Rocky Mountain Journal of Mathematics

Infinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growth

Lixia Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we consider the existence of infinitely many sign-changing solutions for the following Hardy-Sobolev-Maz'ya equation \[ \left \{\begin{aligned} &-\triangle u-\frac {\mu u}{|y|^2}=\lambda u+\frac {|u|^{2^{\ast }(t)-2}u}{|y|^t} \quad \mbox {in } \Omega , \\ &u=0 \quad \qquad \qquad \qquad \qquad \qquad \qquad \,\mbox {on } \partial \Omega , \end{aligned} \right . \] where $\Omega $ is an open bounded domain in $\mathbb {R}^N, \mathbb {R}^N=\mathbb {R}^k\times \mathbb {R}^{N-k}$, $\lambda >0$, $0\leq \mu \lt {(k-2)^2}/{4}$ when $k>2$, $\mu =0$ when $k=2$ and $2^{\ast }(t)={2(N-t)}/({N-2})$. A point $x\in \mathbb {R}^N$ is denoted as $x=(y,z)\in \mathbb {R}^k\times \mathbb {R}^{N-k}$, and the points $x^0=(0,z^0)$ are contained in $\Omega $. By using a compactness result obtained previously, we prove the existence of infinitely many sign changing solutions by a combination of the invariant sets method and the Ljusternik-Schnirelman type minimax method.

Article information

Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1371-1390.

First available in Project Euclid: 29 August 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations

Hardy-Sobolev-Maz'ya equation sign-changing solutions critical growth


Wang, Lixia. Infinitely many sign-changing solutions for the Hardy-Sobolev-Maz'ya equation involving critical growth. Rocky Mountain J. Math. 49 (2019), no. 4, 1371--1390. doi:10.1216/RMJ-2019-49-4-1371.

Export citation


  • A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
  • G. Arioli, F. Gazzola, H.C. Grunau and E. Sassone, The second bifurcation branch for radial solutions of the Brézis-Nirenberg problem in dimension four, Nonlin. Diff. Eqs. Appl. 15 (2008), 69–90.
  • F.V. Atkinson, H. Brézis and L.A. Peletier, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Diff. Eqs. 85 (1990), 151–170.
  • M. Badiale and G. Tarantello, A Sobolev Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Rat. Mech. Anal. 163 (2002), 259–293.
  • T. Bartsch, Z.L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Part. Diff. Eqs. 29 (2004), 25–42.
  • J. Batt, W. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal. 93 (1986), 159–183.
  • M. Bhakta and K. Sandeep, Poincaré -Sobolev equations in the hyperbolic space, Calc. Var. Part. Diff. Eqs. 44 (2012), 247–269.
  • H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
  • D.M. Cao and P.G. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Diff. Eqs. 205 (2004), 521–537.
  • A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Poincare Anal. 2 (1985), 463–470.
  • D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Diff. Eqs. 246 (2009), 1187–1206.
  • G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. Poincare Anal. 1 (1984), 341–350.
  • G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), 289–306.
  • Z.J. Chen and W.M. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Diff. Eqs. 252 (2012), 969–987.
  • M. Clapp and T. Weth, Multiple solutions for the Brézis-Nirenberg problem, Adv. Diff. Eqs. 10 (2005), 463–480.
  • G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Diff. Eqs. 7 (2002), 1257–1280.
  • ––––, A multiplicity result for elliptic equations at critical growth in low dimension, Comm. Contemp. Math. 15 (2003), 171–177.
  • D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh 105 (1987), 205–213.
  • D. Ganguly, Sign changing solutions of the Hardy-Sobolev-Maz'ya equation, Adv. Nonlin. Anal. 3 (2014), 187–196.
  • M. Gazzini and R. Musina, Hardy-Sobolev-Maz'ya inequalities: Symmetry and breaking symmetry of extremals, Comm. Contemp. Math. 11 (2009), 993–1007.
  • E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Eqs. 156 (1999), 407–426.
  • L. Li, J.J. Sun and S. Tersian, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian, Fract. Calc. Appl. Anal. 20 (2017), 1146–1164.
  • S.J. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002), 3207–3227.
  • Y.Y. Li, On the existence and symmetry properties of finite total mass solutions on Matukuma equation, the Eddington equation and their generalization, Arch. Rat. Mech. Anal. 108 (1989), 175–194.
  • ––––, On the positive solutions of the Matukuma equation, Duke Math. J. 70 (1993), 575–589.
  • Y.Y. Li and W.N. Ni, On conformal scalar curvature equations in $\mathbb{R}^N$, Duke Math. J. 57 (1988), 895–924.
  • Z.L. Liu, F.A. van Heerden and Z.-Q. Wang, Nodal type bound states of Schrödinger equations via invariant set and minimax methods, J. Diff. Eqs. 214 (2005), 358–390.
  • G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy Sobolev operator, J. Diff. Eqs. 224 (2006), 258–276.
  • G. Mancini and K. Sandeep, On a semilinear elliptic equation in $H^n$, Ann. Soc. Norm. Sup. Pisa 7 (2008), 635–671.
  • V.G. Maz'ja, Sobolev spaces, Springer, Berlin (1985).
  • R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlin. Anal. 68 (2008), 3927–3986.
  • P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65 (1986).
  • M. Schechter and W.M. Zou, On the Brézis-Nirenberg problem, Arch. Rat. Mech. Anal. 197 (2010), 337–356.
  • S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sovolev space, Ann. Inst. Poincare Anal. 12 (1995), 319–337.
  • M. Struwe, Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, Springer, Berlin (2000).
  • J.J. Sun and S.W. Ma, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem, Comm. Pure Appl. Ananl. 13 (2014), 2317–2330.
  • A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Diff. Int. Eqs. 22 (2009), 913–926.
  • A.K. Tertikas and K. Tintarev, On the existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura Appl. 186 (2007), 645–662.
  • C.H. Wang and T.J. Wang, Infinitely many solutions for Hardy-Sobolev-Maz'ya equation involving critical growth, Comm. Contemp. Math. 14 (2012), 1250044.
  • Y.Z. Wu and Y.S. Huang, Infinitely many sign-changing solutions for $p$-Laplacian equation involving the critical Sobolev exponent, Bound. Val. Probl. 2013 (2013).
  • J. Zhang and S.W. Ma, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving Hardy potential, Acta Math. Sci. 36 (2016), 527–536.