Infinitely many nodal solutions for a class of quasilinear elliptic equations

Infinitely many nodal solutions for a class of quasilinear elliptic equations

Xiaolong Yang

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P. R. China

Received 19 December 2020, appeared 11 April 2021 Communicated by Patrizia Pucci

Abstract. In this paper, we study the existence of infinitely many nodal solutions for the following quasilinear elliptic equation

(−∇ ·φ⁰(|∇u|²)∇u

+|u|^α−2u= f(u), x∈R^N, u(x)→0, as|x| →_∞,

where N ≥ 2, φ(t) behaves liket^q/2 for small tand t^p/2 for large t, 1 < p < q < N, f ∈ C¹(_R⁺_,R) is of subcritical, q ≤ α ≤ p^∗q⁰/p⁰, let p^∗ = _N−p^{N p} _, p⁰ and q⁰ be the conjugate exponents respectively ofpandq. For any given integerk≥0, we prove that the equation has a pair of radial nodal solution with exactlyknodes.

Keywords: quasilinear elliptic equation, nodal solutions, multiple solutions.

2020 Mathematics Subject Classification: 35A15, 35J62, 46E30.

1 Introduction

In this paper, we consider the following quasilinear elliptic equation

− ∇ ·φ⁰(|∇u|²)∇u

+|u|^α−2u= f(u), x∈_R^N, (1.1) where N ≥ 2, φ ∈ C²(_R+,R+) has a different growth near zero and infinity. Quasilinear equation of form (1.1) can be transformed into different differential equations corresponding to various types of φ. For example, when φ(t) = 2[(1+t)¹² −1] and α = 2, equation (1.1) corresponds to the prescribed mean curvature equation or the capillary surface equation

−∇ · ∇u p1+|∇u|²

+u= f(u), x∈_R^N.

Such problem has been deeply studied since last century, under different assumptions on the nonlinearity f, the existence and nonexistence of solutions have been investigated by many authors, see [3,5,8,27] for example.

BEmail: yangxiaolong@mails.ccnu.edu.cn

Equation (1.1) also related to (p,q)-Laplacian equations. In fact, ifφ(t) = ²_pt^2p + ²_qt^q2, then equation (1.1) becomes

−_∆pu−_∆qu+|u|^α−2u= f(u) inR^N, (1.2) where∆pu= div(|∇u|^p−2∇u), 1< p <q< Nandα>2 satisfies some conditions. Equation (1.2) originates from the following reaction diffusion system

∂u

∂t =div

D(u)∇u

+c(x,u), (1.3)

where D(u) = (|∇u|^p−2+|∇u|^q−2). This system has a wide range of application in physics and related sciences such as biophysics, plasma physics and chemical reaction design. In such applications, the functionu describes a concentration; the first term on the right hand side of (1.3) corresponds to diffusion with a diffusion coefficientD(u), whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological appli- cations, the reaction termc(x,u)has a polynomial form with respect to the concentration u.

For more mathematical and physical background of equations (1.2)–(1.3), we refer the reader to the papers [9,24,25,31] and the references therein. In particular, when p = q = α = 2, equation (1.2) reduced to

−_∆u+u= f(u) _inR^N_{. (1.4)}

There has been plenty of results on the existence, nonexistence and multiplicity of positive or sign-changing solutions for equation (1.4), see [2,6,7,10,17] and the references therein.

If p = q = α 6= 2, then equation (1.2) becomes into the following general p-Laplacian equation

−_∆pu+|u|^p−2u= f(u) inR^N, (1.5) which was studied by many authors. Many results for equation (1.4) has been extended to equation (1.5). Deng, Guo and Wang in [12] proved the existence of nodal solutions for p- Laplacian equations with critical growth. Recently in [13], Deng, Li and Shuai studied the existence of solutions for a class of p-Laplacian equations with critical growth and potential vanishing at infinity.

Recently, Azzollini et al. [1] studied the following quasilinear elliptic equation (−∇ ·[φ⁰(|∇u|²)∇u] +|u|^α−2u=|u|^s−2u, x∈_R^N,

u(x)→0, as |x| →_∞, ^(1.6)

where N ≥ 2, φ(t) behaves like t^q/2 for small t and t^p/2 for large t, 1 < p < q < N, 1 <

α ≤ p^∗q⁰/p⁰ and max{q,α} < s < p^∗ = _N^{N p}_−p, with being p⁰, q⁰ are the conjugate exponents of p, q respectively. The authors in [1] found a sort of Orlicz–Sobolev space in which the energy functional is well defined. They also proved that the Orlicz–Sobolev space compactly embedded into certain Lebesgue spaces. Then, they obtained the existence of a sequence of nontrivial radial solutions for equation (1.6) besides a nontrivial non-negative radial solution.

