A critical Kirchhoff type problem involving a nonlocal operator

Alessio Fiscella1, Enrico Valdinoci2
1Dipartimento di Matematica “Federigo Enriques”, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
2Weierstraß Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 11A, D-10117, Berlin, Germany

Tóm tắt

Từ khóa


Tài liệu tham khảo

Di Nezza, 2012, Hitchhiker’s guide to the fractional Sobolev spaces, Sci. Math. B., 136, 521, 10.1016/j.bulsci.2011.12.004

A. Fiscella, Saddle point solutions for non-local elliptic operators, paper. Available online at http://arxiv.org/abs/1210.8401 submitted for publication.

R. Servadei, E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent. Preprint available at http://www.math.utexas.edu/mp_arc-bin/mpa?yn=12-58.

Servadei, 2013, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc.

Alves, 2010, On a class of nonlocal elliptic problems with critical growth, Differential Equations & Applications, 2, 409, 10.7153/dea-02-25

Chen, 2013, Multiple solutions for the nonhomogeneous Kirchhoff equation on RN, Nonlinear Anal. RWA, 14, 1477, 10.1016/j.nonrwa.2012.10.010

Figueiredo, 2013, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401, 706, 10.1016/j.jmaa.2012.12.053

Figueiredo, 2012, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differential Integral Equations, 25, 853

Liang, 2013, Soliton solutions to Kirchhoff type problems involving the critical growth in RN, Nonlinear Anal., 81, 31, 10.1016/j.na.2012.12.003

Autuori, 2013, A priori estimates for solutions of p-Kirchhoff systems under dynamic boundary conditions, vol. 594, 59

Autuori, 2010, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73, 1952, 10.1016/j.na.2010.05.024

Autuori, 2012, Lifespan estimates for solutions of polyharmonic Kirchhoff systems, Math. Models Methods Appl. Sci., 22, 1150009, 10.1142/S0218202511500096

Autuori, 2010, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196, 489, 10.1007/s00205-009-0241-x

Colasuonno, 2011, Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equations, Nonlinear Anal., 74, 5962, 10.1016/j.na.2011.05.073

Yang, 2012, Finite-dimensional attractors for the Kirchhoff models with critical exponents, J. Math. Phys., 53, 15

Autuori, 2013, Elliptic problems involving the fractional Laplacian in RN, J. Differential Equations, 255, 2340, 10.1016/j.jde.2013.06.016

Barrios, 2012, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252, 6133, 10.1016/j.jde.2012.02.023

B. Barrios, E. Colorado, R. Servadei, F. Soria, A critical fractional equation with concave–convex power nonlinearities, paper. Available online at http://arxiv.org/abs/1306.3190 submitted for publication.

Capella, 2011, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10, 1645, 10.3934/cpaa.2011.10.1645

G. Molica Bisci, R. Servadei, Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent, paper. Available online at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=13-24 submitted for publication.

Servadei, 2013, A Brezis–Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12, 2445, 10.3934/cpaa.2013.12.2445

Tan, 2011, The Brezis–Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36, 21, 10.1007/s00526-010-0378-3

Servadei, 2012, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389, 887, 10.1016/j.jmaa.2011.12.032

Servadei, 2013, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33, 2105, 10.3934/dcds.2013.33.2105

Palatucci, 2013, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 10.1007/s00526-013-0656-y

Rabinowitz, 1986, vol. 65

Brézis, 2011

Carrier, 1945, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math., 3, 157, 10.1090/qam/12351

Kirchhoff, 1876

Kirchhoff, 1883

Carrier, 1949, A note on the vibrating string, Quart. Appl. Math., 7, 97, 10.1090/qam/28511

Oplinger, 1960, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32, 1529, 10.1121/1.1907948

Caffarelli, 2010, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63, 1111, 10.1002/cpa.20331

Ambrosio, 2011, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134, 377, 10.1007/s00229-010-0399-4

Caffarelli, 2011, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41, 203, 10.1007/s00526-010-0359-6