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Uniqueness results for nonlinear elliptic equations with a lower order term. (English) Zbl 1125.35343

Summary: We consider the functional \[ I(\lambda,u)= \tfrac12 \int_\Omega|\nabla u|^2-\lambda\log \biggl( \frac{1}{|\Omega|} \int_\Omega e^{it}\biggr), \quad u\in H_0^1(\Omega). \] When \(\lambda=8\pi N\) (\(N\) any positive integer), \(l(\lambda,\cdot)\) admits unbounded Palais-Smale sequences. To overcome this lack of compactness, we propose a new deformation lemma.

MSC:

35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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[1] Artola, M., Sur une classe de problèmes paraboliques quasi-linéaires, Boll. Un. Mat. Ital., 5-B, 51-70 (1986) · Zbl 0605.35044
[2] Artola, M.; Tartar, L., Un résultat d’unicité pour une classe de problèmes paraboliques quasi-linéaires, Ricerche Mat., 44, 409-420 (1995) · Zbl 0918.35068
[3] Barles, G.; Blanc, A. P.; Georgelin, C.; Kobylanski, M., Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28, 381-404 (1999) · Zbl 0940.35078
[4] Barles, G.; Murat, F., Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Rational Mech. Anal., 133, 77-101 (1995) · Zbl 0859.35031
[5] Betta, M. F.; Mercaldo, A.; Murat, F.; Porzio, M. M., Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in \(L^1(\Omega)\), ESAIM Control Optim. Calc. Var., 8, 239-272 (2002), (special issue dedicated to the memory of Jacques-Louis Lions) · Zbl 1092.35032
[6] Betta, M. F.; Mercaldo, A.; Murat, F.; Porzio, M. M., Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure, J. Math. Pures Appl., 82, 90-124 (2003)
[7] L. Boccardo, Uniqueness of solutions for some nonlinear Dirichlet problems, Proceedings of the Conference to Celebrate the 65th Birthday of M. Artola, Bordeaux, 1997, to appear.; L. Boccardo, Uniqueness of solutions for some nonlinear Dirichlet problems, Proceedings of the Conference to Celebrate the 65th Birthday of M. Artola, Bordeaux, 1997, to appear.
[8] Boccardo, L.; Gallouët, T.; Murat, F., Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris Sér. I, 315, 1159-1164 (1992) · Zbl 0789.35056
[9] Bottaro, G.; Marina, M. E., Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati, Boll. Un. Mat. Ital., 8, 46-56 (1973) · Zbl 0291.35021
[10] Carrillo, J.; Chipot, M., On some nonlinear elliptic equations involving derivatives of nonlinearity, Proc. Roy. Soc. Edinburgh A, 100, 281-294 (1985) · Zbl 0586.35044
[11] Chipot, M.; Michaille, G., Uniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22, 137-166 (1989) · Zbl 0699.35113
[12] Del Vecchio, T.; Porzio, M. M., Existence results for a class of non coercive Dirichlet problems, Ricerche Mat., 44, 421-438 (1995) · Zbl 0925.35067
[13] Del Vecchio, T.; Posteraro, M. R., An existence result for nonlinear and noncoercive problems, Nonlinear Anal., 31, 191-206 (1998) · Zbl 0911.35045
[14] Droniou, J., Non-coercive linear elliptic problems, Potential Anal., 17, 181-203 (2002) · Zbl 1161.35362
[15] Leray, J.; Lions, J.-L., Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93, 97-107 (1965) · Zbl 0132.10502
[16] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969.; J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969.
[17] Lopez-Pouso, O.; Porzio, M. M., Linear elliptic problems with mixed boundary conditions related to radiation heat transfer, Math. Models Methods Appl. Sci., 12, 541-565 (2002) · Zbl 1290.35060
[18] A. Mokrane, F. Murat, The Lewy-Stampacchia’s inequality for bilateral problems, Ricerche Mat. 53 (2004) 139-182.; A. Mokrane, F. Murat, The Lewy-Stampacchia’s inequality for bilateral problems, Ricerche Mat. 53 (2004) 139-182. · Zbl 1121.35066
[19] Porretta, A., Uniqueness and homogenization for a class of operators in divergence form, Atti Sem. Mat. Fis. Univ. Modena, 46, 915-936 (1998), (supplement dedicated to Professor C. Vinti) · Zbl 0914.35049
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