Existence theorems for nonlinear elliptic problems (Q5952858)

Let \(Z\subset \mathbb{R}^N\) be a bounded domain with \(C^1\)-boundary \(\Gamma \). In the first part of the article the existence of a solution of the Dirichlet problem \(-\operatorname {div} (|Dx(z)|^{p-2}Dx(z))+\partial j(z,x(z))\) a.e. on \(Z\), \(x=0\) on \(\Gamma \), is proved. Here \(\partial j\) denotes the subdiffererential in the sense of Clarke of \(j(z,.)\). In the second part a nontrivial solution of the Neumann problem \(-\operatorname {div} (|Dx(z)|^{p-2}Dx(z))=f(z,x(z))-h(z)\) a.e. on \(Z\), \(-(|Dx(z)|^{p-2}Dx(z),n(z))\in \beta (z,\tau x(z))\) a.e. on \(\Gamma \), is found. Here \(\tau \) is the trace operator on \(W^{1,p}(Z)\) and \(n(z)\) is the outward normal at \(z\in \Gamma \).