Abstract
Using the bounded mountain pass lemma and the Ekeland variational principle, we prove a bounded version of the Pucci-Serrin three critical points result in the intersection of a ball with a wedge in a Banach space. The localization constraints are overcome by boundary and invariance conditions. The result is applied to obtain multiple positive solutions for some semilinear problems.
Authors
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Patrizia Pucci
Dipartimento di Matematica e Informatica (DMI) Università di Perugia 06100 Perugia, ITALY
Csaba Varga
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
?
Paper coordinates
R. Precup, P. Pucci, C. Varga, A three critical point result in a bounded domain of a Banach space and applications, Differential Integral Equations 30 (2017), no. 7-8, 555-568.
About this paper
Journal
Differential Integral Equations
Publisher Name
DOI
Print ISSN
Online ISSN
google scholar link
[2] C. Azizieh and P. Clément, A priori estimates and continuous methods for positive solutions of p–Laplace equations, J. Differential Equations, 179 (2002), 213–245. MR1883743
[3] C. Chidume, “Geometric Properties of Banach Spaces and Nonlinear Iterations,” Lecture Notes in Mathematics, xviii, Springer–Verlag London, (1965). MR2504478
[4] I. Ciorănescu, “Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems,” Mathematics and its Applications, 62 Kluwer, Dordrecht, (1990). MR1079061
[5] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of a p–Laplace equations, 1<p<2, via the moving plane method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 689–707. MR1648566
[6] G. Dinca, P. Jebelean, and J. Mawhin, Variational and topological methods for Dirichlet problems with p–Laplacian, Port. Math., 58 (2001), 339–378. MR1856715
[7] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 443–474. MR0346619
[8] H. Lisei, R. Precup, and C. Varga, A Schechter type critical point result in annular conical domains of a Banach space and applications, Discrete Contin. Dyn. Syst., 36 (2016), 3775–3789. MR3485852
[9] R. Precup, The Leray–Schauder boundary condition in critical point theory, Nonlinear Anal., 71 (2009), 3218–3228. MR2532844
[10] R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal., 75 (2012), 834–851. MR2847461
[11] R. Precup and C. Varga, Localization of positive critical points in Banach spaces and applications, Topol. Methods Nonlinear Anal., to appear. cf. MR3670487
[12] P. Pucci and J. Serrin, Extensions of mountain pass theorem, J. Funct. Anal., 59 (1984), 185–210. MR0766489
[13] P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142–149. MR0808262
[14] P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1–66. MR2025185
[15] M. Schechter, “Linking Methods in Critical Point Theory”, Birkhäuser, Boston, (1999). MR1729208
[16] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24 (1996). MR1400007