Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Abstract

In this paper we study the existence, nonexistence and multiplicity of positive solutions for the family of problems $-\Delta u = f_\lambda (x,u)$, $u \in H^1_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3$ and $\lambda>0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti-Brezis-Cerami type in a more general form, namely $\lambda a (x)u^q + b(x) u^p$ , where $0 \leq q<1<p\leq 2^{\ast}-1$. The coefficient a(x) is assumed nonnegative but b(x) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in \cite{dF-G-U} are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.