Mutiplicity of solutions for almost critical elliptic problems via Orlicz spaces approach*

Abstract: We study the existence of positive solutions of a Dirichlet problem for a class of semilinear superlinear elliptic equations whose nonlinear term is of subcritical nature in a generalized sense and involves indefinite nonlinearities. More precisely, given Ω ⊂ R^N , N > 2, a bounded, connected open subset with C^2 boundary ∂Ω, we look for positive solutions to the elliptic pde − ∆u = λu + a(x) h(u) in Ω , u = 0, on ∂Ω, where λ > 0 is a real parameter, a (x) is a regular function and h(u) is a nonlinearity of non power type and slighty critical. The aim is first to prove existence of positive solutions when the parameter λ is less than the first eigenvalue of the laplacian operator and second to use the bifurcations result of Crandall-Rabinowitz to the right of this eigenvalue. In order to deal with this kind of nonlinearity in a variational framework, we introduce some Orliz spaces and discuss some compacteness issues. In this talk, we will also review some old and new results related to superlinear problems. This work is a collaboration with Rosa Pardo.

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