Ground State Solutions for Kirchhoff Type Quasilinear Equations

1 Introduction

We consider the following quasilinear elliptic equation of Kirchhoff type:

where Ω⊂ℝN (N≥3) is a bounded domain with smooth boundary, Δp⁢u=div⁡(|∇⁡u|p-2⁢∇⁡u) is the p-Laplacian operator. We assume the following:

1. 1 < p < N , μ>0, α>1, q>1, α⁢(p+q)<p*=N⁢pN-p.

2. a ∈ C 1 ⁢ ( ℝ , ℝ ) and satisfies the following conditions:

   1. There exists C0>0 such that for t∈ℝ

      C 0 - 1 ⁢ ( 1 + | t | q ) ≤ a ⁢ ( t ) ≤ C 0 ⁢ ( 1 + | t | q ) ,

      C 0 - 1 ⁢ ( 1 + | t | q ) ≤ a ⁢ ( t ) + 1 p ⁢ t ⁢ a ′ ⁢ ( t ) ≤ C 0 ⁢ ( 1 + | t | q ) .

   2. The function

      g a ⁢ ( t ) = a ⁢ ( t ) + 1 p ⁢ t ⁢ a ′ ⁢ ( t ) | t | q

      is increasing in (-∞,0) and decreasing in (0,∞).

3. b ∈ C 1 ⁢ ( ℝ , ℝ ) and satisfies the following conditions:

   1. There exists C0>0 such that for t∈ℝ

      C 0 - 1 ⁢ ( 1 + | t | q ) ≤ b ⁢ ( t ) ≤ C 0 ⁢ ( 1 + | t | q ) ,

      C 0 - 1 ⁢ ( 1 + | t | q ) ≤ b ⁢ ( t ) + 1 p ⁢ t ⁢ b ′ ⁢ ( t ) ≤ C 0 ⁢ ( 1 + | t | q ) .

   2. The function

      g b ⁢ ( t ) = b ⁢ ( t ) + 1 p ⁢ t ⁢ b ′ ⁢ ( t ) | t | q

      is increasing in (-∞,0) and decreasing in (0,∞).

   3. b is multiplicative convex, that is, for s,t∈ℝ, s⁢t≥0, it holds that

      b ⁢ ( s ⁢ t ) ≤ b ⁢ ( s ) ⋅ b ⁢ ( t ) .

4. f ∈ C ⁢ ( ℝ , ℝ ) and satisfies the following conditions:

   1. lim t → 0 ⁡ f ⁢ ( t ) | t | p - 1 = 0 .

   2. There exist C0>0, p+q<ν<(1+qp)⁢p* such that for t∈ℝ

      | f ⁢ ( t ) | ≤ C 0 ⁢ ( 1 + | t | ν - 1 ) .

   3. lim | t | → ∞ ⁡ f ⁢ ( t ) ⁢ t | t | α ⁢ ( p + q ) = + ∞ .

   4. The function f⁢(t)⁢t|t|α⁢(p+q) is decreasing in (-∞,0) and increasing in (0,∞).

Set

A function u∈X is called a (weak) solution of (1.1) if, for all φ∈C0∞⁢(Ω), it holds that

Formally, the problem (1.1) has a variational structure, given by the functional

where F⁢(t)=∫0tf⁢(τ)⁢d⁢τ. Given u∈X, assume φ∈X with the property that

for example, φ∈C0∞⁢(Ω) or φ=u,u+ and u-, where u+=max⁡{u,0},u-=min⁡{u,0}. We define the derivative of I in the direction φ at u, denoted by 〈D⁢I⁢(u),φ〉 as

Hence u is a weak solution of (1.1) if and only if the derivative 〈D⁢I⁢(u),φ〉 at u is zero for every direction φ∈C0∞⁢(Ω). If u is a weak solution of (1.1), we say that u is a critical point of I and c=I⁢(u) is a critical value of I.

In this paper, we shall use the Nehari method. For u∈X, define

and

Here are our main results.

Condition (c) says that

We consider the case s>0, t>0. Let s=ex and t=ey. Then condition (c) can be rewritten as

That is, the function g⁢(x)=ln⁡b⁢(ex), x∈ℝ, is convex. The sufficient and necessary condition is g′′≥0 or g′ is increasing. We have g′⁢(x)=ex⁢b′⁢(ex)b⁢(ex); g′ is increasing in x∈ℝ if and only if t⁢b′⁢(t)b⁢(t) is increasing in t∈(0,+∞).

