Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

1 Introduction and Main Results

This paper is concerned with the following quasilinear elliptic equation:

where Δp⋅=div(|∇⋅|p-2∇⋅) is the p-Laplacian operator with 2<p<N.

Such types of equations have been derived as models of several physical phenomena and have been the subject of extensive study in recent years. For example, solutions to (1.1) for p=2 are related to the solitary wave solutions for quasilinear Schrödinger equations of the form

where z:ℝ×ℝN→ℂ, W:ℝN→ℝ is a given potential, κ is a real constant and f,h:ℝ+→ℝ are suitable functions. The quasilinear equation (1.2) appears more naturally in mathematical physics and has been derived as a model for several physical phenomena corresponding to various types of h⁢(s). For instance, the case h⁢(s)=s models the time evolution of the condensate wave function in super-fluid film [17, 18], and are called the superfluid film equation in fluid mechanics by Kurihara [17]. We can refer to [3, 7, 16, 30, 31] and references therein for more physical motivations and applications.

Considering the case h⁢(s)=s, κ>0, and taking z⁢(t,x)=exp⁡(-i⁢E⁢t)⁢u⁢(x) in (1.2), we find that u⁢(x) solves the following elliptic equation:

where V⁢(x)=W⁢(x)-E is a new potential.

Our paper was motivated by the quasilinear Schrödinger equation (1.3), on which much concentration has been focused in the past several years. The case f⁢(s2)⁢s=|s|q-2⁢s (4≤q<22*, 2*=2⁢NN-2, N≥3) is called subcritical growth in the spirit of [22], and this case was studied extensively recently; we can refer to the references [10, 13, 23, 24, 21, 22, 30, 29, 19], where positive or sign-changing solutions were obtained by using a constrained minimization argument, or a Nehari method, or a technique of changing variables. We remark that among the above three methods, the last one, which was first proposed in [22], is most effective for the power nonlinearity case, since this argument can transform the quasilinear problem to a semilinear one, and hence an Orlitz space framework can be used. The case q=22* corresponds to a critical growth for (1.3), since it was shown in [22] that (1.3) has no positive solutions in H1⁢(ℝN) with u2⁢|∇⁡u|2∈L1⁢(ℝN) if f⁢(u2)⁢u=|u|p-2⁢u, p≥22*, and V satisfies ∇⁡V⁢(x)⋅x≥0 in ℝN. Concerning this case, very few results can be found. Positive solutions were obtained in [4] via the mountain pass lemma, and nonnegative nontrivial solutions were given by Moameni in [27]. By a constructing argument, the authors considered problem (1.3) with f⁢(u2)⁢u=|u|22*-2⁢u+λ⁢|u|q-2⁢u, and infinitely many node solutions were given in [14].

It should be mentioned that Liu et al. recently considered the existence of positive solutions and sign-changing solutions for general quasilinear elliptic equations like (1.3), which cannot be transformed into a semilinear form by a perturbation method. Particularly, some interesting results when the nonlinearity f⁢(u2)⁢u has critical growth were obtained in [20] and [25].

Here, our main purpose is to construct infinitely many nodal solutions for (1.1) with the following assumptions:

1. k⁢(u)=λ⁢|u|q-2⁢u+|u|2⁢p*-2⁢u, λ>0, N>p>2, 2⁢p<q<2⁢p*, where p*=p⁢NN-p is the Sobolev critical exponent,

2. V∈C1⁢(ℝN,ℝ)∩L∞⁢(ℝN), V⁢(0)>0 and V⁢(x)=V⁢(r), V′⁢(r)≥0 for all r=|x|∈(0,∞).

A function u:ℝN→ℝ is called a weak solution of (1.1) if u∈W1,p⁢(ℝN)∩Lloc∞⁢(ℝN), and for all φ∈C0∞⁢(ℝN), we have

We point out that we can not apply directly variational methods here because the natural functional corresponding to (1.1), given by

where

satisfying K′⁢(s)=k⁢(s), is not well defined in W1,p⁢(ℝN), since for u∈W1,p⁢(ℝN)∖L∞⁢(ℝN), it may hold that ∫ℝN|u|p|∇u|p=+∞.

To overcome this difficulty, we generalize an argument developed in [22] for p=2 (see also [32]). We make the change of variables v=f-1⁢(u), where f is defined by

After the change of variables, I⁢(u) can be reduced to the following functional:

which is C1 on the usual Sobolev space W1,p⁢(ℝN) under suitable assumptions on the potential V⁢(x) and the nonlinearity K⁢(s). Moreover, critical points of the functional J correspond to weak solutions of the following equation:

For convenience, we rewrite equation (1.4) in the following form:

and the related variational functional is

where

and

For any k∈{0,1,2,…}, u± is said to be a pair of k-node solution if u± is a radial solution with the following properties:

1. u-⁢(0)<0<u+⁢(0),

2. u± possess exactly k nodes ri with 0<r1<r2<⋯<rk<+∞, and u±⁢(ri)=0, i=1,2,…,k.

