×

Infinitely many solutions for a critical Grushin-type problem via local Pohozaev identities. (English) Zbl 1448.35134

Summary: 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.

MSC:

35J15 Second-order elliptic equations
35B09 Positive solutions to PDEs
35B33 Critical exponents in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beckner, W., On the Grushin operator and hyperbolic symmetry, Proc. Am. Math. Soc., 129, 1233-1246 (2001) · Zbl 0987.47037
[2] Badiale, M.; Tarantello, G., A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163, 259-293 (2002) · Zbl 1010.35041
[3] Benci, V.; Cerami, G., Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}\) in \(\mathbb{R}^N\), J. Funct. Anal., 88, 90-117 (1990) · Zbl 0705.35042
[4] Castorina, D.; Fabbri, I.; Mancini, G.; Sandeep, K., Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differ. Equ., 246, 1187-1206 (2009) · Zbl 1173.35053
[5] Cao, D.; Heinz, HP, Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations, Math. Z., 243, 599-642 (2003) · Zbl 1142.35601
[6] Cao, D.; Peng, S.; Yan, S., On the Webster scalar curvature problem on the CR sphere with a cylindrical-type symmetry, J. Geom. Anal., 23, 1674-1702 (2013) · Zbl 1277.32041
[7] Cerami, G.; Devillanova, G.; Solimini, S., Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial. Differ. Equ., 23, 139-168 (2005) · Zbl 1078.35113
[8] Chen, W.; Wei, J.; Yan, S., Infinitely many solutions for the Schrödinger equations in \(\mathbb{R}^N\) with critical growth, J. Differ. Equ., 252, 2425-2447 (2012) · Zbl 1235.35104
[9] Deng, Y.; Lin, CS; Yan, S., On the prescribed scalar curvature problem in \(\mathbb{R}^N\), local uniqueness and periodicity, J. Math. Pures Appl., 104, 1013-1044 (2015) · Zbl 1328.53045
[10] del Pino, M.; Felmer, P.; Musso, M., Two-bubble solutions in the super-critical Bahri-Coron’s problem, Calc. Var. Partial. Differ. Equ., 16, 113-145 (2003) · Zbl 1142.35421
[11] Gheraibia, B.; Wang, C.; Yang, J., Existence and local uniqueness of bubbling solutions for the Grushin critical problem, Differ. Integral Equ., 32, 49-90 (2019) · Zbl 1449.35069
[12] Guo, Y.; Li, B., Infinitely many solutions for the prescribed curvature problem of polyharmonic operator, Calc. Var. Partial. Differ. Equ., 46, 809-836 (2013) · Zbl 1262.53035
[13] Guo, Y.; Nie, J.; Niu, M.; Tang, Z., Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in \(\mathbb{R}^N\), Calc. Var. Partial. Differ. Equ., 56, 118 (2017) · Zbl 1376.35045
[14] Jerison, D.; Lee, JM, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Am. Math. Soc., 1, 1-13 (1988) · Zbl 0634.32016
[15] Li, YY, On a singularly perturbed elliptic equation, Adv. Differ. Equ., 2, 955-980 (1997) · Zbl 1023.35500
[16] Li, YY, On \(-\Delta u=K(x)u^5\) in \(\mathbb{R}^3\), Commun. Pure Appl. Math., 46, 303-340 (1993) · Zbl 0799.35068
[17] Li, YY; Wei, J.; Xu, H., Multi-bump solutions of \(-\Delta u=K(x)u^{\frac{n+2}{n-2}}\) on lattices in \(\mathbb{R}^n\), J. Reine Angew. Math., 743, 163-211 (2018) · Zbl 1410.35047
[18] Lin, FH; Ni, WM; Wei, J., On the number of interior peak solutions for a singularly perturbed Neumann problem, Commun. Pure Appl. Math., 60, 252-281 (2007) · Zbl 1170.35424
[19] Lucia, M.; Tang, Z., Multi-bump bound states for a Schrödinger system via Lyapunov-Schmidt reduction, Nonlinear Differ. Equ. Appl., 24, 65 (2017) · Zbl 1383.35067
[20] Mancini, G.; Fabbri, I.; Sandeep, K., Classification of solutions of a critical Hardy-Sobolev operator, J. Differ. Equ., 224, 258-276 (2006) · Zbl 1208.35054
[21] Monti, R.; Morbidelli, D., Kelvin transform for Grushin operators and critical semilinear equations, Duke Math. J., 131, 167-202 (2006) · Zbl 1094.35036
[22] Peng, S.; Wang, C.; Yan, S., Construction of solutions via local Pohozaev identities, J. Funct. Anal., 274, 2606-2633 (2018) · Zbl 1392.35148
[23] Rey, O., Boundary effect for an elliptic Neumann problem with critical nonlinearity, Commun. Partial Differ. Equ., 22, 1055-1139 (1997) · Zbl 0891.35040
[24] Rey, O.; Wei, J., Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc., 7, 449-476 (2005) · Zbl 1129.35406
[25] Vétois, J.; Wang, S., Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, Adv. Nonlinear Anal., 8, 715-724 (2019) · Zbl 1419.35012
[26] Wang, ZQ, Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonlinear Anal., 27, 1281-1306 (1996) · Zbl 0862.35040
[27] Wang, C.; Wang, Q.; Yang, J., On the Grushin critical problem with a cylindrical symmetry, Adv. Differ. Equ., 20, 77-116 (2015) · Zbl 1311.35169
[28] Wang, L.; Wei, J.; Yan, S., A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture, Trans. Am. Math. Soc., 362, 4581-4615 (2010) · Zbl 1204.35093
[29] Wang, L.; Wei, J.; Yan, S., On Lin-Ni’s conjecture in convex domains, Proc. Lond. Math. Soc., 102, 1099-1126 (2011) · Zbl 1236.35077
[30] Wei, J.; Yan, S., Infinitely many positive solutions for the nonlinear Schrödinger equations in \(\mathbb{R}^N\), Calc. Var. Partial. Differ. Equ., 37, 423-439 (2010) · Zbl 1189.35106
[31] Wei, J.; Yan, S., Infinitely many solutions for the prescribed scalar curvature prolem on \(S^N\), J. Funct. Anal., 258, 3048-3081 (2010) · Zbl 1209.53028
[32] Yan, S.; Yang, J., Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents, Calc. Var. Partial. Differ. Equ., 48, 587-610 (2013) · Zbl 1280.35048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.