Construction of solutions for a critical problem with competing potentials via local Pohozaev identities

doi:10.1142/S0219199720500716

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	Construction of solutions for a critical problem with competing potentials via local Pohozaev identities
	He, Qihan 1,3; Wang, Chunhua 2,4; Wang, Da-Bin 5
刊名	COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
	2022-02-01
卷号	24 期号:1
关键词	A critical elliptic problem competing potentials local Pohozaev identities the stable critical point
ISSN号	0219-1997
DOI	10.1142/S0219199720500716
英文摘要	In this paper, we consider the following critical equation: -Delta u + V(y)u = K(y)u (N + 2/N - 2), u > 0, u is an element of H-1 (R-N), where (y', y '') is an element of R-2 x RN-2, V(vertical bar y'vertical bar, y '') and K(vertical bar y'vertical bar, y '') are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if N >= 5, K(r, y '') has a stable critical point (r(0), y(0)'') with r(0) > 0, K(r(0), y(0)'') > 0 and B-1 := V(r(0), y(0)'') integral(RN) U-0,1(2) dy - Delta K (r(0), y(0)'')/2* N integral(RN) vertical bar y vertical bar 2U(0,1)(2)* dy > 0, then the above equation has infinitely many positive solutions, where U-0,U- 1 is the unique positive solution of -Delta u = u(N + 2/N -2) with u(0) = max(y is an element of RN) u(y). Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606-26331, it implies that the role of stable critical points of K(r, y '') in constructing bump solutions is more important than that of V(r, y '') and that V(r(0), y(0)'') can influence the sign of Delta K(r(0), y(0)''), i.e. Delta K(r(0), y(0)'') can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of K (r, y ''), which include the case of a saddle point.
WOS研究方向	Mathematics
语种	英语
出版者	WORLD SCIENTIFIC PUBL CO PTE LTD
WOS记录号	WOS:000752945700011
内容类型	期刊论文
源URL	[http://ir.lut.edu.cn/handle/2XXMBERH/154806]
专题	理学院
作者单位	1.Guangxi Univ, Guangxi Ctr Math Res, Nanning 530003, Peoples R China; 2.Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China; 3.Guangxi Univ, Coll Math & Informat Sci, Nanning 530003, Peoples R China; 4.Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China; 5.Lanzhou Univ Technol, Dept Appl Math, Lanzhou 730050, Peoples R China
推荐引用方式 GB/T 7714	He, Qihan,Wang, Chunhua,Wang, Da-Bin. Construction of solutions for a critical problem with competing potentials via local Pohozaev identities[J]. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS,2022,24(1).
APA	He, Qihan,Wang, Chunhua,&Wang, Da-Bin.(2022).Construction of solutions for a critical problem with competing potentials via local Pohozaev identities.COMMUNICATIONS IN CONTEMPORARY MATHEMATICS,24(1).
MLA	He, Qihan,et al."Construction of solutions for a critical problem with competing potentials via local Pohozaev identities".COMMUNICATIONS IN CONTEMPORARY MATHEMATICS 24.1(2022).