Multiplicity of solutions for a fourth order equation with power-type nonlinearity
Multiplicity of solutions for a fourth order equation with power-type nonlinearity
复制标题
DOI：
10.1007/s00208-009-0476-8
复制
发表时间：
2010-09
影响因子：
1.4
通讯作者：
J. Dávila;I. Flores;Ignacio Guerra
中科院分区：
文献类型：
--
作者：
J. Dávila;I. Flores;Ignacio Guerra
设B为内的单位球，N≥ 3，n为边界上的外单位法向量。我们考虑$\Delta^2 u = \lambda（1+ {\rm sign}（p）u）^{p} \quad {\rm in} \，B，\quad u = 0，\quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \，\partial B$$的径向解，其中λ≥ 0.对于正p，我们假设5 ≤N≤ 12和，或N ≥ 13和，其中pc是依赖于N的常数。对于负p，我们假设4 ≤N≤ 12且p <pc，或N = 3，其中是常数。证明了存在唯一的λS> 0，使得当λ=λ S时，存在径向弱奇异解.当λ=λ S时，存在无穷多个径向正则解，且当λ→λS时，径向正则解的个数趋于无穷多.
LetBbe the unit ball in,N≥ 3 andnbe the exterior unit normal vector on the boundary. We consider radial solutions to$$\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B$$whereλ≥ 0. For positivepwe assume 5 ≤N≤ 12 and, orN≥ 13 and, wherepcis a constant depending onN. For negativepwe assume 4 ≤N≤ 12 andp<pc, orN= 3 and, whereis a constant. We show that there is a uniqueλS> 0 such that ifλ=λSthere exists a radial weakly singular solution. Forλ=λSthere exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity asλ→λS.