# American Institute of Mathematical Sciences

May  2014, 34(5): 1829-1840. doi: 10.3934/dcds.2014.34.1829

## On a functional satisfying a weak Palais-Smale condition

 1 Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via dell'Ateneo Lucano 10, I-85100 Potenza

Received  February 2013 Revised  June 2013 Published  October 2013

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.
Citation: Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829
##### References:
 [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar [2] A. Azzollini, P. d'Avenia and A. Pomponio, Quasilinear elliptic equations in $\mathbbR^N$ via variational methods and Orlicz-Sobolev embeddings,, Calc. Var. (to appear)., ().  doi: 10.1007/s00526-012-0578-0.  Google Scholar [3] A. Azzollini and A. Pomponio, On the Schrödinger equation in $\mathbbR^N$ under the effect of a general nonlinear term, Indiana Univ. Math. Journal, 58 (2009), 1361-1378. doi: 10.1512/iumj.2009.58.3576.  Google Scholar [4] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y.  Google Scholar [5] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorem and applications to some nonlinear problems with "strong'' resonance at infinity, Nonlin. Anal. TMA, 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.  Google Scholar [6] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar [7] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.  Google Scholar [8] G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Rc. Ist. lomb. Sci. Lett., 112 (1978), 332-336.  Google Scholar [9] T. D'Aprile and G. Siciliano, Magnetostatic solutions for a semilinear perturbation of the Maxwell equations, Adv. Differential Equations, 16 (2011), 435-466.  Google Scholar [10] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: Mountain pass and symmetric mountain pass approaches, TMNA, 35 (2010), 253-276.  Google Scholar [11] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbbR^N$, Proc. R. Soc. Edinb., Sect. A, Math., 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar [12] E. H. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.  Google Scholar [13] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581. doi: 10.1007/BF02567432.  Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar [2] A. Azzollini, P. d'Avenia and A. Pomponio, Quasilinear elliptic equations in $\mathbbR^N$ via variational methods and Orlicz-Sobolev embeddings,, Calc. Var. (to appear)., ().  doi: 10.1007/s00526-012-0578-0.  Google Scholar [3] A. Azzollini and A. Pomponio, On the Schrödinger equation in $\mathbbR^N$ under the effect of a general nonlinear term, Indiana Univ. Math. Journal, 58 (2009), 1361-1378. doi: 10.1512/iumj.2009.58.3576.  Google Scholar [4] M. Badiale, L. Pisani and S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 369-405. doi: 10.1007/s00030-011-0100-y.  Google Scholar [5] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorem and applications to some nonlinear problems with "strong'' resonance at infinity, Nonlin. Anal. TMA, 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.  Google Scholar [6] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar [7] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.  Google Scholar [8] G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Rc. Ist. lomb. Sci. Lett., 112 (1978), 332-336.  Google Scholar [9] T. D'Aprile and G. Siciliano, Magnetostatic solutions for a semilinear perturbation of the Maxwell equations, Adv. Differential Equations, 16 (2011), 435-466.  Google Scholar [10] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: Mountain pass and symmetric mountain pass approaches, TMNA, 35 (2010), 253-276.  Google Scholar [11] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbbR^N$, Proc. R. Soc. Edinb., Sect. A, Math., 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar [12] E. H. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.  Google Scholar [13] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581. doi: 10.1007/BF02567432.  Google Scholar
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