American Institute of Mathematical Sciences

Advanced
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

[1]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[2]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[3]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[4]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[5]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[6]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[7]

Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785
[8]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[9]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[10]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[11]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[12]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627
[13]

Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603
[14]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[15]

Maicon Sônego. Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5981-5988. doi: 10.3934/dcdsb.2019116
[16]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295
[17]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095
[18]

Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030
[19]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045
[20]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences