American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[2]

G. Bonanno and S. A. Marano, Positive solutions of elliptic equations with discontinuous nonlinearities, Topol. Methods Nonlinear Anal., 8 (1996), 263-273.  Google Scholar

[3]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliqueées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[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.  Google Scholar

[5]

H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963. doi: 10.1002/cpa.3160440808.  Google Scholar

[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.  Google Scholar

[7]

F. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc., 76 (1979), 186-188.  Google Scholar

[8]

F. Clarke, Periodic solutions to Hamiltonian inclusions, J. Differential Equations, 40 (1981), 1-6. doi: 10.1016/0022-0396(81)90007-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609.   Google Scholar

[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.  Google Scholar

[16]

R. Palais, Lusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132. doi: 10.1016/0040-9383(66)90013-9.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

G. Bonanno and S. A. Marano, Positive solutions of elliptic equations with discontinuous nonlinearities, Topol. Methods Nonlinear Anal., 8 (1996), 263-273.  Google Scholar

[3]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliqueées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[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.  Google Scholar

[5]

H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963. doi: 10.1002/cpa.3160440808.  Google Scholar

[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.  Google Scholar

[7]

F. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc., 76 (1979), 186-188.  Google Scholar

[8]

F. Clarke, Periodic solutions to Hamiltonian inclusions, J. Differential Equations, 40 (1981), 1-6. doi: 10.1016/0022-0396(81)90007-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609.   Google Scholar

[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.  Google Scholar

[16]

R. Palais, Lusternik-Schnirelmann theory on Banach manifolds, Topology, 5 (1966), 115-132. doi: 10.1016/0040-9383(66)90013-9.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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