一类椭圆方程解的存在性

万保成, 李 健, 李士军

(吉林农业大学 信息技术学院, 长春 130118)

考虑如下拟线性椭圆Dirichlet边值问题:

(1)

(H1) 存在C>0和q∈(p,p*)(如果1

事实上, 文献[4]在非线性项超线性增长时结合Ambrosetti-Rabinowitz(AR)条件或在渐近线性增长且在无穷远处关于λ1的共振条件满足时分别得到了一个非平凡解. 文献[5-6]在非线性项满足超线性增长但(AR)条件不满足的情形下, 结合非线性项在零点处的渐近性态, 得到了问题(1)非平凡解的存在性.

2) 当t→+∞时,J(tφ1)→-∞.

引理2在定理1的假设下, 函数J满足(C)c条件.

J(un)→c∈R, (1+‖un‖)‖J′(un)‖→0,n→+∞.

(2)

(3)

由(H4)知, 对任意的>0, 存在M3>0, 使得

(4)

令wn=un/‖un‖p, 则存在{wn}的子列(不妨仍记为{wn})及w0∈W01,p(Ω), 使得wn⇀w0, 并且wn(x)→w0(x) a.e.x∈Ω. 由式(4)可得1≤(η+). 因此, 存在Ω的正测度子集Ω0, 使得w0(x)≠0 a.e.x∈Ω0. 从而对a.e.x∈Ω0, 有un(x)→∞(n→∞). 由(H3)和Fatou引理可得这与式(3)矛盾. 因此{un}在中有界. 从而存在使得当n→∞时, ‖un‖→‖u0‖.

由引理1与引理2, 并应用推广形式的山路定理[8], 即可完成定理1的证明.

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