequence vnk of vn such that for any positive integer k,

meas(

x ∈ RN : |vn(x)| ≥ k)≥ ε0 > 0.

Set Θnk := x ∈ RN : |vnk(x)| ≥ k. By (9) in Lemma 2.1, we have

S2nk≥∫

RNV(|x|)g2(vnk)dx ≥

∫Θnk

V(|x|)g2(vnk)dx ≥ C9kε0 → +∞,

as k → ∞, which is a contradiction. Hence the assertion is true. Notice that as |vn(x)| ≤ C8,by (9) and (10) in Lemma 2.1, we have

C10v2n ≤ g2(

1C8

vn) ≤ C11g2(vn).

Existence of infinitely many radial and non-radial solutions 11

Thus∫RN\Θn

V(|x|)ω2ndx ≤ C12

∫RN\Θn

V(|x|) g2(vn)

‖vn‖2E

dx ≤ C12

∫RN

V(|x|)hn(x)dx → 0. (2.19)

At last, by virtue of the integral absolutely continuity, there exists ε > 0 such that wheneverA′ ⊂ RN and meas(A′) < ε, ∫

A′V(|x|)ω2

ndx ≤ 12

. (2.20)

It follows from (2.19) and (2.20) that∫RN

V(|x|)ω2ndx =

∫RN\Θn

V(|x|)ω2ndx +

∫Θn

V(|x|)ω2ndx ≤ 1

2+ on(1).

This yields that 1 ≤ 12 , which is a contradiction. This implies that (2.18) holds. Hence vn is

bounded in E.

Lemma 2.4. Suppose that (V ′), ( f1), ( f2), ( f ′3) and ( f ′4) are satisfied. Then ` satisfies (C)c-condition.

Proof. By Lemma 2.3, it can conclude that vn is bounded in E. Going if necessary to asubsequence, we can assume that vn v in E. By the embedding, we have vn → v in Ls(R3)

for all 2 < s < 2∗.Firstly, we prove that there exists C13 > 0 such that∫

RN

(|∇(vn − v)|2 + V(|x|)(g(vn)g′(vn)− g(v)g′(v))(vn − v)

)dx ≥ C13‖vn − v‖2

E. (2.21)

Indeed, we may assume vn 6= v (otherwise the conclusion is trivial). Set

ωn =vn − v‖vn − v‖E

and hn =g(vn)g′(vn)− g(v)g′(v)

vn − v.

We argue by contradiction and assume that∫RN|∇ωn|2 + V(|x|)hn(x)ω2

ndx → 0.

Byddt(g(t)g′(t)) = g(t)g′′(t) + (g′(t))2 =

1(1 + 2g2(t))2 > 0,

then g(t)g′(t) is strictly increasing and for each C14 > 0 there is δ > 0 such that

ddt(g(t)g′(t)) ≥ δ

as |t| ≤ C14. Thus we can see that hn(x) is positive. Hence∫RN|∇ωn|2dx → 0,

∫RN

V(|x|)hn(x)ω2ndx → 0 and

∫RN

V(|x|)ω2n(x)dx → 1.

By a similar fashion as (2.19) and (2.20), we can get a contradiction. This implies that (2.21)holds.

12 J. Chen, X. Tang, J. Zhang and H. Luo

Secondly, by (2), (3), (8), (11) in Lemma 2.1 and (2.10), we have∣∣∣∣∫RN

(f (|x|, g(vn))g′(vn)− f (|x|, g(v))g′(v)

)(vn − v)dx

∣∣∣∣≤∫

RN

(ε|g(vn)||g′(vn)|+ Cε|g(vn)|p−1|g′(vn)|

+ε|g(v)||g(v)|+ Cε|g(v)|p−1|g′(v)|)|vn − v|dx

≤ εC15 + Cε

(‖vn‖

p−22

p2

+ ‖v‖p−2

2p2

)‖vn − v‖ p

2= on(1).

(2.22)

Hence together with (2.21) and (2.22), we get

on(1) = 〈`(un)− `(u), un − u〉

=∫

R3|∇(un − u)|2dx +

∫RN

V(|x|)(

g(vn)g′(vn)− g(v)g′(v))(vn − v)dx

−∫

RN

(f (|x|, g(vn))g′(vn)− f (|x|, g(v))g′(v)

)(vn − v)dx

≥ C13‖vn − v‖2E + on(1).

