Quasilinear equations involving nonlinear Neumann boundary conditions. (English) Zbl 1265.35075

Summary: We study the multiplicity of positive solutions of the problem \[ -\Delta _pu+| u| ^{p-2}u=0 \] in a bounded smooth domain \(\Omega \subset \mathbb {R}^N\) with a nonlinear boundary condition given by \[ | \nabla u| ^{p-2}\frac {\partial u}{\partial \nu }=\lambda f(u)+\mu \varphi (x)| u| ^{q-1}u, \] where \(f\) is continuous and satisfies some kind of \(p\)-superlinear condition at 0 and \(p\)-sublinear condition at infinity, \(0<q<p-1\) and \(\varphi \) is \(L^{\beta }(\partial \Omega )\) for some \(\beta >1\). In addition, we consider the case \(q=0\), where the nonlinear boundary condition becomes an elliptic inclusion. Our approach allows us to show that these problems have at least six nontrivial solutions, three positive and three negative, for some positive parameters \(\lambda \) and \(\mu \). The proof is based on variational arguments.


35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
35B09 Positive solutions to PDEs