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

Roberto Livrea, Pasquale Candito, Carl, Roberto Livrea

Risultato della ricerca: Article

4 Citazioni (Scopus)

### Abstract

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.
Lingua originale English 397-410 14 Dynamic Systems and Applications 22 Published - 2013

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Pseudomonotone Operator
Nonlinear Elliptic Problems
Elliptic Problems
Dependent
Discontinuous Nonlinearities
Nonsmooth Analysis
Operator Theory
Multiplicity Results
Differential Inequalities
P-Laplacian
Nontrivial Solution
Variational Principle
Truncation
Dirichlet Problem
Bounded Domain
Critical point
Regularity
Fixed point
Denote
Partial

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cita questo

Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems. / Livrea, Roberto; Candito, Pasquale; Carl; Livrea, Roberto.

In: Dynamic Systems and Applications, Vol. 22, 2013, pag. 397-410.

Risultato della ricerca: Article

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abstract = "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. {\circledC}Dynamic Publishers, Inc.",
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T1 - Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems

AU - Livrea, Roberto

AU - Candito, Pasquale

AU - Carl, null

AU - Livrea, Roberto

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - p-Laplacian, Dirichlet problem

UR - http://hdl.handle.net/10447/258581

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VL - 22

SP - 397

EP - 410

JO - Dynamic Systems and Applications

JF - Dynamic Systems and Applications

SN - 1056-2176

ER -