This implies vn → v in E and this completes the proof.

3 Proof of Theorem 1.4 and Theorem 1.5

To prove our results, we state the following symmetric mountain pass theorem.

Lemma 3.1 ([2, 26]). Let X be an infinite dimensional Banach space, X = Y ⊕ Z, where Y is finitedimensional. If ` ∈ C1(X, R) satisfies (C)c-condition for all c > 0, and

(`1) `(0) = 0, `(−u) = `(u) for all u ∈ X;

(`2) there exist constants ρ, α > 0 such that `|∂Bρ∩Z ≥ α;

(`3) for any finite dimensional subspace X ⊂ X, there is R = R(X) > 0 such that `(u) ≤ 0 onX\BR;

then ` possesses an unbounded sequence of critical values.

Let ej is a total orthonormal basis of E and define Xj = Rej,

Yk =k⊕

j=1

Xj, Zk =∞⊕

j=k+1

Xj, ∀ k ∈ Z.

Then E = Yn ⊕ Zn, Yn is a finite dimensional space.

Lemma 3.2. Suppose that (V ′), ( f1), ( f2), ( f ′3) and ( f ′4) are satisfied. Then there exist constant ρ, α > 0such that

`|Sρ∩Zm ≥ α.

Existence of infinitely many radial and non-radial solutions 13

Proof. From (2.1), (3) and (8) in Lemma 2.1, for u ∈ Zm and p ∈ (4, 22∗), we can choose ε smallenough such that

`(v) =12

∫RN

[|∇v|2 + V(|x|)g2(v)]dx−∫

RNF(|x|, g(v))dx

≥ C14

2‖g(v)‖2

E −ε

2

∫RN|g(v)|2dx− Cε

p

∫RN|g(v)|pdx

≥ C14

2‖g(v)‖2

E −ε

2

∫RN|g(v)|2dx− C′ε

p

∫RN|g(v)|

p2 dx

≥ C15

(12‖g(v)‖2

E −14‖g(v)‖2

E −14‖g(v)‖

p2E

)≥ C15

4‖g(v)‖2

E

(1− ‖g(v)‖

p−42

E

)> 0.

This completes the proof.

Lemma 3.3. Suppose that (V ′), ( f1), ( f2), ( f ′3) and ( f ′4) are satisfied. Then for any finite dimensionalsubspace E ⊂ E, there exists constant R = R(E) > 0 such that

`(v)→ −∞, ‖v‖E → ∞, v ∈ E.

Proof. Arguing indirectly, assume that for some sequence vn ⊂ E with ‖vn‖E → ∞, there isM > 0 such that `(vn) ≥ −M for all n ∈N. On the one hand, let ωn = vn

‖vn‖E, then ‖ωn‖E = 1.

Since E is finite dimensional, passing to a subsequence, then we assume that

ωn ω in E,

ωn → ω in Ls(RN) for 2 < s < 2∗,

ωn → ω a.e. RN ,

and so ‖ω‖E = 1, which implies that ω 6= 0. Let Λ = x ∈ RN : ω(x) 6= 0, then meas(Λ) > 0.Since |vn| = |ωn|‖vn‖E, by ‖vn‖E → ∞ and (4) in Lemma 2.1, then we have |g(vn)| → ∞.Therefore, by (2) in Lemma 2.1, we have

S2n =

∫RN

(|∇vn|2 + V(|x|)g2(vn)

)dx

≥∫

Λ

(|∇g(vn)|2 + V(|x|)g2(vn)

)dx

→ ∞.

On the other hand, set g(vn) =g(vn)

Sn, then ‖g(vn)‖E ≤ 1. Passing to a subsequence, we may

assume that g(vn) σ in E, g(vn) → σ in Ls(RN) for all 2 < s < 2∗, g(vn) → σ a.e. on RN ,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 ′), ( f1), ( f2), ( f ′3) and ( f ′4) are satisfied. Then for any E ⊂ E, thereexists R = R(E) > 0, such that

`(v) ≤ 0, ‖v‖E ≥ R, ∀ v ∈ E.