General quasilinear elliptic problems of (1.1) have been intensively studied, see for example, [1,11,15,16,18,28] and the references therein.

Motivated by the above results, in this paper, we intend to find nodal solutions for the following quasilinear elliptic equation

(−∇ ·φ⁰(|∇u|²)∇u

+|u|^α−2u= f(u)_, x∈_R^N_,

u(x)→0, as |x| →_∞, ^(1.7)

where N ≥2, φ(t)behaves like t^q/2 for small tand t^p/2 for large t, 1< p < q < N,q ≤ α≤ p^∗q⁰/p⁰, and the function f satisfies some conditions given by(f₁)-(f3)in this paper. Similar as [1], we impose some restrictions onφ, letφ∈ C²(_R+,R+)such that

(Φ1) φ(0) =0;

(_Φ2) there exists a positive constantCsuch that

(Ct^p2−1 ≤φ⁰(t), ift≥1, Ct^q2−1≤φ⁰(t), if 0≤t≤1;

(_Φ3) there exists a positive constantCsuch that (

φ(t)≤ Ct^p2, if t≥1, φ(t)≤ Ct^2q, if 0≤t≤_1;

(_Φ4) there existsα<θsuch thatφ⁰(t)/t^θ−22 is strictly decreasing for allt >0;

(_Φ5) the mapt7→ φ(t²)is convex.

We also assume the nonlinearity f satisfies:

(f₁) f(t) =o(t^α−1), ast →0⁺; (f2) f(t) =o(t^p∗−1), ast →+_∞;

(f₃) there existsα<θ such that

0< (θ−1)f(t)≤ f⁰(t)t, for all t>0.

Before we present our main result, we give some notions and definitions. In the following, we usekuk_qto denote the L^q(_R^N)norm.

Definition 1.1 (See [1, Definition 2.1]). Let 1 < p < qandΩ ⊂ _R^N. Denote L^p(_Ω) +L^q(_Ω) the completion of C_c^∞(_Ω,R)in the norm

kuk_Lp(_Ω)+L^q(_Ω)=inf

kvk_Lp(_Ω)+kwk_Lq(_Ω)|v∈ L^p(_Ω), w∈L^q(_Ω),u= v+w .

Next, we denote kuk_p,q = kuk_Lp(_R^N)+L^q(_R^N). Moreover, in [4], it has shown that L^p(_Ω) + L^q(_Ω)can be characterized as an Orlicz spaces.

Definition 1.2 (See [1, Definition 2.3]). Let α > 1, the Orlicz–Sobolev space W(_R^N) is the completion ofC_c^∞(_R^N,R)in the norm

kuk=kuk_α+k∇uk_p,q.

By Theorem 2.8 of [1], the spaceW(_R^N)can be precise description by W(_R^N) =u∈ L^α(_R^N)∩L^p∗(_R^N)| ∇u∈ L^p(_R^N) +L^q(_R^N) . In the following, we define

C_c^∞(_R^N_,R)

r=u∈ C_c^∞(_R^N_,R)|uis radially symmetric .

ThenW_r(_R^N)is the completion of C_c^∞(_R^N,R)

rin the normk · k, namely W_r(_R^N) = C_c^∞(_R^N,R)

r k·k.

Thus, W_r(_R^N)coincides with the set of radial functions ofW(_R^N). Define the energy func- tional I corresponding to equation (1.7) by

I(u) = ¹ 2

Z

R^Nφ(|∇u|²)dx+ ¹ α

Z

R^N|u|^αdx−

Z

R^NF(u)dx, u∈ W_r(_R^N), where F(u) = Ru

0 f(z)dz. The well-posedness and regularity of I(u) are given by Proposi- tion 3.1 in [1] and hypotheses(f₁)–(f₂).

A functionu∈ W_r(_R^N)is called a weak solution of equation (1.7) if for allϕ∈ C₀^∞(_R^N_,R)_,

it holds _Z

R^Nφ⁰(|∇u|²)∇u∇ϕdx+

Z

R^N|u|^α−2uϕdx−

Z

R^N f(u)ϕdx=0.

In particular, foru∈ W_r(_R^N), we denote γ(u) =hI⁰(u),ui=

Z

R^Nφ⁰(|∇u|²)|∇u|²dx+

Z

R^N|u|^αdx−

Z

R^N f(u)udx.

Now we state our main result. We denoteu⁺=max{u, 0}andu⁻ =min{u, 0}.

