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Existence of nonradial solutions for Hénon type biharmonic equation involving critical Sobolev exponents. (English) Zbl 1474.35280

Summary: We prove the existence of nonradial solutions under some conditions for a semilinear biharmonic Dirichlet problem involving critical Sobolev exponents.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B33 Critical exponents in context of PDEs
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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[1] Grunau, H.-Ch., Polyharmonische Dirichletprobleme: Positivität, kritische Exponenten und kritische Dimensionen (1996), Habilitationsschrift, Universität Bayreuth
[2] Gazzola, F.; Grunau, H.; Squassina, M., Existence and nonexistence results for critical growth biharmonic elliptic equations, Calculus of Variations and Partial Differential Equations, 18, 2, 117-143 (2003) · Zbl 1290.35063 · doi:10.1007/s00526-002-0182-9
[3] Bernis, F.; García Azorero, J.; Peral, I., Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Advances in Differential Equations, 1, 2, 219-240 (1996) · Zbl 0841.35036
[4] Bartsch, T.; Weth, T.; Willem, M., A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calculus of Variations and Partial Differential Equations, 18, 3, 253-268 (2003) · Zbl 1059.31006 · doi:10.1007/s00526-003-0198-9
[5] Bahri, A.; Coron, J., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Communications on Pure and Applied Mathematics, 41, 3, 253-294 (1988) · Zbl 0649.35033 · doi:10.1002/cpa.3160410302
[6] Berchio, E.; Gazzola, F.; Weth, T., Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, Journal für die Reine und Angewandte Mathematik, 620, 165-183 (2008) · Zbl 1182.35109 · doi:10.1515/CRELLE.2008.052
[7] Ni, W. M., A nonlinear Dirichlet problem on the unit ball and its applications, Indiana University Mathematics Journal, 31, 6, 801-807 (1982) · Zbl 0515.35033 · doi:10.1512/iumj.1982.31.31056
[8] Byeon, J.; Wang, Z., On the Hénon equation: asymptotic profile of ground states, I, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 23, 6, 803-828 (2006) · Zbl 1114.35071 · doi:10.1016/j.anihpc.2006.04.001
[9] Byeon, J.; Wang, Z., On the Hénon equation: asymptotic profile of ground states, II, Journal of Differential Equations, 216, 1, 78-108 (2005) · Zbl 1114.35070 · doi:10.1016/j.jde.2005.02.018
[10] Cao, D.; Peng, S., The asymptotic behaviour of the ground state solutions for Hénon equation, Journal of Mathematical Analysis and Applications, 278, 1, 1-17 (2003) · Zbl 1086.35036 · doi:10.1016/S0022-247X(02)00292-5
[11] Smets, D.; Willem, M.; Su, J., Non-radial ground states for the Hénon equation, Communications in Contemporary Mathematics, 4, 3, 467-480 (2002) · Zbl 1160.35415 · doi:10.1142/S0219199702000725
[12] Serra, E., Non radial positive solutions for the Hénon equation with critical growth, Calculus of Variations and Partial Differential Equations, 23, 3, 301-326 (2005) · Zbl 1207.35147 · doi:10.1007/s00526-004-0302-9
[13] Luckhaus, S., Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order, Journal für die Reine und Angewandte Mathematik, 306, 192-207 (1979) · Zbl 0395.35026 · doi:10.1515/crll.1979.306.192
[14] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solution of elliptic partial differential equations satisfying general boundary conditions I, Communications on Pure and Applied Mathematics, 12, 623-727 (1959) · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[15] Gazzola, F.; Grunau, H.; Sweers, G., Polyharmonic Boundary Value Problems. Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics, 1991 (2010), Springer · Zbl 1239.35002 · doi:10.1007/978-3-642-12245-3
[16] Grunau, H.; Sweers, G., Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Mathematische Annalen, 307, 4, 589-626 (1997) · Zbl 0892.35031 · doi:10.1007/s002080050052
[17] Tintarev, K.; Fieseler, K.-H., Concentration Compactness, Functional-Analytic Grounds and Applications (2007), London, UK: Imperial College Press, London, UK · Zbl 1118.49001
[18] Swanson, C. A., The best Sobolev constant, Applicable Analysis, 47, 4, 227-239 (1992) · Zbl 0739.46026 · doi:10.1080/00036819208840142
[19] Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, 88, 3, 486-490 (1983) · Zbl 0526.46037 · doi:10.2307/2044999
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