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

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).


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
Full Text: DOI


[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
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.