A multiplicity result for hemivariational inequalities
Let \(\Omega\) be an unbounded domain in \(R^{N},X\) be a Banach space (compactly embedded in some space \(L^{r}(\Omega),A\) the duality mapping on \(X\) induced by the weight function \(t\rightarrow t^{p-1}(1<p<N),F:\mathbb{R}\rightarrow \mathbb{R}\) a locally Lipschitz function with generalized derivative \(F^{0}:\) \[ F^{0}(x;y)=\lim_{z\rightarrow x}\sup_{\tau\rightarrow 0+}\frac{F(z+\tau y)-F(z)}{\tau}, \] \(b:\Omega \rightarrow \mathbb{R}\) a non-negative function. In this work it is proved the existence such \(u_0\in X\) and \(\lambda>0\) that the hemivariational inequality \[ <A(u-u_0)>+\lambda \int_{\Omega}b(x)F^{0}(u(x;-v(x)))dx\geq 0,\forall v\in X \] admits at least three solutions in \(X\). As applications of the general result in Sobolev spaces some variational inequalities are considered.
scientific article; zbMATH DE number 5141611