一类退化拟线性椭圆型方程的无穷多个非平凡的广义解

Infinitely Many Nontrivial Generalized Solutions of a Class of Quasilinear Degenerated Elliptic Equation

文献[1]证明了下述Dirichlet问题:{-D_i(g(|Du|~2)D_iu)=f(x,u) x∈Ω u=0 x∈Ω存在无穷多个非平凡广义解。其中要求f(x,ζ)对第二个变量满足增长性条件|(f(x,ζ)|≤C_1+C_2|ζ|~S,S<(n+2)/(n-2),本文对这一条件作了些改进,给出了一个更一般的条件,它允许f(x,ζ)关于ζ有更快的增长性,即f(x,ζ)的增长性条件被改进为如下的条件: 存在[0,+∞)上非减的连续函数φ(t),它满足: 并且我们仍然得到上述Dirichlet问题存在着无穷多个非平凡广义解的结果。

Paper [1] deals with the Dirichlet proble m, i. e. has infinitely many nontrivial generalized solutions,where f(xξ) satisfy growth condition In this paper, above condition is improved as follows. There exists a wondecrease continuous function f(x) on an interval satisfy the condition (i) there exists α,β (ii) lim (iii) where such that βφ (t), such that the function f(x) satisfy the relation Finall. the same result be obtained.