Multiple solutions for a coercive quasilinear elliptic equation via Morse theory

We study the quasilinear elliptic problem which is resonant at zero. By using Morse theory, we obtain five nontrivial solutions for the equation with coercive nonlinearities.


Introduction
Let be a bounded domain in R N (N ≥ 1) with smooth boundary ∂ . We study the following quasilinear elliptic problem: where 2 < p < ∞ and p denotes the p-Laplacian operator defined by In what follows, we denote by 0 < λ 1 < λ 2 ≤ · · · ≤ λ k ≤ · · · the eigenvalues of -in W 1,2 0 ( ), and we let μ 1 > 0 be the first eigenvalue ofp in W 1,p 0 ( ) (see [9]). Moreover, we make the following assumptions: (f 1 ) f ∈ C 1 ( × R, R) with f (x, 0) = 0, and satisfies the following condition: f (x, t) ≤ c 1 + |t| q-2 , ∀t ∈ R, x ∈ , for some constants c > 0 and q ∈ [2, p * ), where p * = Np/(Np) if p < N and p * = +∞ if N ≤ p, (f 2 ) there exist M > 0 and λ < λ 1 2 such that where F(x, t) = t 0 f (x, s) ds, (f -) there exist α > 0 and k ≥ 3 such that Under the conditions above, from [14, Theorem 1.2] we know that Eq. (1.1) has at least four nontrivial solutions. It is worth pointing out that, using similar conditions, the authors in [10, Theorem 3.2] not only obtain four nontrivial solutions, but also prove that two of them are sign changing. Moreover, when the nonlinearity f is resonant at infinity and non-resonant at zero, using variational methods, together with truncation and comparison techniques and Morse theory, the paper [11] can get the existence of six nontrivial solutions (two of them are sign changing).
The aim of this paper is to obtain the existence of another solution. Specifically, our result reads as follows. Remark 1.2 (1) In our proof, we first obtain a nontrivial solution near zero inspired by papers [13,15]. Then we use the estimation of critical groups to distinguish the new solution from the known solutions of [10,14]. The method of estimating critical groups comes from [5], which has studied the bifurcation problem of semilinear elliptic equations at zero, and obtained six nontrivial solutions of the equation with coercive nonlinearities.
(2) Checking the proof below, our result is also true when p = 2. So as far as we know, our theorem is new even for the semilinear elliptic equation. This paper is organized as follows. In Sect. 2, by Morse theory the existence of a new nontrivial solution and the estimation of its critical groups are given. In Sect. 3, we give the proof of Theorem 1.1. In the sequel, the letter C will be used indiscriminately to denote a suitable positive constant whose value may change from line to line.

A solution near zero
For any λ ∈ R, let f (x, u) = λ k u + g(x, u) and G(x, u) = u 0 g(x, s) ds, then we consider the C 2 functional I λ : W 1,p 0 ( ) → R defined by setting where W 1,p 0 ( ) is the Sobolev space endowed with the norm u = ∇u p = |∇u| p dx 1/p . By (f 1 ), weak solutions of Eq. (1.1) correspond to critical points of functional I λ k , which is also defined in the following form: By [6, Page 277], the second order differential of I λ k in isolated critical point u 0 is given by In addition, if we assume that I λ k (u 0 ) = c ∈ R, and U is an isolated neighborhood of u 0 , then the group is called the th critical group of the functional I λ k at u 0 , where I c λ k = {u ∈ W 1,p 0 ( ) : I λ k (u) ≤ c}, and H (·, ·) are the singular relative homological groups with a coefficient group F (see [1,Definition 4

.1, Chapter I]).
Before stating our results, we recall the following result concerning critical groups estimates.
Proof The proof will be divided into several steps.

Proof of theorem
Now we can give the proof of our theorem as follows.
Proof of Theorem 1.1 Under our assumptions, Ref. [14] has proved that I λ k satisfies the Palais-Smale condition, and there are three nontrivial solutions u i (i = 1, 2, 3). Moreover, two of them are local minima such that and u 3 is the mountain pass solution such that (see [11] or [10, Page 412]) From Lemma 2.2 and Lemma 2.3, we know that Reasoning by contradiction, when u i ∈ W , from Lemma 2.2, Lemma 2.3 and Remark 2.4, there is ρ > 0 small enough such that C (I λ k , u i ) = {0}, ∀ ≤ d k-1 -1, i = 1, 2, 3, which is in contradiction with (3.1) and (3.2) because of k ≥ 3. Then the claim holds.
If I λ k has only four nontrivial critical points: v 0 and u i for i = 1, 2, 3, then, for a < inf I λ k (K(I λ k )), [14,Lemma 4.1] gives the th critical group of I λ k at infinity:

Conclusions
Using the critical groups estimates, our theorem can get more nontrivial solutions. The main results presented in this paper improve and generalize the results in [10,14].