### Multiple solutions for asymptotically linear resonant elliptic problems

DOI: http://dx.doi.org/10.12775/TMNA.2003.014

#### Abstract

In this paper we establish the existence of multiple solutions for the

semilinear elliptic problem

$$\alignedat 2

-\Delta u&=g(x,u) &\quad&\text{in } \Omega, \\

u&=0 &\quad&\text{on } \partial\Omega,

\endalignedat

\tag 1.1

$$

where $\Omega \subset {\mathbb R}^N$ is a bounded domain with smooth

boundary $\partial \Omega$,

a function $g\colon\Omega\times{\mathbb R}\to {\mathbb R}$

is of class $C^1$ such that $g(x,0)=0$ and

which is asymptotically linear at infinity.

We considered both cases,

resonant and nonresonant.

We use critical groups to distinguish the critical points.

semilinear elliptic problem

$$\alignedat 2

-\Delta u&=g(x,u) &\quad&\text{in } \Omega, \\

u&=0 &\quad&\text{on } \partial\Omega,

\endalignedat

\tag 1.1

$$

where $\Omega \subset {\mathbb R}^N$ is a bounded domain with smooth

boundary $\partial \Omega$,

a function $g\colon\Omega\times{\mathbb R}\to {\mathbb R}$

is of class $C^1$ such that $g(x,0)=0$ and

which is asymptotically linear at infinity.

We considered both cases,

resonant and nonresonant.

We use critical groups to distinguish the critical points.

#### Keywords

Cerami condition; multiplicity of solutions; double resonance; sign changing solution

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