Elliptic problems involving natural growth in the gradient and general absorption terms

Abstract

In this paper, we treat the existence of solutions for a class of general elliptic problems whose prototype is the following: \begin{equation} \begin{cases}-\Delta _{p}u+h(x)|u|^{q-1}u=\beta |\nabla u|^{p}+\lambda f(x) &\mbox {in } \Omega , \\ u=0 &\mbox {on } \partial \Omega ,\end{cases} \end{equation} where $\Omega $ is a bounded open subset of $\mathbb {R}^{N}$ with $N>1$, $1\lt p\lt N$, $q\geq 1$, $\lambda \in \mathbb {R}$, $\beta \in \mathbb {R}$, $h\in L^{1}(\Omega )$ with $h\geq 0$ and $f\in L^{1}(\Omega )$. Assuming that the source term $f$ satisfies $$\lambda _{1}(f)=\inf \bigg \{ \frac {\int _{\Omega }\vert \nabla w\vert ^{p}dx}{\int _{\Omega }|f|\vert w\vert ^{p}dx}:w\in W_{0}^{1,p} (\Omega )\setminus \{ 0\} \bigg \}>0,$$ we obtain the existence of a solution $u\in W_{0}^{1,p}(\Omega )$ when $|\lambda |$ is sufficiently small.