Abstract

This article shows the existence of weak solutions in $W_0^1(\Omega )$ to a class of Dirichlet problems of the form $$ - \hbox{div}({a({x,\nabla u} )})= \lambda_1 |u|^{p - 2} u + f(x,u)-h $$ in a bounded domain $\Omega$ of $\mathbb{R}^N$. Here a satisfies $$ |{a({x,\xi } )}| \leq c_0 \big({h_0 (x)+ h_1 (x )|\xi|^{p - 1}}\big) $$ for all $\xi \in \mathbb{R}^N$, a.e. $x \in \Omega$, $h_0 \in L^{\frac{p}{p - 1}} (\Omega )$, $h_1 \in L_{loc}^1 ( \Omega )$, $h_1(x) \geq 1$ for a.e. x in $x \in \Omega$; $\lambda_1$ is the first eigenvalue for $-\Delta_p$ on $x \in \Omega$ with zero Dirichlet boundary condition and g, h satisfy some suitable conditions.