Theorem 1.3. Suppose1 < p < q < min{N,p^∗}, q ≤ α ≤ p^∗q⁰/p⁰, (_Φ1)–(_Φ5) and(f₁)–(f₃) hold, then there exists a pair of radial solutions u^±_k of equation(1.7)with the following properties:

(i) u⁻_k (0)<0<u⁺_k (0),

(ii) u^±_k possess exactly k nodes r_i with 0 < r₁ < r₂ < · · · < r_k < +_∞, and u^±_k (x)|_|x|=r

i = 0, i=1, 2, . . . ,k.

Remark 1.4. The solutionsu_kobtained in Theorem1.3, as we will see, is the least energy radial solution of equation (1.7) and changes sign exactlyk(k ∈ {0, 1, 2, . . .})times. We should point out thatα< p^∗. The existence ofu₀had been proved by the Mountain Pass Theorem in [1].

Remark 1.5. Like [1], a specific example of the functionφ(t)is φ(t) = ²

p h

1+t^q2p_q

−1i .

In this paper, we prove by constrained minimization method in a special space in which each function changes signk(k∈ {0, 1, 2, . . .})times. We first prove the existence of minimizer and then verify that the minimizer is indeed a solution to equation (1.7) by analyzing the least energy related to the minimizer. Here, we have to point out that it is hard to obtain radial solutions with a prescribed number of nodes by gluing method as in Bartsch–Willem [6] and Cao–Zhu [10]. Because, we obtain that all weak solutions of (1.7) by Lemma 2.7 are only of class C_loc^1,γ(_R^N), and it is not enough to glue the functions in each annuli by matching the normal derivative at each junction point. We will follow the approach explored by Z. Liu and Z.-Q. Wang [21,22], see Section 3 for more details. Moreover, we introduce some new analysis techniques and establish better inequalities.

This paper is organized as follows. In Section 2, we give some preliminary results, which are crucial to prove our main results. In Section 3, we will prove our main theorem.

Throughout this paper, we denote “ → _{” and “} * ” as the strong convergence and the weak convergence, respectively. We useh·,·ito denote the duality pairing between W_r(_R^N) andW_r⁰(_R^N). We employCorC_j, j= 1, 2, . . . to denote the generic constant which may vary from line to line.

2 Some preliminary lemmas

In this section, let us first recall some known facts about (1.7). From [1], we introduce the embedding result on W_r(_R^N)and a uniform decaying estimate on the functions ofW_r(_R^N)_. The proof of lemmas can be found in the corresponding references.

Lemma 2.1(see [1, Remark 2.7]). If1< p <min{q,N}and1< p^∗q_p⁰0, then for everyα∈ 1,p^∗q_p⁰0

, W_r(_R^N)is continuously embedded into L^τ(_R^N)withα≤τ≤ p^∗.

Lemma 2.2(see [1, Theorem 2.11]). If1< p<q< N and1< p^∗q_p⁰0, then for everyα∈ 1,p^∗q_p⁰0

, W_r(_R^N)is compactly embedded into L^τ(_R^N)withα<τ< p^∗.

Lemma 2.3 (see [1, Lemma 2.13]). If 1 < p < q < N, there exists C > 0 such that for every u∈ W_r(_R^N)

|u(x)| ≤ ^C

|x|^N−qq

k∇uk_p,q, for|x| ≥1.

LetΩbe one of the following domains:

{x∈_R^N :|x|< R₁}, {x ∈_R^N : 0<R₂ ≤ |x|<R₃ <_∞}, {x∈_R^N :|x| ≥R₄ >0}. Thus, we first consider the existence of positive least energy solution for

(−∇ ·[φ⁰(|∇u|²)∇u] +|u|^α−2u= f(u), x∈_Ω, u

∂Ω=0. (2.1)

Define

I_Ω(u) = ¹ 2

Z

Ωφ(|∇u|²)dx+ ¹ α

Z

Ω|u|^αdx−

Z

ΩF(u)dx, γ_Ω(u) =hI_Ω⁰ (u),ui=

Z

Ωφ⁰(|∇u|²)|∇u|²dx+

Z

Ω|u|^αdx−

Z

Ω f(u)udx and

M(_Ω) =u∈ W_r(_Ω):u6≡0,u|_∂Ω=0, γ_Ω(u) =0 . Then we have the following lemmas.

Lemma 2.4. Suppose1 < p < q<min{N,p^∗}, q≤α ≤ p^∗q⁰/p⁰,(_Φ1)–(_Φ5)and(f₁)–(f₃)hold and u ∈ W_r(_Ω). Then there exists a unique t>0such that tu∈M(_Ω).

