×

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 \[ \begin{cases} (-\Delta)^s u = f(x,u) &\text{in }\Omega, \\u=0 &\text{in } {\mathbb R}^n \backslash \Omega. \end{cases} \] As far as we know, all these results are new.

MSC:

35R09 Integro-partial differential equations
35R11 Fractional partial differential equations
PDF BibTeX XML Cite
Full Text: DOI

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
[2] Brézis, H., Analyse fonctionelle. théorie et applications, (1983), Masson Paris
[3] Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. partial differential equations, 32, 1245-1260, (2007) · Zbl 1143.26002
[4] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhikerʼs guide to the fractional Sobolev spaces, preprint, available at: http://arxiv.org/abs/1104.4345; Bull. Sci. Math., in press. · Zbl 1252.46023
[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., vol. 65, (1986), American Mathematical Society Providence, RI · Zbl 0609.58002
[7] Servadei, R.; Valdinoci, E., Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, preprint, available at: · Zbl 1275.49016
[8] Struwe, M., Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, Ergeb. math. grenzgeb. (3), (1990), Springer-Verlag Berlin-Heidelberg · Zbl 0746.49010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.