14 J. Chen, X. Tang, J. Zhang and H. Luo

Proof of Theorem 1.4. Let X = E, Y = Ym and Z = Zm. By Lemmas 2.3, 2.4, 3.2 and Corol-lary 3.4, all conditions of Lemma 3.1 are satisfied. Thus, problem (1.1) possesses has a se-quence of radial solutions vn such that `(vn) → ∞ as n → ∞, where un = g(vn). Thiscompletes the proof.

Proof of Theorem 1.5. Using a similar way as Theorem 1.4, we can complete the proof of Theo-rem 1.5.

4 Proof of Theorem 1.6

In this section, we want to prove Theorem 1.6. Before proving our results, we need thefollowing mountain pass theorem without compactness (see [22], Theorem 1.15)

Lemma 4.1 ([22]). Let X be an Hilbert space, ` ∈ C1(X, R), e ∈ X, and r > 0 such that ‖e‖ > r andinf‖v‖=r `(v) > `(0) ≥ `(e). Then there exists a sequence vn such that `(vn)→ c and `′(vn)→ 0,where

c = infγ∈Γ

maxt∈[0,1]

`(γ(t)) > 0,

and Γ = γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e.

The following lemma, has been proved in [30], which is very useful for the proof of Theo-rem 1.6.

Lemma 4.2 ([30]). Let Ωjj∈N be a sequence of open subsets of R such that

(1) R = ∪j∈NΩj and Ωi ∩Ωj = ∅, if i 6= j.

(2) There exists a constant c0 > 0 such that for j ∈N

‖v‖L

103 (Ωj×R4)

≤ c0‖v‖H1(Ωj×R4), ∀ v ∈ H1(Ωj ×R4).

Let vn be a bounded sequence of H1(R5). If

supj∈N

∫Ωj×R4

|vn|q → 0, when n→ ∞, f or q ∈[

2,103

),

then vn → 0 in Ls(R5), for all 2 < s < 103 .

For the proof of Theorem 1.6, following [19], let G be a subgroup of O(R4). It is obviousthat R4 is compatible with G if for some r > 0,

m(y, r, G) = lim|y|→∞

m(y, r, G) = +∞,

where

m(y, r, G) := supn∈N

n ∈N : ∃ g1, g2, . . . , gn ∈ G such that i 6= j⇒ B(gi(y)) ∩ B(gj(y)) = ∅

.

Note that R4 is compatible with O(R4) and O(R2) × O(R2) (see [22]). For simplicity ofnotation, we denote G := O(R2)×O(R2). We consider the action of G on H1(R5), defined by

(lu)(x, y) = u(x, l−1y), where (x, y) ∈ R×R4, l ∈ O(R4).

Existence of infinitely many radial and non-radial solutions 15

LetH1

G(R5) :=

u ∈ H1(R5) : lu = u, ∀ l ∈ G

and ς ∈ O(N) be the involution in R5 = R×R2 ×R2 given by ς(x1, x2, x3) = (x2, x1, x3). Wedefine an action of the group G1 := id, ς on H1(R5) by

hu(x) =

u(x), if h = id,

−u(h−1x), if h = ς.

LetH1

G1(R5) :=

u ∈ H1(R5) : lu = u, ∀ l ∈ G1

.

Set E := H1G(R

5) ∩ H1G1(R5). It is clear that u = 0 is only radial function in E, which is a

Hilbert space with the inner product of H1(R5).The following compactness result is due to [19].

Lemma 4.3 ([19]). The imbedding H1G(R

5) → Ls(Ω×R4) is compact, where Ω is a bounded subsetof R, s ∈ (2, 10

3 ).

Proof of Theorem 1.6. Similar to Lemma 3.2 and 3.3, it is easy to verify that ` satisfies the con-ditions of Lemma 4.1. Then, there exists a (C)c-sequence vn ⊂ E i.e., vn satisfies

`(vn)→ c and (1 + ‖vn‖)`′(vn)→ 0,

where c is the Mountain Pass value given in Lemma 4.1. Similarly as in [30,37], by Lemma 2.1,we have

on(1) = ‖`′(vn)‖(1 + ‖vn‖)≥ 〈`′(vn), vn〉

=∫

R5|∇vn|2 +

∫R5

V(|x|)g(vn)g′(vn)vn −∫

R5f (|x|, g(vn))g′(vn)vn

≥ C‖vn‖2 − C∫

R5|vn|

p2 .

