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Nontrivial solutions of semilinear elliptic equations with continuous or discontinuous nonlinearities

Author: Noriko Mizoguchi
Journal: Proc. Amer. Math. Soc. 116 (1992), 513-520
MSC: Primary 35J65; Secondary 47H15, 58E05
MathSciNet review: 1098403
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Abstract: In this paper, we are concerned with the boundary value problem of the form $ - \Delta u = g(u)$ in $ \Omega ,u{\vert _{\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 [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{g} (u),\bar g(u)]$ in $ \Omega ,u{\vert _{\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).

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Keywords: Semilinear elliptic equation, resonance, discontinuous nonlinearity
Article copyright: © Copyright 1992 American Mathematical Society

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