On fractional Schro{sup ¨}dinger equation in R{sup N} with critical growth
- Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023 (China)
- School of Science, Jiangnan University, Wuxi 214122 (China)
In this paper, we study the following nonlinear fractional Schro{sup ¨}dinger equation with critical exponent h{sup 2α}(−Δ){sup α}u+V(x)u=|u|{sup 2{sub α}{sup *−2}}u+λ|u|{sup q−2}u,x∈R{sup N}, where h is a small positive parameter, 0 < α < 1, 2<2{sub α}{sup *}, 2{sub α}{sup *}=(2N)/(N−2α) is the critical Sobolev exponent, and N > 2α, λ > 0 is a parameter. The potential V:R{sup N}→R is a positive continuous function satisfying some natural assumptions. By using variational methods, we obtain the existence of solutions in the following case: if 2<2{sub α}{sup *}, there exists λ{sub 0} > 0 such that for all λ ⩾ λ{sub 0}, we show that it has one nontrivial solution and there exist at least cat{sub Λ{sub δ}}(Λ) nontrivial solutions; if max(2,(4α)/(N−2α) )<2{sub α}{sup *}, then there is one nontrivial solution and there exist at least cat{sub Λ{sub δ}}(Λ) nontrivial solutions for all λ > 0.
- OSTI ID:
- 22251777
- Journal Information:
- Journal of Mathematical Physics, Vol. 54, Issue 12; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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