Multiple variational solutions to nonlinear Steklov problems
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 \(\lambda \).
scientific article; zbMATH DE number 6113127
scientific article