Multiplicity of solutions for resonant and non-resonant asymptotically linear elliptic problems

TL;DR

This paper investigates the existence and multiplicity of solutions for asymptotically linear elliptic problems, establishing conditions under which multiple solutions exist based on the nonlinearity's behavior at infinity.

Contribution

It provides new existence and multiplicity results for elliptic problems with weaker nonlinear assumptions than previously considered.

Findings

Existence of a signed ground state solution.

Multiple solutions when the nonlinearity is odd.

The number of solutions relates to the asymptotic behavior of the nonlinearity.

Abstract

Results about existence of a signed ground state solution and multiple solutions (if $f$ is odd with respect to the second variable) are proven for a class of asymptotically linear elliptic problems involving a Carath\'eodory type nonlinearity satisfying assumptions weaker than (fF) in \cite{GLZ} or $(F_{2})_{+}$ in \cite{CM}. A close relation between the behaviour of $\lim_{|t|\to \infty}f(x, t)/|t|$ and the number of solutions is stablished.