RN中一类临界非局部问题的正解和无穷多对解
Positive Solutions and Infinity Many Pairs Distinct Solutions for a Critical Nonlocal Problems in RN
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摘要: 研究如下一类带临界指数的非局部问题：{-（a+b∫RN|▽u|2dx）Δu=μ|u|2*-2u+λf（x）|u|q-2u x∈RNu∈D1，2（RN）其中a ≥ 0，b，μ>0，N ≥ 4，1 ≤ q ≤ 2，2*=（2N）/（N-2），系数函数f∈L（2*）/（2*-q）（RN）满足一定的条件.当1 ≤ q N ≥ 4时，利用变分方法和临界点理论获得了该问题的无穷多对解；当q=2，N=4时，利用山路引理获得了该问题的1个正解.Abstract: In this article, we consider a class of nonlocal problem with critical growth{-(a+b∫RN|▽u|2dx)Δu=μ|u|2*-2u+λf(x)|u|q-2u x∈RNu∈D1,2(RN)where a ≥ 0, b,μ>0, N ≥ 4, 1 ≤ q ≤ 2, 2*=(2N)/(N-2),f∈L(2*)/(2*-q)(RN) satisfies some certain conditions. When 1 ≤ q N ≥ 4, infinitely many pairs solutions are obtained by the critical point theorem; while q=2, N=4, by using the Mountain Pass Lemma, one positive solution is obtained.
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Key words:
- critical exponent /
- nonlocal problem /
- infinitely many solutions /
- critical point theorem /
- Mountain Pass Lemma .
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