【Author】

Baishun LAI;Zhengxiang YAN;Yinghui ZHANG;Institute of Contemporary Mathematics, Henan University;Xinyang Vocational and Technical College;Department of Mathematics, Hunan Institute of Science and Technology;

【Institution】

Institute of Contemporary Mathematics, Henan University;Xinyang Vocational and Technical College;Department of Mathematics, Hunan Institute of Science and Technology;

【Abstract】

Let B R~n be the unit ball centered at the origin. The authors consider the following biharmonic equation:{?~2u = λ(1 + u)~p in B,u =?u/?ν= 0 on ?B, where p >n+4/ n-4and ν is the outward unit normal vector. It is well-known that there exists a λ*> 0 such that the biharmonic equation has a solution for λ∈ (0, λ*) and has a unique weak solution u*with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies u*≤ r~(-4/ (p-1)) - 1 on the unit ball, which actually solve a part of the open problem left in [D`avila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193] .

【Keywords】

Minimal solutions;;Regularity;;Stability;;Fourth order

To explore the background and basis of the node document

Total: 17 articles

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- [2] Michael G. Crandall;;Paul H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Archive for Rational Mechanics and Analysis, 1975 (3)
- [3] Alberto Ferrero;;Hans-Christoph Grunau;;Paschalis Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Annali di Matematica Pura ed Applicata, 2009 (1)
- [4] Craig Cowan;;Pierpaolo Esposito;;Nassif Ghoussoub;;Amir Moradifam, The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity, Archive for Rational Mechanics and Analysis, 2010 (3)

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