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Construction of solutions for a critical problem with competing potentials via local Pohozaev identities
Communications in Contemporary Mathematics ( IF 1.6 ) Pub Date : 2020-10-23 , DOI: 10.1142/s0219199720500716
Qihan He 1 , Chunhua Wang 2 , Da-Bin Wang 3
Affiliation  

In this paper, we consider the following critical equation: Δu + V (y)u = K(y)uN+2 N2,u > 0,u H1(N), where (y,y) 2 × N2, V (|y|,y) and K(|y|,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 (r0,y0) with r0 > 0,K(r0,y0) > 0 and B1 := V (r0,y0) NU0,12dy ΔK(r0,y0) 2N N|y|2U 0,12dy > 0, then the above equation has infinitely many positive solutions, where U0,1 is the unique positive solution of Δu = uN+2 N2 with u(0) =maxyNu(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) 2606–2633], 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 (r0,y0) can influence the sign of ΔK(r0,y0), i.e. ΔK(r0,y0) 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.

中文翻译:

通过本地 Pohozaev 身份构建具有竞争潜力的关键问题的解决方案

在本文中,我们考虑以下关键方程: -Δ + (是的) = ķ(是的)ñ+2 ñ-2, > 0, H1(ñ), 在哪里(是的',是的) 2 × ñ-2, (|是的'|,是的)ķ(|是的'|,是的)是两个非负有界函数。使用有限维约简参数和局部 Pohozaev 类型的恒等式,我们证明如果ñ 5,ķ(r,是的)有一个稳定的临界点(r0,是的0)r0 > 0,ķ(r0,是的0) > 01 = (r0,是的0) ñü0,12d是的 -Δķ(r0,是的0) 2*ñ ñ|是的|2ü 0,12*d是的 > 0,则上式有无穷多个正解,其中ü0,1是的唯一正解 - Δ = ñ+2 ñ-2(0) =最大限度是的ñ(是的). 结合 [S. Peng, C. Wang 和 S. Wei,通过局部 Pohozaev 恒等式构造规定的标量曲率问题的解决方案,出现在J. 微分方程; S. Peng、C. Wang 和 S. Yan,通过本地 Pohozaev 身份构建解决方案,J. 功能。肛门。 274(2018) 2606-2633],这意味着稳定临界点的作用ķ(r,是的)在构建凹凸解决方案中比 (r,是的)然后 (r0,是的0)可以影响符号Δķ(r0,是的0), IEΔķ(r0,是的0)可以是非负的,不同于 [S. Peng, C. Wang 和 S. Wei,通过局部 Pohozaev 恒等式构造规定的标量曲率问题的解决方案,出现在J. 微分方程]。溶液的浓度点位于稳定临界点附近ķ(r,是的),其中包括鞍点的情况。
更新日期:2020-10-23
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