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 [9] are essential in this more general framework. The techniques used in the proofs are lower and upper solutions and variational methods.