Topological Methods in Nonlinear Analysis

A multiplicity result for a degenerate elliptic equation with critical growth on noncontractible domains

Elisa Garagnani and Francesco Uguzzoni

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Abstract

In this paper we consider the semilinear problem with critical growth in the Heisenberg group $-\Delta_{_{\mathbb{H}^n}} u = u^{(Q+2)/(Q-2)}+\lambda u$ in $\Omega$, $u> 0$ in $\Omega$, $u=0$ in $\partial\Omega$, and we provide a multiplicity existence result involving Lusternik-Schnirelmann category.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 1 (2003), 53-68.

Dates
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475266317

Mathematical Reviews number (MathSciNet)
MR2037266

Zentralblatt MATH identifier
1070.35011

Citation

Garagnani, Elisa; Uguzzoni, Francesco. A multiplicity result for a degenerate elliptic equation with critical growth on noncontractible domains. Topol. Methods Nonlinear Anal. 22 (2003), no. 1, 53--68. https://projecteuclid.org/euclid.tmna/1475266317


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References

  • V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u + a(x)u = u^{(n+2)/(n-2)}$ in $\R^N$ , J. Funct. Anal., 88(1990), 90–117 \ref\key 2
  • I. Birindelli, I. Capuzzo Dolcetta and A. Cutr\`i, Liouville theorems for semilinear equations on the Heisenberg group , Ann. Inst. H. Poincaré Anal. Non Linéaire, 14(1997), 295–308 \ref\key 3 ––––, Indefinite semi-linear equations on the Heisenberg group: a priori bounds and existence , Comm. Partial Differential Equations, 23(1998), 1123–1157 \ref\key 4
  • I. Birindelli and A. Cutr\`i, A semi-linear problem for the Heisenberg Laplacian , Rend. Sem. Mat. Univ. Padova, 94(1995), 137–153 \ref\key 5
  • S. Biagini, Positive solutions for a semilinear equation on the Heisenberg group , Boll. Un. Mat. Ital. B (7), 9 (1995), 883–900 \ref\key 6
  • H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math., 36(1983), 437–477 \ref\key 7
  • L. Brandolini, M. Rigoli and A. G. Setti, On the existence of positive solutions of Yamabe-type equations on the Heisenberg group , Duke Math. J., 2 (1996), 101–107 \ref\key 8
  • G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian , Ann. Mat. Pura Appl., 169(1995), 375–392 \ref\key 9
  • G. Citti and F. Uguzzoni, Critical semilinear equations on the Heisenberg group: the effect of the topology of the domain , Nonlinear Anal., 46(2001), 399–417 \ref\key 10
  • I. Ekeland, On the variational principle , J. Math. Anal. Appl., 47(1974), 324–353 \ref\key 11
  • N. Gamara, The CR Yamabe conjecture: the case $n=1$ , J. Eur. Math. Soc., 3(2001), 105–137 \ref\key 12
  • N. Gamara and R. Yacoub, CR Yamabe conjecture – the conformally flat case , Pacific J. Math., 201(2001), 121–175 \ref\key 13
  • N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group , Indiana Univ. Math. J., 41(1992), 71–98 \ref\key 14
  • D. S. Jerison and J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem , J. Differential Geom., 29(1989), 303–343 \ref\key 15 ––––, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem , J. Amer. Math. Soc., 1(1988), 1–13 \ref\key 16
  • M. Lazzo, Multiple positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , C. R. Acad. Sci. Paris Sér. I Math. I (1992), 61–64 \ref\key 17
  • E. Lanconelli and F. Uguzzoni, Asymptotic behaviour and non-existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group , Boll. Unione Mat. Ital. B (8), 1 (1998), 139–168 \ref\key 18
  • G. Lu and J. Wei, On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups , Electron. Res. Announc. Amer. Math. Soc., 3(1997), 83–89 \ref\key 19
  • A. Malchiodi and F. Uguzzoni, A perturbation result for the Webster scalar curvature problem on the CR sphere , J. Math. Pures Appl., 81(2002), 983–997 \ref\key 20
  • D. Passaseo, Multiplicity of positive solutions for the equation $\Delta u + \l u + u^{2^*-1}=0$ in noncontractible domains , Topol. Methods Nonlinear Anal., 2(1993), 343–366 \ref\key 21
  • O. Rey, A multiplicity result for a variational problem with lack of compactness , Nonlinear Anal., 13(1989), 1241–1249 \ref\key 22
  • M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities , Math. Z., 187(1984), 511–517 \ref\key 23