On a functional satisfying a weak Palais-Smale condition

Antonio Azzollini

doi:10.3934/dcds.2014.34.1829

Article Contents

2014, Volume 34, Issue 5: 1829-1840. Doi: 10.3934/dcds.2014.34.1829

This issue Previous Article When are the invariant submanifolds of symplectic dynamics Lagrangian? Next Article Lyapunov spectrum for geodesic flows of rank 1 surfaces

On a functional satisfying a weak Palais-Smale condition

Antonio Azzollini^1,

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: May 2014

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35j62; Secondary: 46E30, 46E35.

Citation:

Related Papers

Cited by

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitate, Rc. Ist. lomb. Sci. Lett., 112 (1978), 332-336.
[9]	T. D'Aprile and G. Siciliano, Magnetostatic solutions for a semilinear perturbation of the Maxwell equations, Adv. Differential Equations, 16 (2011), 435-466.
[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.
[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.
[12]	E. H. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001.
[13]	M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581.doi: 10.1007/BF02567432.