Proof. For fixedu∈ W_r(_Ω)_withu6≡_0,tuis contained in M(_Ω)if and only if γ_Ω(tu) =

Z

Ωφ⁰(|t∇u|²)|t∇u|²dx+

Z

Ω|tu|^αdx−

Z

Ω f(tu)tudx=0. (2.2) Hence, the problem is reduced to verify that there is only one solution of equation (2.2) with t>0. Since 1< p<q≤αand

φ(t²)'

(t^p, if|t| 1, t^q, if|t| 1.

By(f₁)–(f₂), for anyε>0, there exists a constantC_ε >0 andα<s < p^∗ such that

f(u)u≤ε|u|^α+C_ε|u|^s_{. (2.3)}

It is easy to see that I_Ω(tu) → 0 as t → 0 and I_Ω(tu) → −_∞ as t → +∞. We have that I_Ω possesses a global maximum pointt∈(0,+_∞), i.e.,tu∈M(_Ω).

It remains to show the uniqueness oft. We shall divide our proof into two cases.

Case 1. u∈M(_Ω). First of all, we note that it follows fromγ_Ω(u) =0 that Z

Ωφ⁰(|∇u|²)|∇u|²dx+

Z

Ω|u|^αdx−

Z

Ω f(u)udx =0. (2.4) We now prove thatt =1 is the unique number such thattu∈M(_Ω). In fact, ift >0 such that γ_Ω(tu) =0, then we have

Z

Ωφ⁰(|t∇u|²)|t∇u|²dx+

Z

Ω|tu|^αdx−

Z

Ω f(tu)tudx=0. (2.5) Furthermore, combining equation (2.4) and (2.5), we have

Z

Ω

φ⁰(t²|∇u|²)t²|∇u|²−t^θφ⁰(|∇u|²)|∇u|²dx +

Z

Ω

(t^α−t^θ)|u|^α+ t^θf(u)− f(tu)tu

dx=0.

(2.6)

On one hand, by(f₃), we can get that

f(t) t^θ−1

is increasing for allt >0. On the other hand, by(_Φ4), we can deduce that φ⁰(t²)

t^θ−2

is strictly decreasing for allt>0. Assumet>1 for a while, then we get f(u)

u^θ−1 ≤ ^f(tu)

|tu|^θ−1,

φ⁰(t²|∇u|²) t^θ−2|∇u|^θ−2 < ^φ

0(|∇u|²)

|∇u|^{θ−2 ,} that is

t^θf(u)− f(tu)tu≤_{0 (2.7)}

and

φ⁰(t²|∇u|²)t²|∇u|²−t^θφ⁰(|∇u|²)|∇u|² <_{0. (2.8)} Since α < θ, the left side of equation (2.6) is negative, which gives a contradiction. With a similar argument, the caset<1 is also contradictory. Thus we deduce thatt=1.

Case 2. u6∈M(_Ω). If there existt₁,t₂>0 such thatt₁u,t₂u∈ M(_Ω), we have t2

t₁(t₁u) =t₂u∈M(_Ω).

Noticingt₁u∈ M(_Ω), by Case 1, we obtaint₁=t2. This completes the proof of Lemma2.4.

Lemma 2.5. Suppose1< p< q<min{N,p^∗}, q≤ α≤ p^∗q⁰/p⁰,(_Φ1)–(_Φ5)and(f₁)–(f₃)hold and u∈_M(_Ω), t∈(_0,∞)and t6=1, then I_Ω(tu)< I_Ω(u).

Proof. Define a function in(0,∞)byg(t) = I_Ω(tu) g(t) =I_Ω(tu) = ¹

2 Z

Ωφ(t²|∇u|²)dx+ ^t

α

α Z

Ω|u|^αdx−

Z

ΩF(tu)dx.

Then

g⁰(t) =

Z

Ωtφ⁰(t²|∇u|²)|∇u|²dx+t^α−1 Z

Ω|u|^αdx−

Z

Ω f(tu)udx.

By the factu∈M(_Ω), i.e., Z

Ωφ⁰(|∇u|²)|∇u|²dx+

Z

Ω|u|^αdx−

Z

Ωf(u)udx=0,

using a similar argument to Lemma2.4, we obtain g⁰(t)> 0 for 0 < t < 1 and g⁰(t) < 0 for t>1. Henceg(t)<g(1), that isI_Ω(tu)< I_Ω(u)fort∈ (0,∞)andt 6=1.

Next we consider the following minimization problem

˜ c= inf

M(_Ω)I_Ω(u).

