Abstract
We consider quasilinear strongly resonant problems with discontinuous right‐hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais‐Smale (PS)‐condition implies the coercivity of the functional, extending this way a well‐known result of the “smooth” case.
Highlights
In [16, 17], we studied quasilinear elliptic problems at resonance and near resonance with discontinuous right-hand side
In [16], we investigated the resonant problem and using a variational approach, we proved the existence of a nontrivial solution
In [17], we considered problems near resonance with the parameter λ approaching from the left the first eigenvalue λ1 of the p-Laplacian. For such problems we prove the existence of at least three nontrivial solutions
Summary
In [16, 17], we studied quasilinear elliptic problems at resonance and near resonance with discontinuous right-hand side. In [17], we considered problems near resonance with the parameter λ approaching from the left the first eigenvalue λ1 of the p-Laplacian For such problems we prove the existence of at least three nontrivial solutions. All the aforementioned works deal with semilinear equations which have a continuous right-hand side and prove the existence of one nontrivial solution. The first multiplicity result for the quasilinear resonant problem was obtained recently by Alves et al [4], who studied an equation with the p-Laplacian and a continuous right-hand side. This allows us to extend the well-known result in the “smooth” context to the present case
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