Quasilinear elliptic boundary value problems with discontinuous nonlinearities (Q2721910)

This paper deals with a discontinuous nonlinear nonmonotone elliptic boundary value problem, that is a quasilinear elliptic hemivariational inequality NEWLINE\[NEWLINE\begin{cases} Au+e(x)= f(x),\quad & x\in\Omega\\ u(x)=0,\quad & x\in \partial \Omega\\ e(x)\in \partial J\bigl(u(x) \bigr),\quad & x\in\Omega \end{cases} \tag{1}NEWLINE\]NEWLINE where \(Au=-\sum^n_{j=1} {\partial\over \partial x_j}a_j(x,u, \nabla u)+a_0(x,u, \nabla u)\) and the symbol \(\partial J\) designates Clarke's generalized gradient of locally Lipschitz functional \(J\). Under some natural conditions on \(e,a_i,i=0, \dots,n\) the author proves the existence of solutions of (1). A proof of the existence of solutions is based on the theory of pseudomonotone operator theory.