Multiplicity Results for a Class of Asymptotically Linear Elliptic Problems With Resonance and Applications to Problems With Measure Data

Abstract

Abstract We study existence and multiplicity results for solutions of elliptic problems of the type -Δu = g(x; u) in a bounded domain Ω with Dirichlet boundary conditions. The function g(x; s) is asymptotically linear as |s| → +∞. Also resonant situations are allowed. We also prove some perturbation results for Dirichlet problems of the type -Δu = gƐ(x, u) where gε(x, s) → g(x, s) as ε → 0. The previous results find an application in the study of Dirichlet problems of the type -Δu = g(x, u)+μ where μ is a Radon measure. To properly set the above mentioned problems in a variational framework we also study existence and properties of critical points of a class of abstract nonsmooth functional defined on Banach spaces and extend to this nonsmooth framework some classical linking theorems.