Multiple solutions to asymmetric semilinear elliptic problems via Morse theory

Abstract

In this article we study the existence of solutions to the problem $$\displylines{ -\Delta u = g(x,u) \quad \text{in } \Omega; \cr u = 0 \quad\text{on } \partial\Omega, }$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $(N\geq 2)$ and $g:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a differentiable function with g(x,0)=0 for all $x\in\Omega$. By using minimax methods and Morse theory, we prove the existence of at least three nontrivial solutions for the case in which an asymmetric condition on the nonlinearity g is assumed. The first two nontrivial solutions are obtained by employing a cutoff technique used by Chang et al in [9]. For the existence of the third nontrivial solution, first we compute the critical group at infinity of the associated functional by using a technique used by Liu and Shaoping in [19]. The final result is obtained by using a standard argument involving the Morse relation.

Keywords

• Morse theory
• critical groups, local linking