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Nonsmooth critical point theory and nonlinear elliptic equations at resonance. (English) Zbl 0999.58006

Summary: We complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the \(p\)-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In the simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J20 Variational methods for second-order elliptic equations
35R70 PDEs with multivalued right-hand sides
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