Infinitely many solutions for a critical Grushin-type problem via local Pohozaev identities
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Infinitely many solutions for a critical Grushin-type problem via local Pohozaev identities Min Liu1 · Zhongwei Tang1 · Chunhua Wang2 Received: 15 August 2019 / Accepted: 8 December 2019 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we are concerned with a critical Grushin-type problem. By applying Lyapunov– Schmidt reduction argument and attaching appropriate assumptions, we prove that this problem has infinitely many positive multi-bubbling solutions with arbitrarily large energy and cylindrical symmetry. Instead of estimating the corresponding derivatives of the reduced functional in locating the concentration points of the solutions, we employ the local Pohozaev identities to locate them. Keywords Pohozaev identity · Critical Grushin problem · Cylindrical symmetry · Multi-bubbling solution Mathematics Subject Classification 35J15 · 35B09 · 35B33
1 Introduction and main results Consider the following semilinear elliptic equations involving Grushin operators: Q α +2
G α u + (α + 1)2 |y|2α K (x)u = (α + 1)2 u Q α −2 , u > 0, x = (y, z) ∈ Rm 1 × Rm 2 , (1.1)
Zhongwei Tang: Supported by NSFC (11571040, 11671331). Chunhua Wang: Supported by NSFC (11671162) and CCNU18CXTD04.
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Zhongwei Tang [email protected] Min Liu [email protected] Chunhua Wang [email protected]
1
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
2
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People’s Republic of China
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M. Liu et al.
where α ≥ 0, {m 1 , m 2 } ⊂ N+ , K (x) is a function defined in Rm 1 +m 2 , G α := − y − (α + 1)2 |y|2α z is the so-called Grushin operator, Q α := m 1 + (α + 1)m 2 is the appropriate homogeneous α +2 dimension, and the power Q Q α −2 is the corresponding critical exponent. For general case α > 0, the entire positive solutions of (1.1) with K (x) = 0 were discussed by Monti and Morbidelli [21]. For α = 1, the problem (1.1) becomes into m 1 +2m 2 +2
(− y −4|y|2 z )u +4|y|2 K (x)u = 4u m 1 +2m 2 −2 , u > 0, x = (y, z) ∈ Rm 1 ×Rm 2 . (1.2) This case appeared very early in connection with the Cauchy–Riemann Yamabe problem solved by Jerison and Lee [14]. The Sharp Sobolev estimates for the Grushin operator G 1 = − y − 4|y|2 z were calculated by Beckner [1] in low dimension by using hyperbolic symmetry and conformal geometry. However, as far as we know, there is very little literature on the existence of infinitely many solutions for (1.1) and even (1.2), which is one of our motivations to study this type of problem. In this paper, we shall study the critical Grushin-type problem (1.2) by using Lyapunov–Schmidt reduction argument and local Pohozaev identities. Under appropriate assumptions on K (x), we will prove that (1.2) possesses infinitely many multi-bubbling solutions with arbitrarily large energy and cylindrical symmetry. Our first technique is to transform (1.2) into a Hardy–Sobolev-type problem
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