Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity

1.1 Background

Consider the following quasilinear elliptic equations of the form

where κ ∈ ℝ⁺, z : ℝ × Ω → ℂ, W : Ω → ℝ is a given potential and l, h are real functions in ℝ⁺. Of particular interest are solitary wave solutions of (1.1), i.e., z(t, x) = exp(–iEt)u(x), where E ∈ ℝ, u is a real function and satisfies the stationary quasilinear elliptic equation

In particular, equation (1.2) is a special case of the following generalized quasilinear elliptic equations

if ones take

Equation (1.3) corresponds to a large number of elliptic equations which appear in mathematical physics. In the literature, equation (1.3) has been derived as models of several physical phenomena corresponding to various types of φ(s). If φ(s) ≡ const, equation (1.3) is reduced to

which is the well-known elliptic equation in the quantum mechanic and also arises in biological models and propagation of laser beams(Ref. [21, 29]). If φ(s)=1+2κs2, equation (1.3) can be rewritten as follows

which is called the superfluid film equation in plasma physics and fluid mechanics(Ref.[28, 30]). If φ²(s) = κs22(1+s2), then one can get the following equation of the form

which models the self-channeling of a high-power ultrashort laser in matter(Ref. [30]). For the further physical background, we refer the readers to [15, 31, 33, 41, 45] and the reference therein.

1.2 Motivation

In the last decades, quasilinear Schrödinger equations have received a considerable attention by numerous researchers. To the best of our knowledge, the first existence results for quasilinear equations of the form of (1.4) with κ ≠ 0 is due to [33, 41], in which, the main existence results are obtained, through a constrained minimization argument. Actually, in these papers, they obtained solutions in H_V of the problem with an unknown Lagrange multiplier λ:

Here H_V := {u ∈ H¹(ℝ^N) : ∫_ℝ^N V(x)u² < ∞}. To investigate the case with any prescribed λ > 0 in the variational setting, one can formulate this problem as follows: consider the formal energy functional

However, Ĵ is not well defined in H_V, except for N = 1. To overcome this difficulty, a change of variable v = f^–1(u)(see Section 2) was introduced in [31] and Ĵ can be rewritten in a new variable. Then this problem was resolved in an associated Orlicz space. Subsequently, a simpler and shorter proof of some results in [31] was given by M. Colin, L. Jeanjean [9]. Moreover, a dual approach was introduced in [9] so that problems of the form (1.4) can be dealt with in H¹(ℝ^N) instead of the Orlicz space.

Initiated by M. Colin, L. Jeanjean, the dual approach introduced in [9] has been one of main tools in studying problem (1.4) by the variational approach and there have been the extensive results in the literature. By using such dual approach, J. M. do Ó, O. Miyagaki, S. M. Soares [12] considered problem (1.4) in ℝ² involving a critical growth of the Trudinger-Moser type(for instance see [10]). By using the mountain pass theorem and the concentration-compactness principle, a positive solution was obtained. For the semiclassical states of quasilinear problems, E. Gloss [23] considered the following problem in the subcritical case

Under some sort of Berestycki and Lions conditions as in [5], in the framework of J. Byeon and L. Jeanjean[6], the author shows that (1.6) admits positive solutions. Moreover, there solutions exhibit a spike near local minimal points of the potential well V as ε → 0. Later, through the same dual approach, Y. Wang and W. Zou [52] considered the semiclassical states of the critical quasilinear Schrödinger equations (1.6). By the penalization argument by M. del Pino and P. Felmer[13], the authors proved the existence of positive bound states which concentrate around a local minimum point of V as ε → 0. By the Nehari approach and the dual approach above, X. He, A. Qian and W. Zou [25] considered the semiclassical ground states of the critical quasilinear Schröinger equations (1.6). Moreover, the multiplicity was considered by the Ljusternik-Schnirelmann theory as well. In this aspect, we also would like to cite [7, 16, 26, 37, 39, 49, 51]. For the generalized quasilinear equation (1.3), by introducing a new variable replacement v = G(u) = ∫0u φ(s) ds, Y. Shen and Y. Wang [45] reduced (1.3) to a semi-linear elliptic equation

By virtue of the mountain pass theorem, positive solutions were obtained when the nonlinearity is subcritical. Subsequently, by adopting the same change of variable, Y. Deng, S. Peng and S. Yan [15] investigated the generalized quasilinear Schrödinger equations (1.3) involving critical growth. For more related results to quasilinear problems (1.3), we refer the readers to [3, 24] for uniqueness of solutions, [48] for non-degeneracy of solutions, [3, 35, 36, 38] for critical or supercritical exponent, [19, 42] for ground state solutions, [17, 20, 44] for multiple solutions, [46] for quasilinear p-Laplacian problems, [47] for asymptotical problems and [2] for the case κ < 0.