M(_Ω)is nonempty inW_r(_Ω)by Lemma2.4. Here we denote kuk_Ω= kuk_Lα(_Ω)+k∇uk_Lp(_Ω)+L^q(_Ω), and

Λu ={x∈ _Ω:|u|>₁}_{, Λ}^c_u ={x∈ _Ω:|u| ≤₁}_.

Lemma 2.6. Suppose1< p <q<_min{N,p^∗}, q≤ α≤ p^∗q⁰/p⁰,(_Φ1)–(_Φ5)and(f₁)–(f₃)hold, thenc can be achieved by some positive function˜ u which is a solution of equation˜ (2.1).

Proof. We use the minimization method. The proof can be divided into two steps.

Step 1. c˜is attained. By the definition of ˜c, there exists a sequence{u˜n} ⊂M(_Ω)such that I_Ω(u˜n) =c˜+o(1), γ_Ω(u˜n) =0,

i.e.,

I_Ω(u˜_n) = ¹ 2

Z

Ωφ(|∇u˜_n|²)dx+ ¹ α

Z

Ω|u˜_n|^αdx−

Z

ΩF(u˜_n)dx= c˜+o(1), Z

Ωφ⁰(|∇u˜_n|²)|∇u˜_n|²dx+

Z

Ω|u˜_n|^αdx−

Z

Ω f(u˜_n)u˜_ndx=0.

By the Proposition 2.2 of [1], we have

ku˜_nk_Lp(_Ω)+L^q(_Ω)≤max

ku˜_nk_Lp(_Λun˜ ),ku˜_nk_Lq(_Λ^c_un˜ ) .

It follows from (Φ4) that φ⁰⁰(t)t < ^θ−₂²φ⁰(t) for all t > 0. Moreover, φ(0) = 0, we see that φ⁰(t)t< ^θ₂φ(t). There exists 0<µ<1 such that

φ⁰(t)t≤ ^θµ

2 φ(t), for allt ≥0.

Thus, by(_Φ2)and the fact that ˜u_n∈ L^p(_Λu˜n)∩L^q(_Λ^c_u˜

n)(see Proposition 2.2 (iv) in [1]), we get

˜

c+o(1) = I_Ω(u˜n)−¹

θhI_Ω⁰ (u˜n), ˜uni

≥

Z

Ω

h1

2φ(|∇u˜n|²)−¹

θφ⁰(|∇u˜n|²)|∇u˜n|²ⁱdx+¹ α− ¹

θ ^Z

Ω|u˜n|^αdx

≥ ¹−µ 2

Z

Ωφ(|∇u˜_n|²)dx+¹ α−¹

θ ^Z

Ω|u˜_n|^αdx

≥C₁ Z

Λ^c∇un˜

|∇u˜_n|^qdx+C₂ Z

Λ∇un˜

|∇u˜_n|^pdx+¹ α

−¹ θ

^Z

Ω|u˜_n|^αdx

≥Ch min

k∇u˜_nk^q

L^p(_Ω)+L^q(_Ω), k∇u˜_nk^p

L^p(_Ω)+L^q(_Ω) +ku˜_nk^α_Lα(Ω)ⁱ

≥Cku˜_nk^α_Ω.

(2.9)

SinceC> 0, it is easy to verify { ˜u_n}is bounded in M(_Ω). Then by Proposition 2.5 of [1] and Lemma2.1, there exists ˜u ∈ W_r(_Ω)such that

˜

u_n*u,˜ weakly inW_r(_Ω),

˜

u_n→u,˜ inL^s(_Ω),

˜

u_n→u,˜ a.e. inΩ,

whereα< s< p^∗. By Theorem A.2 in [34], we can deduce that f(u˜_n)u˜_n→ f(u˜)u˜ in L¹(_Ω).

Sinceγ_Ω(u˜n) = 0, we first prove ˜u 6≡0. In fact, by equation (2.3) , Lemma2.1 and inequality (2.9), we have

Cεku˜nk^s_Ω+εku˜nk^α_Ω≥

Z

Ω f(u˜n)u˜ndx =

Z

Ωφ⁰(|∇u˜n|²)|∇u˜n|²dx+

Z

Ω|u˜n|^αdx≥Cku˜nk^α_Ω. (2.10) Sinces>α, we haveku˜nk_Ω≥ C₃ >0. Hence

Cεku˜k^s_Ω+εku˜k^α_Ω+o(1)≥ o(1) +

Z

Ω f(u˜)u˜dx =

Z

Ωφ⁰(|∇u˜_n|²)|∇u˜_n|²dx+

Z

Ω|u˜_n|^αdx

≥Cku˜nk^α_Ω ≥C₃, we get ˜u6≡0.

