Existence of solutions for a nonhomogeneous semilinear elliptic equation


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For a bounded domain $\Omega$, a bounded Carathéodory function $g$ in $\Omega\times\mathbb{R}$, $p>1$ and a nonnegative measurable function $h$ in $\Omega$ which is strictly positive in a set of positive measure we prove that, contrarily with the case $h\equiv 0$, there exists a solution of the semilinear elliptic problem $$ \left \{ \begin{array}{rcll} -\Delta u & = & \lambda u +g(x,u)- h |u|^{p-1} u +f, & \mbox{in } \Omega \\ u & = & 0, & \mbox{on } \partial\Omega,\\ \end{array} \right. $$ for every $\lambda\in \mathbb{R}$ and $f\in\ L^2(\Omega)$.