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Applicable Analysis
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Volume 102, 2023 - Issue 15
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Articles

On the existence and multiplicity of solutions for a class of sub-Laplacian problems involving critical Sobolev–Hardy exponents on Carnot groups

Pages 4209-4229 | Received 18 Mar 2021, Accepted 25 Jun 2022, Published online: 08 Aug 2022

References

  • Garofalo N, Lanconelli E. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann Inst Fourier (Grenoble). 1990;40(2):313–356.
  • D'Ambrosio L. Some hardy inequalities on the Heisenberg group. Differ Equ. 2004;40(4):552–564.
  • D'Ambrosio L. Hardy-type inequalities related to degenerate elliptic differential operators. Ann Sc Norm Super Pisa Cl Sci. 2005;5(4):451–486.
  • Cao D, Han P. Solutions for semilinear elliptic equations with critical exponents and hardy potential. J Diff Equ. 2004;205:521–537.
  • Ghoussoub N, Robert F. The effect of curvature on the best constant in the Hardy–Sobolev inequality. Geom Funct Anal. 2006;16:897–908.
  • Ghoussoub N, Robert F. Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth. Int Math Res Pap IMRP. 2006;2006. Article ID 21867, 1–85.
  • Kang D, Peng S. Existence of solutions for elliptic problems with critical Sobolev–Hardy exponents. Israel J Math. 2004;143:281–297.
  • Kang D, Peng S. Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential. Appl Math Lett. 2005;18:1094–1100.
  • Cao D, Kang D. Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms. J Math Anal Appl. 2007;333:889–903.
  • Demengel F, Hebey E. On some nonlinear equations involving the p-Laplacian with critical Sobolev growth. Adv Diff Equ. 1998;3:533–574.
  • Ghoussoub N, Yuan C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and hardy exponents. Trans Amer Math Soc. 2000;352:5703–5743.
  • Han P. Quasilinear elliptic problems with critical exponents and hardy terms. Nonlinear Anal. 2005;61:735–758.
  • Felli V, Terracini S. Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Comm Partial Diff Equ. 2006;31:469–495.
  • Li J. Equation with critical Sobolev–Hardy exponents. Int J Math Math Sci. 2005;20:3213–3223.
  • Abdellaoui B, Felli V, Peral I. Existence and non-existence results for quasilinear elliptic equations involving the p-Laplacian. Boll UMI. 2006;9B:445–484.
  • Filippucci R, Pucci P, Robert F. On a p-Laplace equation with multiple critical nonlinearities. J Math Pures Appl. 2009;91:156–177.
  • Pucci P, Servadei R. Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights. Ann Inst H Poincare ANL. 2008;25:505–537.
  • Loiudice A. Local behavior of solutions to subelliptic problems with hardy potential on Carnot groups. Mediterr J Math. 2018;15. Article Id 81.
  • Bonfiglioli A, Lanconelli E, Uguzzoni F. Stratified lie groups and potential theory for their sub-Laplacians. In: Springer Monographs in Mathematics; Berlin: Springer; 2007.
  • Cowling M, Dooley AH, Koranyi A, et al. H-type groups and Iwasawa decompositions. Adv Math. 1991;87:1–41.
  • Garofalo N, Vassilev D. Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups. Math Ann. 2000;318:453–516.
  • Han Y, Niu P. Hardy–Sobolev type inequalities on the H-type group. Manuscripta Math. 2005;118:235–252.
  • Loiudice A. Critical growth problems with singular nonlinearities on Carnot groups. Nonlinear Anal. 2015;126:415–436.
  • Jerison D, Lee J. Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J Amer Math Soc. 1988;1:1–13.
  • Kajikiya R. A critical point theory related to the symmetric mountain pass lemma and its applications to elliptic equation. J Funct Anal. 2005;225:352–370.
  • Folland G B. Subelliptic estimates and function spaces on nilpotent lie groups. Ark Mat. 1975;13:161–207.
  • Folland G B, Stein E. Hardy spaces on homogeneous groups. In: Mathematical Notes; Vol. 28, Princeton (NJ): University Press; 1982.
  • Lanconelli E. Nonlinear equations on Carnot groups and curvature problems for CR manifolds. Rend Mat Acc Lincei. 2003;14:227–238.
  • Smeta D. A concentration-compactness lemma with applications to singular eigenvalue problems. J Fun Anal. 1999;167:463–480.
  • Brezis H, Lieb E. A relation between pointwise convergence of functions and convergence functionals. Proc Amer Math Soc. 1983;88:486–490.
  • Bartsch T, Liu Z L, Weth T. Sign changing solutions of superlinear Schrödinger equations. Comm Part Diff Equ. 2004;29:25–42.
  • Rabinowitz P. Minimax methods in critical point theory with applications to differential equations, In: CBMS Regional Conference Series in Mathematics; Vol. 65, Providence RI: American Mathematical Society; 1986.
  • Garcia J, Peral I. Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term. Trans Amer Math Soc. 1991;323:941–957.

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