Nontrivial solutions of semilinear elliptic equations with continuous or discontinuous nonlinearities

Abstract:

In this paper, we are concerned with the boundary value problem of the form $- \Delta u = g(u)$ in $\Omega ,u{|_{\partial \Omega }} = 0$, where $g:{\mathbf {R}} \to {\mathbf {R}}$ is a continuous function, under assumptions of relations between $g$ and the eigenvalues of $- \Delta$. If $g$ is piecewise continuous on any bounded closed interval in ${\mathbf {R}}$, the above equation takes the form $- \Delta u \in [\underline {g}(u), \bar {g}(u)]$ in $\Omega ,u{|_{\partial \Omega }} = 0$. We obtain the existence of nontrivial solutions in both resonant and nonresonant cases at 0. Our theorems can be also applied when $g$ is discontinuous (may be discontinuous at 0).