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Mountain pass solutions for non-local elliptic operators. (English) Zbl 1234.35291

Summary: The purpose of this paper is to study the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a non-trivial solution for them using the mountain pass theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. We prove this result for a general integrodifferential operator of fractional type and, as a particular case, we derive an existence theorem for the fractional Laplacian, finding non-trivial solutions of the equation

(-Δ) s u=f(x,u)inΩ,u=0in n Ω·

As far as we know, all these results are new.

MSC:
35R09Integro-partial differential equations
35R11Fractional partial differential equations
References:
[1]Ambrosetti, A.; Rabinowitz, P.: Dual variational methods in critical point theory and applications, J. funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2]Brézis, H.: Analyse fonctionelle. Théorie et applications, (1983) · Zbl 0511.46001
[3]Caffarelli, L.; Silvestre, L.: An extension problem related to the fractional Laplacian, Comm. partial differential equations 32, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306
[4]E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, preprint, available at: http://arxiv.org/abs/1104.4345; Bull. Sci. Math., in press.
[5]Pucci, P.; Radulescu, V.: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. unione mat. Ital. (9) 3, 543-584 (2010) · Zbl 1225.49004
[6]Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS reg. Conf. ser. Math. 65 (1986) · Zbl 0609.58002
[7]Servadei, R.; Valdinoci, E.: Lewy-stampacchia type estimates for variational inequalities driven by (non)local operators
[8]Struwe, M.: Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, Ergeb. math. Grenzgeb. (3) (1990) · Zbl 0746.49010