Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

Abstract

We provide the existence of a solution for quasilinear elliptic equation $-\text{div}(a_{\infty }(x)|\nabla u{|}^{p-2}\nabla u+\tilde{a}(x,|\nabla u|)\nabla u)=\lambda m(x)|u{|}^{p-2}u+f(x,u)+h(x)$ in $\mathrm{\Omega }$ under the Neumann boundary condition. Here, we consider the condition that $\tilde{a}(x,t)=o(t^{p-2})$ as $t\to +\infty$ and $f(x,u)=o(|u{|}^{p-1})$ as $|u|\to \infty$. As a special case, our result implies that the following $p$-Laplace equation has at least one solution: $-{\mathrm{\Delta }}_{p}u=\lambda m(x)|u{|}^{p-2}u+\mu |u{|}^{r-2}u+h(x)$ in $\mathrm{\Omega },\partial u/\partial \nu =0$ on $\partial \mathrm{\Omega }$ for every , $\lambda \in ℝ$, $\mu \ne 0$ and $m,h\in L^{\infty }(\mathrm{\Omega })$ with ${\int }_{\mathrm{\Omega }}mdx\ne 0$. Moreover, in the nonresonant case, that is, $\lambda$ is not an eigenvalue of the $p$-Laplacian with weight $m$, we present the existence of a solution of the above $p$-Laplace equation for every , $\mu \in ℝ$ and $m,h\in L^{\infty }(\mathrm{\Omega })$.