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

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

#### Abstract

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.

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.

#### Keywords

Semilinear equations; Radon measures; critical point theory

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.