1.3 Our problem and main result

In the present paper, we mainly focus on the quasilinear elliptic equations with critical growth. Precisely, we investigate the problem

where Ω ⊂ ℝ^N is a bounded domain, N ≥ 3 and 2^* = 2N/(N – 2). In [32], it turns out that p = 2 ⋅ 2^* behaves as a critical exponent for the quasilinear elliptic equations. So problem (1.7) can be regarded as the counterpart of the Brézis-Nirenberg problem in the quasilinear case. The first celebrated work is due to H. Brézis and L. Nirenberg [4]. They considered the well known Brézis-Nirenberg problem

In particular, they investigated the relation between the existence of positive solutions to (1.8) and μ, N, q. Precisely, they shows that problem (1.8) is solvable for any q ∈ (2, 2^*) and μ > 0 if N ≥ 4. In contrast, in dimension 3, the situation is much delicate. They shows that if Ω ⊂ ℝ³ is strictly starshaped about the origin, problem (1.8) with q ∈ (2, 4] admits a positive solution if μ > 0 large and no positive solution if μ > 0 small. In [22], F. Gazzola and B. Ruf generalized some results in [4] to the semilinear critical elliptic problem with a wide class of lower order terms –Δu = g(x, u) + |u|^2*–2u in Ω ⊂ ℝ^N. In particular, when N = 3, a similar hypothesis to [4] was imposed: there exists an open nonempty set Ω₀ ⊂ Ω such that

Since the pioneering work [4], there have been extensive works on semilinear elliptic equations with critical exponent. Compared to the semilinear case, the quasilinear equation becomes more complicated. In [40], a mountain-pass technique in a suitable Orlicz space is used to prove the existence of soliton solutions to quasilinear Schrödinger equations involving critical exponent in ℝ^N. In [11], a positive solution was obtained by using the concentration-compactness principle and the mountain pass theorem when h(u) in (1.4) amounts to the sum of the two terms, |u|^q–1u and |u|^p–1u, one of which is critical and the other subcritical. In [34], for a class of quasilinear Schrödinger equations with critical exponent, X. Liu, J. Liu, Z.-Q. Wang established the existence of both one-sign and nodal ground states by the Nehari method. It is established in [43] the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in ℝ^N with critical growth. For h(u) = λ|u|^q–2u + |u|^2⋅2*–2u, λ > 0, 4 < q < 2 ⋅ 2^*. Y. Deng, S. Peng, J. Wang [14], they proved the existence of the nodal solution for problem (1.4) by using Nehari technique. In [35], X. Liu, J. Liu, Z.-Q. Wang considered a kind of more general quasilinear elliptic equations. Via a perturbation method, they obtained positive solutions in the critical case.

In [1], C. O. Alves and Y. Deng considered the Brézis-Nirenberg problem involving the p-Laplacian operator. They were concerned with the p-Laplacian problem

where p^* = pN/(N – p). Through the Lusternik-Schnirelman category theory, the authors obtained at least cat_Ω(Ω) positive solutions for μ > 0 small and N ≥ p². Motivated by [1], our main purpose of this paper is to investigate the multiplicity of positive solutions to quasilinear problem (1.7). Precisely, our main result reads as

1.4 Main difficulties

In the following, we summarize some difficulties caused by the quasilinear term Δ(u²)u and critical term |u|^2⋅2*–2u in seeking solutions. The main difficulties of the present paper are two-fold. First, due to the critical growth, the compactness does not hold in general. We adopt the Brézis-Nirenberg argument as in [4](see also [35]) to show that the least energy c_μ is below 12NSN2 if q > q_N, which yields the compactness. Second, the term Δ(u²)u results in the lack of smoothness to the formal energy functional of problem (1.7) in H01 (Ω). To overcome the difficulty, we use the dual approach introduced in [9] through a change of variable. But, due to the lack of homogeneity for the change of variable, the methods in [1] can not be applied in a direct way. So more delicate analyses and new tricks are needed.

1.5 Outline of this paper

This paper is organized as follows. In Section 2, the variational setting is set up and some preliminaries are given. Section 3 is devoted to proving Theorem 1.1 via the Nehari manifold and the Lusternik-Schnirelman category theory.

Notation. C, C₁, C₂, … will denote different positive constants whose exact value is inessential. |A| is the Lebesgue measure of a measurable set A ⊂ ℝ^N. B_ρ(y) : = {x ∈ ℝ^N : |x – y| < ρ}. The usual norm in the Lebesgue space L^p(Ω) is denoted by ∥u∥_p. E denotes the Sobolev space H01 (Ω) with the standard norm

2.1 The dual approach

In this section we introduce a variational framework associated with problem (1.7). Formally (1.7) is the Euler-Lagrange equation associated to the natural energy functional

However, it is not well defined in general in H01 (Ω). To overcome the difficulty, we use an argument developed in [9]. We make a change of variables v : = f^–1(u), where f is defined by

Let us collect some properties of f, which have been proved in [9, 52].

It is easy to see from the proofs in [52] that (9) is strictly increasing.

Therefore, after the change of variables, we consider the following functional

which is well defined in E and belongs to C¹. Moreover,

for all v, w ∈ E and the critical points of I are the weak solutions of the Euler-Lagrange equation given by

Obviously, if v ∈ E is a positive critical point of the functional I_μ, then u = f(v) ∈ E is a solution of (1.7), see [9].

2.2 Nehari manifold

Let

is the Nehari manifold and c_μ := inf_𝓜μI_μ.

We denote by S the best Sobolev constant of the embedding H01 (Ω) ↪ L^2*(Ω) given by S := inf{∥u∥² : u ∈ H01 (Ω), |u|_2^* = 1}. It is known that S is independent of Ω and is never achieved except when Ω = ℝ^N (see Proposition 1.43 in [50]).

For t > 0, let

3.1 Compactness

According to Lemma 2.3, it is standard to prove that the least energy value c_μ has a minimax characterization given by

For μ₁ ≥ μ₂ ≥ 0, I_μ1(u) = Iμ2(u)−μ1−μ2q ∫_Ωf^q(u⁺). Hence max_t>0I_μ1(tu) ≤ max_t>0I_μ2(tu) and therefore c_μ1 ≤ c_μ2. Moreover, for every λ > 0, we have by Lemma 2.3-(2),

Denote by I_𝓜μ the restriction of I_μ on 𝓜_μ.

3.2 Asymptotic behavior of c_μ

Define