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 $-\Delta u=f_{\lambda }(x,u)$, $u\in H_0^1(\Omega )$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, $N\ge 3$ 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 $0\le q<1<p\le 2^{\ast }-1$. 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 [9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.