Existence of solutions to resonant elliptic boundary value problems with unbounded discontinuous nonlinearities (Q642055)

This paper is concerned with existence results for semiregular solutions to resonant elliptic problems with discontinuous sublinear nonlinearities. More exactly, the author investigates the elliptic equation \({\mathcal L}u+g(x,u)=0\) posed in a bounded and smooth domain \(\Omega\) of \(R^n\), \(n\geq 2\). Here \({\mathcal L}\) is a general uniformly elliptic and self-adjoint differential operator and \(g\) is a Borel discontinuous function satisfying \(g(x,u)\leq a(x)+b|u|^r\) for a.e. \(x\in \Omega\) where \(0\leq r<1\). The equation is complemented with a linear combination with variable coefficients between Dirichlet and Neumann conditions on the boundary. The requirement for \(g\) suggested by the author as a generalized Landesman-Lazer condition is \(\lim_{\|\psi\|\rightarrow 0, \psi\in \text{Ker}({\mathcal L})}\frac{1}{\|\psi\|^{2r}}\int_{\Omega}dx\int 0^{\psi(x)} g(x,s)ds=+\infty\).