(4.1)

Hence, without loss of generality, we can choose δ > 0 such that for each n∫R5|vn|

p2 ≥ ‖vn‖2 ≥ δ. (4.2)

Otherwise, (4.1) implies that vn → 0 in E and hence c = 0, which leads to a contradiction. LetΩj = (j, j + 1), then R = ∪j∈NΩj. We may claim that there exists $ > 0 such that

supj∈N

∫Ωj×R4

vn(x, y)dxdy ≥ 2$ > 0.

Otherwise, by Lemma 4.2, we have vn → 0 in in Ls(R5), where 2 < s < 103 , which contradicts

(4.2) since 2 < p2 < 2∗. Hence, for every n, there exists jn such that∫

Ωjn×R4vn(x, y)dxdy ≥ $ > 0.

Making the change of variable x = x′ + jn, one has∫Ω×R4

vn(x′ + jn, y)dx′dy ≥ $ > 0.

16 J. Chen, X. Tang, J. Zhang and H. Luo

where Ω = (0, 1). Let wn(x′, y) = vn(x′ + jn, y), then∫Ω×R4

wn(x′, y)dx′dy ≥ $ > 0. (4.3)

Note that vn is also a (C)c sequence of `. Hence

wn w in E,

wn → w in Lsloc(R

5), where 2 < s <103

,

wn w a.e. on R5.

It is standard to prove that w is a critical point of `. Moreover, (4.3) implies w 6= 0, i.e., theproblem (1.1) has a nonradial solution v = g(w). This completes the proof.

Acknowledgements

The authors thank the editor and the referees for their valuable comments and suggestions.This work was supported by National Natural Science Foundation of China (Grant No.11461043, 11571370 and 11601525), and supported partly by the Provincial Natural ScienceFoundation of Jiangxi, China (20161BAB201009) and the Science and Technology Project ofEducational Commission of Jiangxi Province, China (150013) and Hunan Provincial Innova-tion Foundation For Postgraduate (Grant No. CX2016B037).

References

[1] C. O. Alves, G. M. Figueiredo, Multiple solutions for a quasilinear Schrödinger equationon RN , Acta. Appl. Math. 136(2015), 91–117. MR3318825; url

[2] T. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applicationsto some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7(1983),241–273. MR0713209; url

[3] T. Bartsch, M. Willem, Infinitely many radial solutions of a semilinear elliptic problemon RN , Arch. Ration. Mech. Anal. 124(1993), 261–276. MR1237913; url

[4] T. Bartsch, Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equa-tion, Z. Angew. Math. Phys. 51(2000), 366–384. MR1762697; url

[5] T. Bartsch, M. Willem, Infinitely many nonradial solutions of a Euclidean scalar fieldequation, J. Funct. Anal. 117(1993), 447–460. MR1244943; url

[6] T. Bartsch, Z. L. Liu, T. Weth, Sign changing solutions of superlinear Schrödinger equa-tions, Comm. Partal differential equations 29(2005), 25–42. MR2038142; url

[7] H. Berestycki, P. L. Lions, Nonlinear scalar field equations I, Arch. Ration. Mech. Anal.82(1983), 313–345. MR0695535; url

[8] H. Berestycki, P. L. Lions, Nonlinear scalar field equations II, Arch. Ration. Mech. Anal.82(1983), 347–375. MR695536; url

Existence of infinitely many radial and non-radial solutions 17

[9] B. T. Cheng, X. H. Tang, High energy solutions of modified quasilinear fourth-orderelliptic equations with sign-changing potential, Comput. Math. Appl. 73(2017), 27–36.MR3591225; url

[10] S. Cingolani, M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equa-tions with competing potential functions, J. Differential Equations 160(2000), 118–138.MR1734531; url

[11] M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual ap-proach, Nonlinear Anal. 56(2004), 213–226. MR2029068; url

[12] Y. Ding, S. Luan, Multiple solutions for a class of nonlinear Schrödinger equations, J. Dif-ferential Equations 207(2004), 423–457. MR2102671; url

[13] Y. Ding, C. Lee, Multiple solutions of Schrödinger equations with indefinite linear partand super or asymptotically linear terms, J. Differential Equations 222(2006), 137–163.MR2200749; url

[14] A. Kristály, G. Moros, anu, D. O’Regan, A dimension-depending multiplicity result fora perturbed Schrödinger equation, Dynam. Systems Appl. 22(2013), 325–335. MR3100207