Recently, the Kirchhoff problem, both on bounded domains and on the whole space ℝN or both semilinear equations and quasilinear equations, has received increasing attention. First, the classical Kirchhoff type equation

which is related to the stationary analogue of the equation

proposed by Kirchhoff in [19] as an extension of the classical D’Alembert wave equations for free vibration of elastic strings. In the pioneering work of Lions [25], an abstract functional analysis framework was proposed to (1.4):

Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Notice that, in (1.5), u denotes the displacement, f⁢(x,u) is the external force and c2 is the initial tension while a is related to the intrinsic properties of the string, such as Young’s modulus. It is pointed out that the problem (1.5) models several physical and biological systems, where u describes a process that depends on the average of itself, for example, population density. For more physical background of this Kirchhoff problem, see [19, 3, 5, 9] and the references therein. Mathematically, (1.3) is a nonlocal problem as the appearance of the nonlocal term ∫Ω|∇⁡u|2⁢d⁢x⁢Δ⁢u, which implies that (1.3) is not a pointwise identity. This leads to some mathematical difficulties, which attract the attention of many mathematicians. Recently, there have been a lot of results on this problem, and several methods have been developed, see [8, 15, 23, 28, 34]. Especially, the existence of positive solutions, multiple solutions, ground states and semiclassical states, sign-changing solutions for the Kirchhoff type problem have been established by the variational method; see, for example, [4, 1, 11, 24, 20, 26, 29, 27, 17, 31] for the bounded domain and [2, 16, 22, 30, 32, 10] for the whole space.

If p≠2, a,b are constants and f satisfies the subcritical growth condition. Guo and Nie [12] considered the equation

They obtained nontrivial weak solutions whenever λ>0 is sufficiently large by variational methods. Han, Ma and He [14] investigated the existence of sign-changing solutions of equation (1.6) for λ=1. In [33], the authors dealt with the existence, multiplicity and asymptotic behavior of solutions for critical Kirchhoff type p-Laplacian problems. Also, there are some papers on the existence of solutions for Kirchhoff type quasilinear equations; see [18, 13] and the references therein.

In addition, for related work, we can refer to [21, 7]. The authors considered the generalized quasilinear Schrödinger equation with a Kirchhoff type perturbation

where λ>0, g∈C1⁢(ℝ,ℝ+), V⁢(x) and K⁢(x) are both positive continuous functions. Under some suitable assumptions on V and K, by using a change of variables, the authors obtained the existence of both the ground state signed and the ground state sign-changing solutions for (1.7). Moreover, the convergence property and the nodal property for solutions were established in [7]. Notice that their method only can be used to treat one integral term ∫Ωg⁢(u)⁢|∇⁡u|2⁢d⁢x in the corresponding functional, so that they can make a change of variables as ∇⁡v=g12⁢(u)⁢∇⁡u (or d⁢v=g12⁢(u)⁢d⁢u); then ∫Ωg⁢(u)⁢|∇⁡u|2⁢d⁢x=∫Ω|∇⁡v|2⁢d⁢x; thus the quasilinear equation is reduced to the semilinear equation. So, essentially, they discussed semilinear elliptic equations. If there is the integral term ∫Ω∑i,j=1Nai⁢j⁢(u)⁢∂i⁡u⁢∂j⁡u⁢d⁢x or two integral terms ∫Ωg⁢(u)⁢|∇⁡u|2⁢d⁢x and ∫Ωf⁢(u)⁢|∇⁡u|2⁢d⁢x in the corresponding functional, then the quasilinear equation cannot be reduced to the semilinear equation by applying the idea of change of variables.

In this paper, we will consider the existence of solutions of (1.1). Due to the appearance of two integral terms 1p⁢∫Ωa⁢(u)⁢|∇⁡u|p⁢d⁢x and μα⁢p⁢(∫Ωb⁢(u)⁢|∇⁡u|p⁢d⁢x)α at the same time, the problem (1.1) cannot be transformed into a semilinear equation by a change of variables. However, in the terms 12⁢∫Ωg⁢(u)⁢|∇⁡u|2⁢d⁢x and 14⁢(∫Ωg⁢(u)⁢|∇⁡u|2⁢d⁢x)2, g⁢(u) is the same function in [21, 7]. Hence the quasilinear problem (1.1) is different from (1.7). So it is rather difficult to obtain the existence of solutions for the problem (1.1). We will utilize the Nehari method to directly treat the quasilinear equation of Kirchhoff type (1.1), obtain the existence of ground state signed solutions and sign-changing solutions and compare the critical values, corresponding to signed solutions and sign-changing solutions.

The paper is organized as follows: In Section 2, we prove Theorem 1.1; in Section 3, we prove Theorem 1.2. Finally, in Section 4, we consider what happens when the parameter μ→0+ in equation (1.1) and prove some convergence property.

2 Ground State Signed Solutions

For presentation convenience, for u∈X, we define

Then I⁢(u)=I1⁢(u)+I2⁢(u)+I3⁢(u).