Multiple variational solutions to nonlinear Steklov problems (Q692796)

The authors consider the problem of finding a harmonic function \(u\) in a bounded domain \({\Omega \subset {\mathbb R}^N}\), \(N \geq 2,\) satisfying a nonlinear boundary condition of the form \[ \partial_{\nu}u(x)=\lambda\,(\eta(x)u(x)+\mu(x)h(u(x)),\,\, x\in \partial \Omega, \] where \(\mu\) and \(\eta\) are bounded functions and \(h\) is a \({\mathcal{C}^1}\) odd function with subcritical growth at infinity and such that \(\lim_{s\rightarrow \infty} h^{\prime}(s) = +\infty\). By using variants of the mountain pass lemma based on index theory, the authors discuss existence and multiplicity of non trivial solutions to the problem for every value of \(\lambda \).