### Multiple solutions for quasilinear equation involving Hardy critical Sobolev exponents

DOI: http://dx.doi.org/10.12775/TMNA.2019.117

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T. Bartsch, S. Peng and Z. Zhang, Existence and non–existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities, Calc. Var. Partial Differential Equations 30 (2007), 113–136.

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math. 36 (1983), 437–477.

D. Cao and P. Han, Solutions to critical elliptic equation with multi-singular inverse square potentials, J. Differential Equations 224 (2006), 332–372.

D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Funct. Anal. 262 (2012), 2861–2902.

D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations 38 (2010), 471–501.

J. Chern and C. Lin, Minimizers of Caffarell–Kohn–Nirenberg inequalities with the singularity on the boundary, Arch. Ration. Mech. Anal. 197 (2010), 401–432.

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. Theor. Meth. App. 56 (2004), 213–226.

F. Gao and Y. Guo, Multiple solutions for a critical quasilinear equation with Hardy potential, Discrete Contin. Dyn. Syst. Ser. S 12 (2019), 1977–2003.

Y. Deng, Y. Guo and J. Liu, Existence of solutions for quasilinear elliptic equations with Hardy potential, J. Math. Phys. 57 (2016), 031503, 15 pp.

Y. Deng, Y. Guo and S. Yan, Multiple solutions for critical quasilinear elliptic equations, Calculus Var. 59 (2019) Art. 2, 26 pp.

G. Divillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations 7 (2002), 1257–1280.

Y. Guo, J. Liu and Z. Wang, On a Brezis–Nirenburg type quasilinear problem, J. Fixed Point Theory Appl. 19 (2017), 719–753.

N. Ghoussoub and X. Kang, Hardy Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 767–793.

N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy–Sobolev inequalities, Geom. Funct. Anal. 16 (2006), 1201–1245.

N. Ghoussoub and N. Yuan, Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), 5703–5743.

C. Hsia, C. Lin and H. Wadade, Revisiting an idea of Brezis and Nirenberg, J. Funct. Anal. 259 (2010), 1816–1849.

Y.Y. Li and C.S. Lin, A nonlinear elliptic PDE with two Sobolev–Hardy critical exponents, Arch. Ration. Mech. Anal. 203 (2012), 943–968.

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equation I, Proc. Amer. Math. Soc. 131 (2003), 441–448.

J. Liu, Y. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equation II, J. Differential Equations 187 (2003), 473–493.

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differerential Equations 14 (2002), 329–344.

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. 80 (2013), 194–201.

C. Tintarev, Concentration analysis and cocompactness, Concentration Analysis and Applications to Partial Differential Equations, Trends Math., Birkhäuser/Springer, Basel, 2013, pp. 117–141.

C. Tintarev and K. H. Fineseler, Concentration Compactness, Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.

C. Wang and C. Xiang, Infinitely many solutions for quasilinear elliptic equations involving double critical terms and boundary geometry, Ann. Acad. Sci. Fenn. Math. 41 (2016), 973–1004.

S. Yan and J. Yang, Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents, Calc. Var. Partial Differerential Equations 48 (2013), 587–610.

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