Multiplicity Results for a Class of Asymptotically Linear Elliptic Problems With Resonance and Applications to Problems With Measure Data
Alberto Ferrero, Claudio Saccon · Advanced Nonlinear Studies · 2010
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 = g Ɛ (x, u) where g ε (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.