According to Lemma2.4, there exists a unique ¯t>0 which satisfiesγ_Ω(t^¯u˜) =0. Using the condition(_Φ5)_{, then}

1 2

Z

Ωφ(^¯t²|∇u˜|²)dx ≤lim inf

n→_∞

1 2

Z

Ωφ(t^¯2|∇u˜n|²)dx.

By the definition of ˜cand equation (2), we have

˜

c≤ I_Ω(t^¯u˜) = ¹ 2

Z

Ωφ(t^¯2|∇u˜|²)dx+^t¯

α

α Z

Ω|u˜|^αdx−

Z

ΩF(t^¯u˜)dx

≤lim inf

n→_∞ Z

Ω

1

2φ(^¯t²|∇u˜n|²) + ^t¯

α

α|u˜n|^α−F(t^¯u˜n)

dx

≤lim inf

n→_∞ I_Ω(^¯tu˜_n)≤lim inf

n→_∞ I_Ω(u˜_n) =c.˜

Thus we get

I_Ω(^¯tu˜) =c,˜ and ˜cis attained by ¯tu.˜

Step 2. In the following, we prove that ¯tu˜is a radial solution of equation(2.1), which is similar to the Lemma 2.7 of [14]. For simplicity, we denote ˜u to ¯tu. Suppose ˜˜ u ∈ M(_Ω), I_Ω(u˜) = c,˜ but the conclusion of the lemma is not true. Then we can find a function ϕ ∈ W_r⁰(_R^N)such that

hI_Ω⁰ (u˜),ϕi=

Z

Ωφ⁰(|∇u˜|²)∇u˜∇ϕdx+

Z

Ω|u˜|^α−2uϕ˜ dx−

Z

Ω f(u˜)ϕdx≤ −1. (2.11) Choosing ε>0 small enough such that

I_Ω⁰ (tu˜+σ ϕ),ϕ

≤ −¹

2, ∀ |t−1|+|σ| ≤ε.

Letηbe a cut-off function such that

η(t) =

(1, |t−1| ≤ ¹₂ε, 0, |t−1| ≥ε.

We estimate

sup

t

I_Ω(tu˜+εη(t)ϕ). If|t−1| ≤ε, then

I_Ω(tu˜+εη(t)ϕ) = I_Ω(tu˜) +

Z ₁

0

hI_Ω⁰ (tu˜+σεη(t)ϕ)_,εη(t)ϕidσ

≤ I_Ω(tu˜)− ¹ 2εη(t).

(2.12)

For |t−1| ≥ε,η(t) =0, and the above estimate is trivial. Now, since ˜u∈M(_Ω), fort6=1, we get I_Ω(tu˜)< I_Ω(u˜)by Lemma2.5. Hence it follows from equation(2.12)that

I_Ω(tu˜+εη(t)ϕ)≤

(I_Ω(tu˜)< I_Ω(u˜), t6=1, I_Ω(u˜)− ¹₂εη(1) =I_Ω(u˜)−¹₂ε, t=1.

In any case, we have I_Ω(tu˜+εη(t)ϕ)< I_Ω(u˜) =c. In particular,˜ sup

0≤t≤2

I_Ω(tu˜+εη(t)ϕ)<c.˜ Since ˜u∈M(_Ω), we have

Z

Ωφ⁰(|∇u˜|²)|∇u˜|²dx+

Z

Ω|u˜|^αdx−

Z

Ω f(u˜)u˜dx=0. (2.13) Let

h(t) =

Z

Ω

φ⁰(|∇(tu˜+εη(t)ϕ)|²)|∇(tu˜+εη(t)ϕ)|²+|tu˜+εη(t)ϕ|^α

− f(tu˜+εη(t)ϕ)(tu˜+εη(t)ϕ)dx.

Without loss of generality, we assumeε< ¹₄. Fort =2, we haveη(2) =0, thus from (2.7)-(2.8) and (2.13)

h(₂) =

Z

Ω

4φ⁰(₄|∇u˜|²)|∇u˜|²+₂^α|u˜|^α− f(_{2 ˜u})_{2 ˜udx}

=

Z

Ω

4φ⁰(4|∇u˜|²)|∇u˜|²−2^θφ⁰(|∇u˜|²)|∇u˜|²dx+

Z

Ω(2^α−2^θ)|u˜|^αdx +

Z

Ω

2^θf(u˜)u˜− f(2 ˜u)2 ˜u dx

<0.

