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次临界分数阶Laplace方程具有两个bubbles的变号解存在性
Two-Bubble Nodal Solutions for Slightly Subcritical Fractional Laplacian
考虑次临界分数阶Laplace问题
(-△)su=|u|p-1-εu，x∈Ω，
u=0，x∈∂Ω
具有两个bubbles的变号解的存在性，其中Ω是RN中的有界光滑区域，N>2s，0<s<1，p=N+2s/N-2s，ε>0充分小.这个工作可以看作Bartsch，Micheletti，Pistoia在文[On the existence and the profile of nodal solutionsof elliptic equations involving critical growth,Calc.Var.Partial Differential Equations,2006，3：265-282]结果的一种非局部形式的推广.
We consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian
(-△)su=|u|p-1-εu，x∈Ω，
u=0，x∈∂Ω
where Ω is a smooth bounded domain in RN, N>2s, 0< s< 1, p=N+2s/N-2s and ε>0 is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti, Pistoia[On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Differential Equations, 2006, 3:265-282].
分数阶Laplace方程 / 变号解 / 次临界问题 {{custom_keyword}} /
fractional Laplacian / nodal solutions / slightly subcritical problem {{custom_keyword}} /
[1] Bahri A., Li Y., Rey O., On a variational problem with lack of compactness:the topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, 1995, 3:67-93.
[2] Barrios B., Colorado E., de Palbo A., et al., On some critical problems for the fractional Laplacian operator, J. Differential Equations, 2012, 252:6133-6162.
[3] Bartsch T., D'Aprile T., Pistoia A., Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2013, 30:1027-1047.
[4] Bartsch T., D'Aprile T., Pistoia A., On the profile of sign changing solutions of an almost critical problem in the ball, Bull. London Math. Soc., 2013, 45:1246-1258.
[5] Bartsch T., Micheletti A. M., Pistoia A., On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Differential Equations, 2006, 3:265-282.
[6] Ben Ayed M., El Mehdi K., Pacella F., Classification of low energy sign-changing solutions of an almost critical equations, J. Funct. Anal., 2007, 50:343-373.
[7] Brándle C., Colorado E., de Pablo A., et al., A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 2013, 143:39-71.
[8] Brézis H., Peletier L. A., Asymptotic for Elliptic Equations Involving Critical Growth, Partial Differential Equations and the Calculus of Variations, vol. I, Progr. Nonlinear Diff. Equ. Appl. Birhäuser, Boston, MA 1, 1989:149-192.
[9] Cabré X., Tan J., Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 2010, 224:2052-2093.
[10] Caffarelli L., Silvestre L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 2007, 32:1245-1260.
[11] Capella A., Dávila J., Dupaigne L., et al., Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 2011, 36:1353-1384.
[12] Carlen E. A., Loss M., Extremals of functionals with competing symmetries, J. Funct. Anal., 1990, 88:437-456.
[13] Chen W., Li C., Ou B., Classification of solutions for an integral equation, Comm. Pure Appl. Math., 2006, 59:330-343.
[14] Chio W., Kim S., Lee K. A., Asymptotic behavior of solutions for nonlinear elliptic problem with the fractional Laplacian, J. Funct. Anal., 2014, 11:6531-6598.
[15] Cotsiolis A., Tavoularis N. K., Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 2004, 295:225-236.
[16] Dávila J., del Pino M., Sire Y., Non degeneracy of the bubble in the critical case for nonlocal equations, Proc. Amer. Math. Soc., 2013, 141:3865-3870.
[17] del Pino M., Felmer P., Musso M., Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 2003, 16:113-145.
[18] Frank R. L., Lieb E. H., Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 2010, 39:85-99.
[19] Grossi M., Takahashi F., Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, J. Funct. Anal., 2010, 259:904-917.
[20] Han Z. C., Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1991, 8:159-174.
[21] Kim S., Lee K. A., Hölder estimates for singular non-local parabolic equations, J. Funct. Anal., 2011, 261:3482-3518.
[22] Li Y. Y., Remark on some conformally invariant integral equations:the method of moving spheres, J. Eur. Math. Soc., 2004, 6:153-180.
[23] Li Y. Y., Zhu M., Uniqueness theorems through the method of moving spheres, Duke Math. J., 1995, 80:383-417.
[24] Lieb E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 1983, 118:349-374.
[25] Musso M., Pistoia A., Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent, Indiana Univ. Math. J., 2002, 51:541-579.
[26] Musso M., Pistoia A., Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl., 2010, 93:1-40.
[27] Rey O., The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 1990, 89:1-52.
[28] Rey O., Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential Integral Equations, 1991, 4:1155-1167.
[29] Ríos L., Two problems in nonlinear PDEs:existence in supercritical elliptic equations and symmetry for a hypo-elliptic operator, Universided de 511-540, Chile, Chile, 2014.
[30] Stinga P. S., Torrea J. L., Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 2010, 35:2092-2122.
[31] Tan T., The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 2011, 42:21-41.
[32] Tan T., Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 2013, 33:837-859.
[33] Xiao J., A sharp Sobolev trace inequality for the fractional-order derivatives, Bull. Sci. Math., 2006, 130:87-96.
国家自然科学基金（11271299，11001221）；中央高校基本科研业务费资助项目（3102015ZY069）
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