摘要:
通过变分方法和分析技巧,得到了非二次的椭圆问题{-△u-a(x)u=f(x,u) u∈Ω u=0 u∈aΩ的非平凡解的存在性:定理1 假设f(x,t)满足如下条件:(f1)F(x,t)/(｜t｜2→+∞),F(x,t)/｜t｜2→0(｜t｜→0)在Ω上一致成立;(f2)存在α1＞0.1＜s＜N+2/N-2,使得｜f(x,t)｜≤a1(1+｜t｜s)对所有的(x,t)∈Ω×R成立(f3)存在常数β＞2N、N+2s-1,a2＞0,L＞0,使得tf(x,t)-2F(x,t)≥a2｜t｜β对所有的｜t｜≥L，x∈Ω成立.(如果0是-△+a 的一个特征值(Dirichlet边界条件)且满足条件:(f4)存在δ0,使得(i) F(x, t) ≥ 0,对所有的|t|≤δ x ∈Ω; or或者(ii) F(x, t) ≤ 0, 对所有的|t|≤δ x ∈Ω.则问题(1)有至少一个非平凡解.
Abstract:
We study the existence of a nontrivial solution for the Dirichlet problem {-△u-a(x)u=f(x,u) u∈Ω u=0 u∈δΩ where Ω is a smooth bounded domainin RN.Theorem 1 Assume that f(x, t) satisfies the following conditions: (f1)F(x,t)/|t|2→+∞as|t|→+∞and F(x,t)/|t|2→0 as|t|→0 uniformly on Ω.(f2)There exist a1＞0 and 1＜s＜N+2/N-2 such that|f(x,t)| ≤a1(1+|t|s)for all(x,t)∈Ω×R.(f3)There are constants β＞2N/N+2s-1,a2＞0 and L＞0 such that tf(x,t)-2F(x,t)≥a2|t|β for all|t|≥ L and x∈Ω.If O is an eigenvalue of -△ + a (with Dirichlet boundary condition) and also satisfies the condition that:(f4)There exists δ 0 such that (i) F(x, t) ≥ 0, for all|t|≤δ x ∈Ω; or (ii) F(x, t) ≤ 0, for all |t|≤δ x ∈Ω then problem (1) has at least one nontrivial solution.
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