Existence results for quasilinear elliptic equations with indefinite weight
Summary: We provide the existence of a solution for quasilinear elliptic equations \(-\text{div}(a_\infty(x)|\nabla u|^{p-2} \nabla u + \tilde{a}(x, |\nabla u|)\nabla u) = \lambda m(x)|u|^{p-2}u + f(x, u) + h(x)\) in \(\Omega\) under the Neumann boundary condition. Here, we consider the condition that \(\tilde{a}(x, t) = o(t^{p-2}\)) as \(t \rightarrow +\infty\) and \(f(x, u) = o(|u|^{p-1})\) as \(|u| \rightarrow \infty\). As a special case, our result implies that the following \(p\)-Laplace equation has at least one solution: \(-\Delta_p u = \lambda m(x)|u|^{p-2}u + \mu|u|^{r-2}u + h(x)\) in \(\Omega, \partial u/\partial v = 0\) on \(\partial \Omega\) for every \(1 < r < p < \infty, \lambda \in \mathbb R, \mu \neq 0\) and \(m, h \in L^\infty(\Omega)\) with \(\int_\Omega mdx \neq 0\). 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 \(1 < r < p < \infty, \mu \in \mathbb R\) and \(m, h \in L^\infty(\Omega)\).
scientific article; zbMATH DE number 6078756
scientific article