[15] Z. Balogh, A. Kristály, Lions-type compactness and Rubik actions on the Heisenberggroup, Calc. Var. Partial Differential Equations 48(2013), 89–109. MR3090536; url

[16] X. Y. Lin, X. H. Tang, Existence of infinitely many solutions for p-Laplacian equations inRN , Nonlinear Anal. 92(2013), 72–81. MR3091109; url

[17] J. Q. Liu, Y. Wang, Z. Q. Wang, Solutions for quasilinear Schrödinger equations, II,J. Differential Equations 187(2003), 473–793. MR1949452; url

[18] X. Liu, F. Zhao, Existence of infinitely many solutions for quasilinear equations perturbedfrom symmetry, Adv. Nonlinear Stud. 13(2013), 965–978. MR3115148; url

[19] S. Lorca, P. Ubilla, Symmetric and nonsymmetric solutions for an elliptic equation onRN , Nonlinear Anal. 58(2004), 961–968. MR2086066; url

[20] L. Jeanjean, T. Luo, Z. Q. Wang, Multiple normalized solutions for quasilinearSchrödinger equations, J. Differential Equations 259(2015), 3894–3928. MR3369266; url

[21] J. Marcos do Ó, U. Severo, Solitary waves for a class of quasilinear Schrödingerequations in dimension two, Calc. Var. Partial Differential Equations 38(2010), 275–315.MR2647122; url

[22] M. Willem, Minimax theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. MR1400007;url

[23] X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Dif-ferential Equations 256(2014), 2619–2632. MR3160456; url

[24] J. Wang, D. C. Lu, J. X. Xu, F. B. Zhang, Multiple positive solutions for semilin-ear Schrödinger equations with critical growth in RN , J. Math. Phys. 56(2015), 041503.MR3390937; url

18 J. Chen, X. Tang, J. Zhang and H. Luo

[25] D. D. Qin, X. H. Tang, J. Zhang, Multiple solutions for semilinear elliptic equations withsign-changing potential and nonlinearity, Electron. J. Differential Equations 2013, No. 207,1–9. MR3119061

[26] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differentialequations, CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathe-matical Society, Providence, RI, 1986. MR0845785

[27] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys.55(1977), 149–162. MR0454365; url

[28] M. Struwe, Multiple solutions of differential equations without the Palais–Smale condi-tion, Math. Ann. 261(1982), 399–412. MR0679798; url

[29] X. H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl. 401(2013), 407–415. MR3011282;url

[30] X. Y. Yang, W. B. Wang, F. K. Zhao, Infinitely many radial and non-radial solutions to aquasilinear Schrödinger equation, Nonlinear Anal. 114(2015), 158–168. MR3300791; url

[31] J. Zhang, X. H. Tang, W. Zhang, Infinitely many solutions of quasilinear Schrödingerequation with sign-changing potential, J. Math. Anal. Appl. 420(2014), No. 2, 1762–1775.MR3240105; url

[32] J. Zhang, X. H. Tang, W. Zhang, Existence of infinitely many solutions for a quasilinearelliptic equation, Appl. Math. Lett. 37(2014), 131–135. MR3231740; url

[33] J. Zhang, X. H. Tang, W. Zhang, Existence of multiple solutions of Kirchhoff type equa-tion with sign-changing potential, Appl. Math. Comput. 242(2014), 491–499. MR3239677;url

[34] Q. Zhang, Q. Wang, Multiple solutions for a class of sublinear Schrödinger equations,J. Math. Anal. Appl. 389(2012), 511–518. MR2876516; url

[35] W. Zhang, X. H. Tang, J. Zhang, Infinitely many solutions for fourth-order elliptic equa-tions with sign-changing potential, Taiwanese J. Math. 18(2014), 645–659. MR3188523

[36] W. Zhang, X. H. Tang, J. Zhang, Multiple solutions for semilinear Schrödinger equa-tions with electromagnetic potential, Electron. J. Differential Equations 2016, No. 26, 1–9.MR3466497

[37] W. Zhang, X. H. Tang, J. Zhang, Infinitely many radial and non-radial solutions for afractional Schrödinger equation, Comput. Math. Appl. 671(2016), 737–747. MR3457376; url