Abstract
Two critical point theorems of M. Schechter in a ball of a Hilbert space are extended to uniformly convex Banach spaces by exploiting the properties of the duality mapping. Moreover, the critical points are sought in the intersection of a ball with a wedge, in particular with a cone, making possible applications to positive solutions of variational problems. The extension from Hilbert to Banach spaces not only requires a major refining of reasoning, but also a different statement by adding a third possibility to the original two alternatives from Schechter’s results. The theory is applied to positive solutions for \(p)\-Laplace equations.
Authors
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Csaba Varga
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
Critical point; mountain pass lemma; positive solution; p-Laplace equation
Paper coordinates
R. Precup, C. Varga, Localization of positive critical points in Banach spaces and applications, Topol. Methods Nonlinear Anal. 49 (2017), 817-833.
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[1] M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003), 771–783
[2] C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer, London, 2009
[3] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
[4] K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.
[5] F. Della Pietra and N. Gavitone, Anisotropic elliptic problems involving Hardy-type potential, J. Math. Anal. Appl. 397 (2013), 800–813.
[6] J. Diestel, Geometry of Banach Spaces — Selected Topics, Springer, Berlin, 1975.
[7] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. 58 (2001), 339–378.
[8] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
[9] 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.
[10] R. Kajikiya, Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations, Nonlinear Differential Equations Appl. 19 (2012), 159–175.
[11] D. O’Regan and R. Precup, Theorems of Leray–Schauder Type, Gordon and Breach, Amsterdam, 2001.[1] R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.
[12] R. Precup, The Leray–Schauder boundary condition in critical point theory, Nonlinear Anal. 71 (2009), 3218–3228.
[13] R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal. 75 (2012), 834–851.
[14] M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331 (1992), 681–703.
[15] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999.