Multiple solutions for asymptotically linear resonant elliptic problems

Francisco Odair de Paiva

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.

Keywords


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

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