The Brezis-Nirenberg result for the fractional Laplacian. (English) Zbl 1323.35202

Servadei, Raffaella; Valdinoci, Enrico

Trans. Am. Math. Soc. 367, No. 1, 67-102 (2015).

The authors enhance a well-known result on the existence of a solution of a non-linear differential equation. Precisely, for \(n>2s\) such that \(s\in(0,1)\), they consider the following Dirichlet (in)homogeneous problem in a bounded Lipschitz open set \(\Omega\): \[ (*)_f:\quad \mathcal{L}_Ku-\lambda u=|u|^{\frac{4s}{n-2s}}u+f(x,u)\text{ in }\Omega,\quad u=0\text{ on }\mathbb R^n\setminus \Omega, \] where \(\mathcal{L}_K\) stands for an explicit integro-differential operator of order \(s\), \(K\) is a positive function (which is part of the \(\mathcal{L}_K\) integrand) satisfying meaningful conditions, and \(\lambda\) is a positive factor. Accordingly, by assuming satisfying that \(f\) is a Carathéodory function satisfying some conditions, that \(\lambda\) is bounded by the first eigenvalue of \(\mathcal{L}_K\), that the functional associated to \((*)_f\) is bounded by a function of the fractional critical Sobolev constant-associated to the embedding map \(X_0\hookrightarrow L_{\frac{2n}{n-2s}}(\mathbb R^n)\) such that \(X_0\) is a suitable Sobolev space-, then (through a polarization) they show that \((*)_f\) (resp. \((*)_0\)) has a non-zero solution in \(X_0\) (resp. in \(H^s(\mathbb R^n)\), the fractional Sobolev space (Theorem 1)) (resp. Theorem 2). Then, thanks to the embedding continuous map \(X_0\hookrightarrow H^s(\mathbb R^n)\) (Lemma 7), they apply the previous results for the pseudo-differential operator \((-\Delta)^s\) (Theorems 3 and 4).

Reviewer: Mohammed El Aïdi (Bogotá)

MSC:

35R11	Fractional partial differential equations
35R09	Integro-partial differential equations
49J45	Methods involving semicontinuity and convergence; relaxation
35A15	Variational methods applied to PDEs
35S15	Boundary value problems for PDEs with pseudodifferential operators
47G20	Integro-differential operators
45G05	Singular nonlinear integral equations

Keywords:

mountain pass theorem; critical non-linearities; best critical Sobolev constant; variational techniques; integrodifferential operators; fractional Laplacian

References:

[1]	Ambrosetti, Antonio; Rabinowitz, Paul H., Dual variational methods in critical point theory and applications, J. Functional Analysis, 14, 349-381 (1973) · Zbl 0273.49063
[2]	Brezis, Ha{\`“{\i }}m, Analyse fonctionnelle, Collection Math\'”ematiques Appliqu\'ees pour la Ma\^\i trise. [Collection of Applied Mathematics for the Master’s Degree], xiv+234 pp. (1983), Masson: Paris:Masson · Zbl 0511.46001
[3]	Br{\'e}zis, Ha{\"{\i }}m; Nirenberg, Louis, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 4, 437-477 (1983) · Zbl 0541.35029
[4]	Caffarelli, Luis; Silvestre, Luis, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 7-9, 1245-1260 (2007) · Zbl 1143.26002
[5]	Capozzi, A.; Fortunato, D.; Palmieri, G., An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 2, 6, 463-470 (1985) · Zbl 0612.35053
[6]	Cotsiolis, Athanase; Tavoularis, Nikolaos K., Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295, 1, 225-236 (2004) · Zbl 1084.26009
[7]	[fsvDensity] A. Fiscella, R. Servadei, and E. Valdinoci, Density properties for fractional Sobolev spaces, to appear in Ann. Acad. Sci. Fenn. Math. · Zbl 1346.46025
[8]	de la Llave, Rafael; Valdinoci, Enrico, Symmetry for a Dirichlet-Neumann problem arising in water waves, Math. Res. Lett., 16, 5, 909-918 (2009) · Zbl 1202.35066
[9]	Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 5, 521-573 (2012) · Zbl 1252.46023
[10]	Rabinowitz, Paul H., Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5, 1, 215-223 (1978) · Zbl 0375.35026
[11]	Rabinowitz, Paul H., Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65, viii+100 pp. (1986), published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0609.58002
[12]	Servadei, Raffaella; Valdinoci, Enrico, A multiplicity result for a class of nonlinear variational inequalities, Nonlinear Stud., 12, 1, 37-48 (2005) · Zbl 1072.49007
[13]	[sv] R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam. 29 (2013), no. 3, 1091-1126. · Zbl 1275.49016
[14]	Servadei, Raffaella; Valdinoci, Enrico, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389, 2, 887-898 (2012) · Zbl 1234.35291
[15]	[svlinking] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105-2137. · Zbl 1303.35121
[16]	Struwe, Michael, Variational methods, xiv+244 pp. (1990), Springer-Verlag: Berlin:Springer-Verlag · Zbl 0746.49010
[17]	Tan, Jinggang, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42, 1-2, 21-41 (2011) · Zbl 1248.35078
[18]	Willem, Michel, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, x+162 pp. (1996), Birkh\"auser Boston Inc.: Boston, MA:Birkh\"auser Boston Inc. · Zbl 0856.49001