Existence and multiplicity results for semilinear ellipctic equations with measure data and jumping nonlinearities - Ferrero, Alberto; Saccon, Claudio (Topological Methods in Nonlinear Analysis, 2007)
We study existence and multiplicity results for semilinear elliptic equations of the type -Δ u=g(x,u)-te_1+μ with homogeneous Dirichlet boundary conditions. Here g(x,u) is a jumping nonlinearity, μ is a Radon measure, t is a positive constant and e_1> 0 is the first eigenfunction of -Δ. Existence results strictly depend on the asymptotic behavior of g(x,u) as u→± ∞. Depending on this asymptotic behavior, we prove existence of two and three solutions for t> 0 large enough. In order to find solutions of the equation, we introduce a suitable action functional I_t by mean of an appropriate iterative scheme. Then we apply to I_t standard results from the critical point theory and we prove existence of critical points for this functional.