Fort= ¹₂, we have h

1 2

=

Z

Ω

1 4φ⁰

1 4|∇u˜|²

|∇u˜|²+ ¹

2^α|u˜|^α− f 1

2u˜ 1

2u˜

dx

=

Z

Ω

1 4φ⁰

1 4|∇u˜|²

|∇u˜|²− ¹

2^θφ⁰(|∇u˜|²)|∇u˜|²

dx+

Z

Ω

1 2^α − ¹

2^θ

|u˜|^αdx +

Z

Ω

1

2^θf(u˜)u˜− f 1

2u˜ 1

2u˜

dx

>0.

Consequently, we can find ˜t ∈(¹₂, 2)such thath(t^˜) =0. It implies ˜tu˜+εη(^˜t)ϕ∈M(_Ω), which contradicts with (2.11). From this, ˜uis a solution for equation (2.1).

Ifα ≥ q, we infer that the solution ˜u is positive by Theorem 1 of [30]. Thus, we complete the proof.

We shall show anyW_r(_R^N)-solution of the equation (1.7) isC_loc^1,γ(_R^N)-solution of the equa- tion (1.7).

Lemma 2.7. Assume u be a weak solution of (1.7),1 < _p < _q < _min{N,p^∗}, q ≤ α ≤ p^∗q⁰/p⁰, u∈ W_r(_R^N),(_Φ1)–(_Φ5)and(f₁)–(f₃)hold, then u∈ C_loc^1,γ(_R^N)for some0< γ<1.

Proof. We first prove by the Moser’s iteration that u ∈ L^∞(_R^N), then belongs to C_loc^1,γ(_R^N)_. Sinceu ∈ W_r(_R^N), u ∈ L^p∗(_R^N). For r > 0 to be determined later, taking ϕ = |u^T|^pru as a test function with

u^T =











T, u> T, u, |u| ≤T,

−T, u< −T.

Moreover, without any loss of generality, we shall assume that T > 1. Then ∇u∇ϕ = pr|u^T|^pr|∇u^T|²+|u^T|^pr|∇u|²_,uis a weak solution of equation (1.7), i.e.,

Z

R^Nφ⁰(|∇u|²)∇u∇ϕdx+

Z

R^N|u|^α−2uϕdx =

Z

R^N f(u)ϕdx.

We have (pr+1)

Z

|u|≤Tφ⁰(|∇u|²)|∇u|²|u^T|^prdx +

Z

|u|>T

φ⁰(|∇u|²)|∇u|²|u^T|^prdx+

Z

R^N|u|^α|u^T|^prdx=

Z

R^N f(u)u|u^T|^prdx.

Define A=x∈_R^N :|u| ≤T ∩_Λ^c_∇uandB=x∈_R^N :|u| ≤T ∩_Λ∇u, then Z

R^N f(u)u|u^T|^prdx ≥(pr+1)

Z

|u|≤Tφ⁰(|∇u|²)|∇u|²|u^T|^prdx+

Z

|u|≤T

|u|^α|u^T|^prdx

≥C(1+r)^1−pminn^Z

A

|∇|u|^1+r|^pdx, Z

B

|∇|u|^1+r|^qdxo

+ ¹

T^(α−p)r Z

|u|≤T

||u|^1+r|^αdx

≥C(₁+r)^1−phk∇|u|^1+rk^q

L^p(|u|≤T)+L^q(|u|≤T)+k|u|^1+rk^α_Lα(|u|≤T)ⁱ

≥C(1+r)^1−pk|u|^1+rk^p

L^p∗(|u|≤T)

≥C(1+r)^1−p

Z

|u|≤T

|u|^(1+r)p∗dx_p^p∗

.

Set d = 1+r = ^{N p−(}_(N^N₋⁻_p^p₎⁾⁽_p^s−p) > 1, s ∈ (α,p^∗). Let T → +∞, by equation (2.3) and H ¨older inequality, we have

Z

R^N f(u)u|u^T|^prdx≤C_ε Z

R^N|u|^s−p|u|^pr+pdx+ε Z

R^N|u|^α−p|u|^pr+pdx

≤Cε

^Z

R^N|u|^p∗dx^{p∗ −}_p∗^pdZ

R^N|u|^p∗dx^pd_p∗

+ε ^Z

R^N|u|^α¯dx^{p∗ −}_p∗^pdZ

R^N|u|^p∗dx^pd_p∗

≤C^Z

R^N|u|^p∗dx^pd_p∗

, whereα<α¯ = ₍⁽_N^α₋^−p_p)(^{)(N p}_s−p⁾₎ < p^∗. Then we get

^Z

R^N|u|^p∗ddx_p^p∗

≤C(1+r)^p−1

Z

R^N f(u)u|u^T|^prdx≤ C(1+r)^p−1

Z

R^N|u|^p∗dx_p^pd∗

. Hence

^Z

R^N|u|^p∗ddx_p¹∗_d

≤C(1+r)

p−1 pd

^Z

R^N|u|^p∗dx_p¹∗

. Therefore

^Z

R^N|u|^p∗dkdx ¹

p∗_dk

≤ _Π^k_i=1Cdⁱ¹

di

^Z

R^N|u|^p∗dx_p¹∗

.

