A Note on the Sign-Changing Solutions for a Double Critical Hardy-Sobolev-Maz'ya Problem

作者：Wang, Chunhua; Yang, Jing^*

来源：Advanced Nonlinear Studies, 2016, 16(3): 519-528.

DOI：10.1515/ans-2015-5043

• Double Critical Hardy-Sobolev-Maz'ya Problem
• Sign-Changing Solutions
• Morse Index

In this paper, we investigate the following double critical Hardy-Sobolev-Maz'ya problem:
{-Delta u = mu vertical bar u vertical bar(2)*((t)-2)u/vertical bar y vertical bar(t) + vertical bar u vertical bar(2)*((s)-2)u/vertical bar y vertical bar(s) + a(x)u in Omega,
u = 0 on partial derivative Omega,
where mu >= 0, a(x) > 0, 2* (t) = 2(N- t)/N-2, 2* (s) = 2(N- s)/N-2, 0 <= t < s < 2, x = (y, z) is an element of R-k x RN-k, 2 <= k < N, (0, z*) is an element of <(Omega)over bar> and Omega is an open bounded domain in R-N. By applying an abstract theorem presented in [42], we prove that if N > 6 + t when mu > 0, and N > 6 + s when mu = 0, and Omega satisfies some geometric conditions, then the above problem has infinitely many sign- changing solutions. The main tool is to estimate the Morse indices of these nodal solutions.

• 出版日期2016-8
• 单位华中师范大学; 江苏科技大学