## 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
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### 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
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