# American Institute of Mathematical Sciences

May  2013, 33(5): 2105-2137. doi: 10.3934/dcds.2013.33.2105

## Variational methods for non-local operators of elliptic type

 1 Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci 31 B, Arcavacata di Rende (Cosenza), 87036 2 Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy

Received  December 2011 Revised  September 2012 Published  December 2012

In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem $$\left\{ \begin{array}{ll} \mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\ u=0 in \mathbb{R}^n \backslash Ω , \end{array} \right.$$ where $\lambda$ is a real parameter and the nonlinear term $f$ satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional $\mathcal J_\lambda$ associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq \lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the operator $-\mathcal L_K$. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $$\left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) in Ω \\ u=0 in \mathbb{R}^n \backslash Ω. \end{array} \right.$$ Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.
Citation: Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105
##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. [2] H. Brézis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983). [3] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. [4] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. [5] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A., (). [6] P. H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 215. [7] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math., (1986). [8] R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12. [9] R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, Rev. Mat. Iberoam., 29 (2013). [10] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, J. Math. Anal. Appl., 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. [11] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., (). [12] M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 (1990). [13] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3. [14] M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996). doi: 10.1007/978-1-4612-4146-1.

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##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. [2] H. Brézis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983). [3] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. [4] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. [5] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A., (). [6] P. H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 215. [7] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math., (1986). [8] R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12. [9] R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, Rev. Mat. Iberoam., 29 (2013). [10] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, J. Math. Anal. Appl., 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. [11] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., (). [12] M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 (1990). [13] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3. [14] M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996). doi: 10.1007/978-1-4612-4146-1.
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