The following theorems are our main results.

To find nontrivial critical points of J, we face two main difficulties: firstly, since the domain is unbounded and the Sobolev critical exponent p* appears, the related Palais–Smale condition may not be satisfied. To regain the compactness of functional J, we should work out a threshold value of energy under which a Palais–Smale sequence is pre-compact. To this end, we need to make a precise estimate on the nonlinearity g⁢(x,v) in (1.5). The second difficulty lies in a new phenomenon in which the nonlinear term g⁢(x,s) satisfies

instead of the usual subcritical condition g⁢(x,s)=o⁢(|s|t) (p<t<p*) at infinity. Furthermore, as we will see later, the functions G⁢(x,v), g⁢(x,v)⁢v and 12⁢g⁢(x,v)⁢v-G⁢(x,v) may change sign. These two new phenomena cause two more obstacles. On the one hand, the usual Ambrosetti–Rabinowitz condition is not satisfied. On the other hand, the usual argument to verify that the functional J satisfies the (PS) condition (see [9, 8]) cannot be employed directly. Hence, we need to analyze the exact asymptotic behavior of g⁢(x,s) and should apply more delicate analysis to the functional J.

To construct nodal solutions for (1.1), we will look for a minimizer for a constrained minimization problem in a special space in which each function changes sign k (k∈{0,1,2,…}) times and then verify that the minimizer is smooth and indeed a solution to (1.1) by analyzing the least energy related to the minimizer. We mention here that the main method to prove our theorem was essentially introduced by Bartsch and Willem in [2], and Cao and Zhu in [8], independently. However, as we have pointed out, the lower order term g⁢(x,s) may cause more difficulties. This argument was also used by the authors in [14] to construct infinitely many nodal solutions for (1.1) with p=2. But, for the case p≠2, the operator Δp is quasilinear, which makes the problem more complicated.

The paper is organized as follows. In Section 2, besides providing some useful lemmas, we will give an exact analysis to the asymptotic behavior of g⁢(x,s). Theorems 1.1 and 1.2 are proved in Section 3 and 4, respectively.

In what follows, if u∈W1,p⁢(ℝN), we shall denote as usual by u+ and u- the functions defined by u+⁢(x)=max⁡{u⁢(x),0} and u-⁢(x)=max⁡{-u⁢(x), 0}.

2 Some Preliminary Lemmas

In this section, we give some lemmas. The proof of some of those lemmas can be found in the corresponding references. For any given set K⊂ℝN, we define

In the sequel, for a radial domain Ω, we set

and

which is equivalent to the usual norm in W1,p⁢(ℝN).

Let f, g and G be as defined in the introduction, we will summarize here some properties of them for completeness.

Lemma 2.1 (7) suggests that the functions G⁢(x,v), g⁢(x,v)⁢v and 12⁢g⁢(x,v)⁢v-G⁢(x,v) may change sign, which will add new obstacles when one tries to obtain nontrivial solutions. Hence, we need to analyze the properties of f⁢(t).

3 The Existence of Positive Solutions

In this section, we prove the existence of positive solutions for problem (1.5) by using the mountain pass lemma [1]. Using the lemmas in Section 2 and proceeding as done in [4], we can verify that the functional J exhibits the mountain pass geometry.

As a consequence of Lemma 3.1 and the mountain pass lemma, for the constant

where

there exists a (PS)c sequence {vn} in W1,p⁢(ℝN) at the level c, that is,

We will verify that the level value c is in an interval where the (PS)c condition holds. To this end, we introduce a well-known fact that the minimization problem S=inf{|∇u|pp:u∈W1,p(ℝN),|u|p*=1} has a solution given by

We can choose A>0 so that wϵ satisfies |∇⁡wϵ|pp=|wϵ|p*p*. Thus,

Let φ∈C0∞⁢(ℝN,[0,1]) be a radial cut-off function such that φ⁢(|x|)=1 for |x|≤ρϵ, φ⁢(|x|)∈(0,1) for ρϵ<|x|<2⁢ρϵ, and φ⁢(|x|)=0 for |x|≥2⁢ρϵ, where ρϵ=ϵτ, τ∈(1p,1). Set ψϵ⁢(x)=φ⁢(x)⁢wϵ⁢(x). We have the following estimations (see [12]).