Localization of positive critical points in Banach spaces and applications
Słowa kluczowe
Critical point, mountain pass lemma, positive solution, p-Laplace equationAbstrakt
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.Bibliografia
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.
C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer, London, 2009.
I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer, Berlin, 1977.
F. Della Pietra and N. Gavitone, Anisotropic elliptic problems involving Hardy-type potential, J. Math. Anal. Appl. 397 (2013), 800–813.
J. Diestel, Geometry of Banach Spaces — Selected Topics, Springer, Berlin, 1975.
G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. 58 (2001), 339–378.
A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, 2003.
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.
R. Kajikiya, Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations, Nonlinear Differential Equations Appl. 19 (2012), 159–175.
D. O’Regan and R. Precup, Theorems of Leray–Schauder Type, Gordon and Breach, Amsterdam, 2001.
R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.
R. Precup, The Leray–Schauder boundary condition in critical point theory, Nonlinear Anal. 71 (2009), 3218–3228.
R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal. 75 (2012), 834–851.
M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc. 331 (1992), 681–703.
M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999.
Pobrania
Opublikowane
Jak cytować
Numer
Dział
Statystyki
Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0
