Nonlinear elliptic differential equations with multivalued nonlinearities

Antonella Fiacca; Nikolaos M. Matzakos; Nikolaos S. Papageorgiou; Raffaella Servadei

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 1, page 135-159
  • ISSN: 0011-4642

Abstract

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In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ . Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ . This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).

How to cite

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Fiacca, Antonella, et al. "Nonlinear elliptic differential equations with multivalued nonlinearities." Czechoslovak Mathematical Journal 53.1 (2003): 135-159. <http://eudml.org/doc/30765>.

@article{Fiacca2003,
abstract = {In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\mathbb \{R\}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\mathbb \{R\}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).},
author = {Fiacca, Antonella, Matzakos, Nikolaos M., Papageorgiou, Nikolaos S., Servadei, Raffaella},
journal = {Czechoslovak Mathematical Journal},
keywords = {upper solution; lower solution; order interval; truncation function; pseudomonotone operator; coercive operator; extremal solution; Yosida approximation; nonsmooth Palais-Smale condition; critical point; eigenvalue problem; upper solution; lower solution; order interval; truncation function; pseudomonotone operator; coercive operator; extremal solution; Yosida approximation; nonsmooth Palais-Smale condition; critical point; eigenvalue problem},
language = {eng},
number = {1},
pages = {135-159},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nonlinear elliptic differential equations with multivalued nonlinearities},
url = {http://eudml.org/doc/30765},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Fiacca, Antonella
AU - Matzakos, Nikolaos M.
AU - Papageorgiou, Nikolaos S.
AU - Servadei, Raffaella
TI - Nonlinear elliptic differential equations with multivalued nonlinearities
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 1
SP - 135
EP - 159
AB - In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\mathbb {R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\mathbb {R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).
LA - eng
KW - upper solution; lower solution; order interval; truncation function; pseudomonotone operator; coercive operator; extremal solution; Yosida approximation; nonsmooth Palais-Smale condition; critical point; eigenvalue problem; upper solution; lower solution; order interval; truncation function; pseudomonotone operator; coercive operator; extremal solution; Yosida approximation; nonsmooth Palais-Smale condition; critical point; eigenvalue problem
UR - http://eudml.org/doc/30765
ER -

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