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

Variational methods for non-local operators of elliptic type

Raffaella Servadei 1, and  Enrico Valdinoci 2,

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.
Keywords: Linking Theorem, fractional Laplacian., Mountain Pass Theorem, variational techniques, integrodifferential operators.
Mathematics Subject Classification: Primary: 35A15, 35A16, 35R09, 35R11, 45K0.
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.   Google Scholar

[2]

H. Brézis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983).   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A., ().   Google Scholar

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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.   Google Scholar

[7]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math., (1986).   Google Scholar

[8]

R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12.   Google Scholar

[9]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, Rev. Mat. Iberoam., 29 (2013).   Google Scholar

[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.  Google Scholar

[11]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., ().   Google Scholar

[12]

M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 (1990).   Google Scholar

[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.  Google Scholar

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M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.   Google Scholar

[2]

H. Brézis, "Analyse Fonctionelle. Théorie et Applications,", Masson, (1983).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A., ().   Google Scholar

[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.   Google Scholar

[7]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math., (1986).   Google Scholar

[8]

R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12.   Google Scholar

[9]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, Rev. Mat. Iberoam., 29 (2013).   Google Scholar

[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.  Google Scholar

[11]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., ().   Google Scholar

[12]

M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 (1990).   Google Scholar

[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.  Google Scholar

[14]

M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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