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 ellipticequations of the type $-Δ u=g(x,u)-te_1+μ$ with homogeneous Dirichletboundary conditions. Here $g(x,u)$ is a jumping nonlinearity, $μ$ is aRadon measure, $t$ is a positive constant and $e_1> 0$ is the firsteigenfunction of $-Δ$. Existence results strictly depend on theasymptotic behavior of $g(x,u)$ as $u→± ∞$. Depending onthis asymptotic behavior, we prove existence of two and three solutions for $t> 0$ large enough. In order to find solutions of the equation, we introducea suitable action functional $I_t$ by mean of an appropriate iterativescheme. Then we apply to $I_t$ standard results from the critical pointtheory and we prove existence of critical points for this functional.