Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values

Abstract

In this paper we study the existence of multiple solutions to the equation \begin{align*} -{\Delta_p} u = f(x,u)-|u|^{p-2}u \end{align*} with the nonlinear boundary condition \begin{align*} |\nabla u|^{p-2} \frac{\partial u}{ \partial \nu} = \lambda |u|^{p-2}u+g(x,u). \end{align*} We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial sign-changing solution when the parameter $\lambda$ is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison principles for nonlinear elliptic differential inequalities. In particular, we make use of variational and topological tools, such as critical point theory, the mountain-pass theorem, the second deformation lemma and variational characterizations of the second eigenvalue of the Steklov eigenvalue problem.