Existence and multiplicity results for semilinear ellipctic equations with measure data and jumping nonlinearities

Alberto Ferrero, Claudio Saccon

DOI: http://dx.doi.org/10.12775/TMNA.2007.018


We study existence and multiplicity results for semilinear elliptic
equations of the type $-\Delta u=g(x,u)-te_1+\mu$ with homogeneous Dirichlet
boundary conditions. Here $g(x,u)$ is a jumping nonlinearity, $\mu$ is a
Radon measure, $t$ is a positive constant and $e_1> 0$ is the first
eigenfunction of $-\Delta$. Existence results strictly depend on the
asymptotic behavior of $g(x,u)$ as $u\rightarrow\pm \infty$. 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.


Semilinear equations; Radon measures; critical point theory

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