SinceΠ^∞_i=1(Cdⁱ)^di1 ≤C^∗ for some constantC^∗>0, we then deduce thatu∈ L^∞(_R^N). Suppose u is a weak solution of the equation (1.7) and u ∈ W_r(_R^N), we have that u ∈ C_loc^1,γ(_R^N) for someγ>0 by Chapter 4 of [19] or [33].

3 Existence of sign-changing solutions

In this section, we construct infinitely many nodal solutions for equation (1.7). For any given knumbersr_j(_j=_{1, . . . ,k})such that 0< _r1 <_r2 <· · ·<_rk <+∞, we denoter0=_0,rk+1=_∞,

Ω¹ =x ∈_R^N _:|x|<r₁ and Ω^j =x ∈_R^N _:r_j−1<|x|<r_j .

We will always extend u_j ∈ W_r(_Ω^j) to W_r(_R^N) by setting u_j ≡ 0 for x ∈ _R^N\_Ω^j for every u_j, j = 1, 2, . . . ,k+1. For convenience, we use I(u_j) to replace I_Ωj(u_j) and γ(u_j)to replace γ_Ωj(u_j). Define

Y_k^±(r₁,r₂, . . . ,r_k+1) =

u∈ W_r(_R^N)|u= ±

k+1 j

∑

(−1)^j−1u_j, u_j ≥0,

u_j 6≡0, u_j ∈ W_r(_Ω^j), j=1, 2, . . . ,k+1

,

M^±_k =u∈ W_r(_R^N)| ∃₀=r₀< r₁ <r₂ <· · ·<r_k <r_k+1 = +_∞,

such thatu∈Y_k^±(r₁,r₂, . . . ,r_k+1)andu_j ∈M(_Ω^j), j=1, 2, . . . ,k+1 . Note that M^±_k 6= ∅^,k =1, 2, . . . In order to prove the existence of non-negative critical points of energy functional I, similar to [6] or [10], we only need to extend f(u)as follows

f⁺(u):=

(f(u)_{, if} u≥_0,

−f(−u), if u<0,

thus the oddness assumption on nonlinear term is actually unnecessary. The function I⁺(u) is defined onW_r(_R^N)by

I⁺(u) = ¹ 2

Z

R^Nφ(|∇u|²)dx+ ¹ α

Z

R^N|u|^αdx−

Z

R^NF⁺(u)dx, c⁺_k = inf_u∈M⁺

k I⁺(u)in the same way as those in [10]. For M⁻_k, we can complete the proof in the same way. By the arguments of the Section 2, it is not difficult to verify that

∀u=

k+1

∑

(−1)^j−1u_j ∈M⁺_k ⇔ I(u) = max

αj>0 1≤j≤k+1

I _k+1

j

∑

α_juˇ_j

,

where ˇu_j = (−1)^j−1u_j. Set

c_k = _inf

u∈M⁺_k

I(u)_, k =1, 2, . . .

Lemma 3.1. c_k is attained provided that1 < p < q< min{N,p^∗}, q ≤ α≤ p^∗q⁰/p⁰, (_Φ1)–(_Φ5) and(f₁)–(f₃)hold.

Proof. We prove by induction that for eachkthere exists ¯u_k ∈_M⁺_{k such that} I(u¯_k) =c_k.

For k = 0 orΩ = _R^N, we can directly derive from Lemma 2.6. We discuss the case k ≥ 1 in the following.

First, we prove I is bounded from below on M⁺_k by a positive constant. Let ¯u ∈M⁺_k, then

¯

u=_∑^k_j=⁺₁¹(−1)^j−1u¯_jand ¯u_j ∈M(_Ω^j), j=1, 2, . . . ,k+1. By the similar arguments of inequality (2.10), we haveku¯_jk_Ωj ≥C_j. It follows from the same computations in (2.9) that

I(u¯) = I _k+1

∑

(−1)^j−1u¯_j

=

k+1 j

∑

I(u¯_j)≥C

k+1 j

∑

ku¯_jk^α_Ω

j ≥C

k+1

∑

C^α_j =C.^¯ (3.1)