Singularity of the Extremal Solution for Supercritical Biharmonic Equations with Power-Type Nonlinearity

【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

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