\null Max-Min characterization of the mountain pass energy level for a class of variational problems

Bellazzini, Jacopo; Visciglia, Nicola

doi:10.1090/S0002-9939-10-10415-8

\null Max-Min characterization of the mountain pass energy level for a class of variational problems
HTML articles powered by AMS MathViewer

by Jacopo Bellazzini and Nicola Visciglia PDF

Proc. Amer. Math. Soc. 138 (2010), 3335-3343 Request permission

Abstract:

We prove the existence of a critical point at the mountain pass energy level for a general class of variational problems. We also provide a Max-Min characterization of the mountain pass energy level. Finally, we present some concrete applications.

References

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI 10.1007/BF00250555
Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
João Marcos do Ó and Everaldo S. Medeiros, Remarks on least energy solutions for quasilinear elliptic problems in $\Bbb R^N$, Electron. J. Differential Equations (2003), No. 83, 14. MR 1995609
Mohammed Guedda and Laurent Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879–902. MR 1009077, DOI 10.1016/0362-546X(89)90020-5
Louis Jeanjean and Kazunaga Tanaka, A remark on least energy solutions in $\textbf {R}^N$, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2399–2408. MR 1974637, DOI 10.1090/S0002-9939-02-06821-1
Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1