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 ...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] .展开更多
基金supported by the National Natural Science Foundation of China(Nos.11201119,11471099)the International Cultivation of Henan Advanced Talents and the Research Foundation of Henan University(No.yqpy20140043)
文摘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] .
