不满足A-R条件的双调和方程无穷多解的存在性

Infinitely many solutions for biharmonic equation without A-R condition

在有界光滑区域ΩRN(N>4)上,研究了双调和方程Δ2u-λu=f(x,u),x∈Ω;u=u/n=0,x∈Ω,其中,f(x,u)是关于u的奇函数,u趋于无穷时是次临界的,并且不满足A-R条件.利用对称的山路引理,证明上面的方程有无穷多解且相应的临界值序列趋于正无穷大.

In this paper,we have studied the following biharmonic problem on a smooth domain Ω C R^N（N〉 4） ∶Δ^2u-λu =f（x,u）,x ∈ Ω; u =Ou/On =0,x ∈δΩ Ω,where the nonlinearity f（x,u） is odd symmetric with respect to u,has subcritical growth at infinity and does not satisfy A-R condition.Using symmetric mountain pass theorem,we prove that the above problem has infinitely many solutions,and the corresponding critical values approach to positive infinity.