zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonexistence results for a class of fractional elliptic boundary value problems. (English) Zbl 1260.35050

Summary: In this paper we study a class of fractional elliptic problems of the form

(-Δ) s u=f(x,u)inΩ,u=0in N Ω,

where s(0,1). We prove nonexistence of positive solutions when Ω is star-shaped and f is supercritical. We also derive a nonexistence result for subcritical f in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the L. Caffarelli and L. Silvestre extension [Commun. Partial Differ. Equations 32, No. 8, 1245–1260 (2007; Zbl 1143.26002)] of a solution of the above problem.

MSC:
35J60Nonlinear elliptic equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35J70Degenerate elliptic equations
26A33Fractional derivatives and integrals (real functions)
References:
[1]Alexandrov, A. D.: A characteristic property of the spheres, Ann. mat. Pura appl. 58, 303-315 (1962)
[2]Berestycki, H.; Nirenberg, L.: On the method of moving planes and the sliding method, Bol. soc. Bras. mat. 22, 1-37 (1991) · Zbl 0784.35025 · doi:10.1007/BF01244896
[3]Birkner, M.; López-Mimbela, J. A.; Wakolbinger, A.: Comparison results and steady states for the Fujita equation with fractional Laplacian, Ann. inst. H. Poincaré anal. Non linéaire 22, No. 1, 83-97 (2005) · Zbl 1075.60081 · doi:10.1016/j.anihpc.2004.05.002 · doi:numdam:AIHPC_2005__22_1_83_0
[4]Bogdan, K.; Żak, T.: On Kelvin transformation, J. theoret. Probab. 19, No. 1, 89-120 (2006) · Zbl 1105.60057 · doi:10.1007/s10959-006-0003-8
[5]C. Brändle, E. Colorado, A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, in press.
[6]Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations, Universitext (2011)
[7]Cabré, X.; Sire, Y.: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
[8]Cabré, X.; Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. math. 224, No. 5, 2052-2093 (2010) · Zbl 1198.35286 · doi:10.1016/j.aim.2010.01.025
[9]Caffarelli, L. A.; Roquejoffre, J. -M.; Savin, O.: Non-local minimal surfaces, Comm. pure appl. Math. 63, No. 9, 1111-1144 (2010)
[10]Caffarelli, L. A.; Salsa, S.; Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. math. 171, No. 2, 425-461 (2008) · Zbl 1148.35097 · doi:10.1007/s00222-007-0086-6
[11]Caffarelli, L.; Silvestre, L.: An extension problem related to the fractional Laplacian, Comm. partial differential equations 32, No. 7-9, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306
[12]Capella, A.; Dávila, J.; Dupaigne, L.; Sire, Y.: Regularity of radial extremal solutions for some non local semilinear equations, Comm. partial differential equations 36, No. 8, 1353-1384 (2011) · Zbl 1231.35076 · doi:10.1080/03605302.2011.562954
[13]Chang, S. -Y.A.; Gonzàlez, M. D. M.: Fractional Laplacian in conformal geometry, Adv. math. 226, 1410-1432 (2011) · Zbl 1214.26005 · doi:10.1016/j.aim.2010.07.016
[14]Chen, W.; Li, C.; Ou, B.: Classification of solutions for an integral equation, Comm. pure appl. Math. 59, No. 3, 330-343 (2006) · Zbl 1093.45001 · doi:10.1002/cpa.20116
[15]A. de Pablo, U. Sánchez, Some Liouville-type results for a fractional equation, preprint.
[16]Dupaigne, L.; Sire, Y.: A Liouville theorem for nonlocal elliptic equations, Contemp. math. 528, 105-114 (2010) · Zbl 1218.35243
[17]Fall, M. M.: On a semilinear elliptic equation with fractional Laplacian and Hardy potential
[18]Frank, R.; Lieb, E. H.; Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. amer. Math. soc. 21, No. 4, 925-950 (2008) · Zbl 1202.35146 · doi:10.1090/S0894-0347-07-00582-6
[19]Fröhlich, A.: The Helmholtz decomposition of weighted lq-spaces for Muckenhoupt weights, Ann. univ. Ferrara sez. 46, No. 1, 11-19 (2000) · Zbl 1034.35089
[20]Gidas, B.; Ni, Wei-Ming; Nirenberg, L.: Symmetry and related problems via the maximum principle, Comm. math. Phys. 68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125
[21]Gidas, B.; Ni, Wei-Ming; Nirenberg, L.: Symmetry of positive solutions of nonlinear equations, Math. anal. Appl. part A adv. Math. suppl. Studies A 7, 369-402 (1981) · Zbl 0469.35052
[22]Grisvard, P.: Elliptic problems in nonsmooth domains, Monogr. stud. Math. 24 (1985) · Zbl 0695.35060
[23]Hu, B.: Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential integral equations 7, 301-313 (1994) · Zbl 0820.35062
[24]Li, Y. Y.; Zhu, M.: Uniqueness theorems through the method of moving spheres, Duke math. J. 80, No. 2, 383-417 (1995) · Zbl 0846.35050 · doi:10.1215/S0012-7094-95-08016-8
[25]Maz’ja, V. G.: Sobolev spaces, (1985)
[26]Reichel, W.: Uniqueness theorems for variational problems by the method of transformation groups, Springer lecture notes in math. 1841 (2004)
[27]Reichel, W.; Zou, H.: Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. differential equations 161, No. 1, 219-243 (2000) · Zbl 0962.35054 · doi:10.1006/jdeq.1999.3700
[28]Serrin, J.: A symmetry theorem in potential theory, Arch. ration. Mech. anal. 43, 304-318 (1971) · Zbl 0222.31007 · doi:10.1007/BF00250468
[29]R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, preprint, 2011.
[30]Sire, Y.; Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. funct. Anal. 256, No. 6, 1842-1864 (2009) · Zbl 1163.35019 · doi:10.1016/j.jfa.2009.01.020
[31]Stein, E. M.; Weiss, G.: Fractional integrals on n-dimensional Euclidean space, J. math. Mech. 7, 503-514 (1958) · Zbl 0082.27201
[32]Tan, J.: The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. var. Partial differential equations 42, 21-41 (2011)