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 - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829
References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (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. 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. 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. 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. 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. 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.

[9]

T. D'Aprile and G. Siciliano, Magnetostatic solutions for a semilinear perturbation of the Maxwell equations,, Adv. Differential Equations, 16 (2011), 435.

[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.

[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., 129 (1999), 787. doi: 10.1017/S0308210500013147.

[12]

E. H. Lieb and M. Loss, Analysis,, Second edition. Graduate Studies in Mathematics, (2001).

[13]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces,, Comment. Math. Helv., 60 (1985), 558. doi: 10.1007/BF02567432.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (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. 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. 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. 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. 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. 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.

[9]

T. D'Aprile and G. Siciliano, Magnetostatic solutions for a semilinear perturbation of the Maxwell equations,, Adv. Differential Equations, 16 (2011), 435.

[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.

[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., 129 (1999), 787. doi: 10.1017/S0308210500013147.

[12]

E. H. Lieb and M. Loss, Analysis,, Second edition. Graduate Studies in Mathematics, (2001).

[13]

M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces,, Comment. Math. Helv., 60 (1985), 558. doi: 10.1007/BF02567432.

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