| Title:
|
Nonlinear elliptic differential equations with multivalued nonlinearities (English) |
| Author:
|
Fiacca, Antonella |
| Author:
|
Matzakos, Nikolas |
| Author:
|
Papageorgiou, Nikolaos S. |
| Author:
|
Servadei, Raffaella |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
135-159 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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). (English) |
| Keyword:
|
upper solution |
| Keyword:
|
lower solution |
| Keyword:
|
order interval |
| Keyword:
|
truncation function |
| Keyword:
|
pseudomonotone operator |
| Keyword:
|
coercive operator |
| Keyword:
|
extremal solution |
| Keyword:
|
Yosida approximation |
| Keyword:
|
nonsmooth Palais-Smale condition |
| Keyword:
|
critical point |
| Keyword:
|
eigenvalue problem |
| MSC:
|
35J20 |
| MSC:
|
35J60 |
| MSC:
|
35R70 |
| idZBL:
|
Zbl 1029.35093 |
| idMR:
|
MR1962005 |
| . |
| Date available:
|
2009-09-24T10:59:58Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127787 |
| . |
| Reference:
|
[1] R. Adams: Sobolev Spaces.Academic Press, New York, 1975. Zbl 0314.46030, MR 0450957 |
| Reference:
|
[2] H. Amann: Order structures and fixed points.Atti Anal. Funz. Appl, Univ. Cosenza, Italy, 1997, pp. 349–381. |
| Reference:
|
[3] A. Ambrosetti and R. Rabinowitz: Dual variational methods in critical point theory and applications.J. Funct. Anal. 14 (1973), 349–381. MR 0370183, 10.1016/0022-1236(73)90051-7 |
| Reference:
|
[4] A. Ambrosetti and R. Turner: Some discontinuous variational problems.Differential Integral Equations 1 (1988), 341–350. MR 0929921 |
| Reference:
|
[5] A. Ambrosetti and M. Badiale: The dual variational principle and elliptic problems with discontinuous nonlinearities.J. Math. Anal. Appl. 140 (1989), 363–373. MR 1001862, 10.1016/0022-247X(89)90070-X |
| Reference:
|
[6] M. Badiale: Semilinear elliptic problems in $\mathbb{R}^N$ with discontinuous nonlinearities.Atti Sem. Mat. Fis. Univ. Modena 43 (1995), 293–305. MR 1366063 |
| Reference:
|
[7] S. Carl and S. Heikkika: An existence result for elliptic differential inclusions with discontinuous nonlinearity.Nonlinear Anal. 18 (1992), 471–472. MR 1152722, 10.1016/0362-546X(92)90014-6 |
| Reference:
|
[8] K.-C. Chang: Variational methods for nondifferentiable functionals and its applications to partial differential equations.J. Math. Anal. Appl. 80 (1981), 102–129. MR 0614246, 10.1016/0022-247X(81)90095-0 |
| Reference:
|
[9] F. H. Clarke: Optimization and Nonsmoooth Analysis.Wiley, New York, 1983. MR 0709590 |
| Reference:
|
[10] D. Costa and J. Goncalves: Critical point theory for nondifferentiable functionals and applications.J. Math. Anal. Appl. 153 (1990), 470–485. MR 1080660, 10.1016/0022-247X(90)90226-6 |
| Reference:
|
[11] E. Dancer and G. Sweers: On the existence of a maximal weak solution for a semilinear elliptic equation.Differential Integral Equations 2 (1989), 533–540. MR 0996759 |
| Reference:
|
[12] J. Deuel and P. Hess: A criterion for the existence of solutions of nonlinear elliptic boundary value problems.Proc. Royal Soc. Edinburg 74 (1974-1975), 49–54. MR 0440191 |
| Reference:
|
[13] D. Gilbarg and N. Trudinger: Elliptic Partial Differential Equations of Second Order.Springer-Verlag, Berlin, 1983. MR 0737190 |
| Reference:
|
[14] J.-P. Gossez and V. Mustonen: Pseudomonotonicity and the Leray-Lions condition.Differential Intgral Equations 6 (1993), 37–46. MR 1190164 |
| Reference:
|
[15] S. Heikkila and S. Hu: On fixed points of multifunvtions in ordered spaces.Appl. Anal. 54 (1993), 115–127. MR 1278995 |
| Reference:
|
[16] S. Hu and N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory.Kluwer, Dordrecht, 1997. MR 1485775 |
| Reference:
|
[17] R. Landes: On Galerkin’s method in the existence theory of quasilinear elliptic equations.J. Funct. Anal. 39 (1980), 123–148. Zbl 0452.35037, MR 0597807 |
| Reference:
|
[18] J. Leray and J. L. Lions: Quelques resultats de Visik sur les problems elliptiques nonlinearairies par methodes de Minty-Browder.Bull. Soc. Math. France 93 (1965), 97–107. MR 0194733 |
| Reference:
|
[19] P. Lindqvist: On the equation $\div (|Dx|^{p-2}Dx)+\lambda |x|^{p-2}x=0$.Proc. Amer. Math. Soc. 109 (1990), 157–164. Zbl 0714.35029, MR 1007505 |
| Reference:
|
[20] M. Marcus and V. Mizel: Alsolute continuity on tracks and mappings of Sobolev spaces.Arch. Rational Mech. Anal. 45 (1972), 294–320. MR 0338765, 10.1007/BF00251378 |
| Reference:
|
[21] J.-J. Moreau: La notion de sur-potentiel et les liaisons unilaterales en elastostatique.CRAS Paris 267 (1968), 954–957. Zbl 0172.49802, MR 0241038 |
| Reference:
|
[22] P. D. Panagiotopoulos: Hemivariational Inequalities. Applications in Mechanics and Engineering.Springer-Verlag, Berlin, 1993. Zbl 0826.73002, MR 1385670 |
| Reference:
|
[23] C. Stuart: Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities.Math. Zeitsch. 163 (1978), 239–249. MR 0513729, 10.1007/BF01174897 |
| Reference:
|
[24] P. Tolksdorf: Regularity for a more general class of quasilinear elliptic equations.J. Differential Equations 51 (1894), 126–150. MR 0727034 |
| Reference:
|
[25] E. Zeidler: Nonlinear Functional Analysis and Its Applications II.Springer-Verlag, New York, 1990. Zbl 0684.47029, MR 0816732 |
| . |
