Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems

Roberto Livrea, Carl, Roberto Livrea, Pasquale Candito

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4 Citations (Scopus)


We study the existence of nontrivial solutions of parameter-dependent quasilinear elliptic Dirichlet problems of the form $-\Delta u = \lambda f(u)$ in $\Omega$, $u = 0$ on $\partial\Omega$, in a bounded domain $\Omega$ with sufficiently smooth boundary, where $\lambda$ is a real parameter and $\Delta_p$ denotes the p-Laplacian. Recently the authors obtained multiplicity results by employing an abstract localization principle of critical points of functional of the form $\Phi-\lambda\Psi$ on open subleveis of $\Phi$, i.e., of sets of the form $\Phi^{-1}(-\infty,r)$, combined with differential inequality techniques and topological arguments. Unlike in those recent papers by the authors, the approach in this paper is based on pseudomonotone operator theory and fixed point techniques. The obtained results are compared with those obtained via the abstract variational principle. Moreover, by applying truncation techniques and regularity results we are able to deal with elliptic problems that involve discontinuous nonlinearities without making use of nonsmooth analysis methods. ©Dynamic Publishers, Inc.
Original languageEnglish
Pages (from-to)397-410
Number of pages14
JournalDynamic Systems and Applications
Publication statusPublished - 2013

All Science Journal Classification (ASJC) codes

  • General Mathematics


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