Noncoercive elliptic equations with subcritical growth

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August 2012, 5(4): 857-864. doi: 10.3934/dcdss.2012.5.857

Noncoercive elliptic equations with subcritical growth

Vicenţiu D. Rădulescu ^1,

1.	Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania, Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Received January 2011 Revised February 2011 Published November 2011

Abstract
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We study a class of nonlinear elliptic equations with subcritical growth and Dirichlet boundary condition. Our purpose in the present paper is threefold: (i) to establish the effect of a small perturbation in a nonlinear coercive problem; (ii) to study a Dirichlet elliptic problem with lack of coercivity; and (iii) to consider the case of a monotone nonlinear term with subcritical growth. This last feature enables us to use a dual variational method introduced by Clarke and Ekeland in the framework of Hamiltonian systems associated with a convex Hamiltonian and applied by Brezis to the qualitative analysis of large classes of nonlinear partial differential equations. Connections with the mountain pass theorem are also made in the present paper.

Keywords: Mountain pass geometry., Nonlinear elliptic equation, Dirichlet boundary condition, Dual variational method.

Mathematics Subject Classification: Primary: 35J60; Secondary: 35A15, 58E05, 58E3.

Citation: Vicenţiu D. Rădulescu. Noncoercive elliptic equations with subcritical growth. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 857-864. doi: 10.3934/dcdss.2012.5.857

References:

[1]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.
[2]	G. Bonanno and S. A. Marano, Positive solutions of elliptic equations with discontinuous nonlinearities, Topol. Methods Nonlinear Anal., 8 (1996), 263-273.
[3]	H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliqueées pour la Maîtrise, Masson, Paris, 1983.
[4]	H. Brezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8 (1983), 409-426. doi: 10.1090/S0273-0979-1983-15105-4.
[5]	H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963. doi: 10.1002/cpa.3160440808.
[6]	S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal., 68 (2008), 2668-2676. doi: 10.1016/j.na.2007.02.013.
[7]	F. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc., 76 (1979), 186-188.
[8]	F. Clarke, Periodic solutions to Hamiltonian inclusions, J. Differential Equations, 40 (1981), 1-6. doi: 10.1016/0022-0396(81)90007-3.
[9]	I. Ekeland, A perturbation theory near convex Hamiltonian systems, J. Differential Equations, 50 (1983), 407-440. doi: 10.1016/0022-0396(83)90069-4.
[10]	R. Filippucci, P. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177. doi: 10.1016/j.matpur.2008.09.008.
[11]	N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 321-330.
[12]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[13]	A. Kristály, V. Rădulescu and Cs. Varga, "Variational Principles in Mathematical Physics, Geometry, and Economics. Qualitative Analysis of Nonlinear Equations and Unilateral Problems," Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010.
[14]	E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/1070), 609-623.
[15]	S. A. Marano and D. Motreanu, Existence of two nontrivial solutions for a class of elliptic eigenvalue problems, Arch. Math. (Basel), 75 (2000), 53-58.
[16]	R. Palais, Lusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132. doi: 10.1016/0040-9383(66)90013-9.
[17]	R. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165-171. doi: 10.1090/S0002-9904-1964-11062-4.
[18]	P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., IX (2010), 543-584.
[19]	P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. doi: 10.1016/0022-1236(84)90072-7.
[20]	P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1.
[21]	P. Pucci and J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc., 299 (1987), 115-132. doi: 10.1090/S0002-9947-1987-0869402-1.
[22]	V. Rădulescu, "Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods," Contemporary Mathematics and Its Applications, 6, Hindawi Publishing Corporation, New York, 2008.
[23]	V. Rădulescu, Remarks on a limiting case in the treatment of nonlinear problems with mountain pass geometry, Universitatis Babes-Bolyai Mathematica, LV (2010), 99-106.
[24]	J. Toland, A duality principle for nonconvex optimisation and the calculus of variations, Arch. Rational Mech. Anal., 71 (1979), 41-61. doi: 10.1007/BF00250669.
[25]	X. M. Zheng, Un résultat de non-existence de solution positive pour une équation elliptique, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 91-96.

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References:

[1]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.
[2]	G. Bonanno and S. A. Marano, Positive solutions of elliptic equations with discontinuous nonlinearities, Topol. Methods Nonlinear Anal., 8 (1996), 263-273.
[3]	H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliqueées pour la Maîtrise, Masson, Paris, 1983.
[4]	H. Brezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc., 8 (1983), 409-426. doi: 10.1090/S0273-0979-1983-15105-4.
[5]	H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963. doi: 10.1002/cpa.3160440808.
[6]	S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal., 68 (2008), 2668-2676. doi: 10.1016/j.na.2007.02.013.
[7]	F. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc., 76 (1979), 186-188.
[8]	F. Clarke, Periodic solutions to Hamiltonian inclusions, J. Differential Equations, 40 (1981), 1-6. doi: 10.1016/0022-0396(81)90007-3.
[9]	I. Ekeland, A perturbation theory near convex Hamiltonian systems, J. Differential Equations, 50 (1983), 407-440. doi: 10.1016/0022-0396(83)90069-4.
[10]	R. Filippucci, P. Pucci and F. Robert, On a $p$-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177. doi: 10.1016/j.matpur.2008.09.008.
[11]	N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 321-330.
[12]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[13]	A. Kristály, V. Rădulescu and Cs. Varga, "Variational Principles in Mathematical Physics, Geometry, and Economics. Qualitative Analysis of Nonlinear Equations and Unilateral Problems," Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010.
[14]	E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/1070), 609-623.
[15]	S. A. Marano and D. Motreanu, Existence of two nontrivial solutions for a class of elliptic eigenvalue problems, Arch. Math. (Basel), 75 (2000), 53-58.
[16]	R. Palais, Lusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132. doi: 10.1016/0040-9383(66)90013-9.
[17]	R. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165-171. doi: 10.1090/S0002-9904-1964-11062-4.
[18]	P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: A mathematical survey, Boll. Unione Mat. Ital., IX (2010), 543-584.
[19]	P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. doi: 10.1016/0022-1236(84)90072-7.
[20]	P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1.
[21]	P. Pucci and J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc., 299 (1987), 115-132. doi: 10.1090/S0002-9947-1987-0869402-1.
[22]	V. Rădulescu, "Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods," Contemporary Mathematics and Its Applications, 6, Hindawi Publishing Corporation, New York, 2008.
[23]	V. Rădulescu, Remarks on a limiting case in the treatment of nonlinear problems with mountain pass geometry, Universitatis Babes-Bolyai Mathematica, LV (2010), 99-106.
[24]	J. Toland, A duality principle for nonconvex optimisation and the calculus of variations, Arch. Rational Mech. Anal., 71 (1979), 41-61. doi: 10.1007/BF00250669.
[25]	X. M. Zheng, Un résultat de non-existence de solution positive pour une équation elliptique, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 91-96.

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