Multiplicity results for fractional $p$-Laplacian problems with Hardy term and Hardy-Sobolev critical exponent in $\mathbb{R}^N$

Hadi Mirzaee

DOI: http://dx.doi.org/10.12775/TMNA.2019.013

Abstract


This paper is devoted to the study of a class of singular fractional $p\hskip0.5pt$-Laplacian Problems of the form $$ (-\Delta)_p^su-\mu\frac{|u|^{p-2}u}{|x|^{ps}} =\alpha\frac{|u|^{ p_{s}^{*}(b)-2 }u}{|x|^b} +\beta f(x)|u|^{q-2}u\quad \text{in }\mathbb{R}^N $$% where $0 < s< 1$, $0\leq b< ps< N$, $1< q< p_{s}^{*}(b)$, $\alpha, \beta> 0$, $\mu\in \mathbb{R}$, and $f(x)$ is a given function which satisfies some appropriate condition. By using variational methods, we prove the existence of infinitely many solutions under different conditions.

Keywords


Hardy term; fractional $p$-Laplacian; critical exponent

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References


G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 20 (2013), 977–1009.

B. Barrios, I.M. Medina and I. Peral, Some remarks on the solvability of nonlocal elliptic problems with the Hardy potential, Commun. Contemp. Math. (2013), DOI: 10.1142/S0219199713500466.

B. Barrios, I.M. Medina and I. Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math. 13 (2015), 390–407.

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl. 195 (2016), 2099–2129.

D.C. Clark, A variant of the Lusternik–Schnirelmann theory, Indiana Univ. Math. J. 22 (1972), 65–74.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhike’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy–Leray potential, Calc. Var. Partial Differential Equations (2016), 55–99, DOI: 10.1007/s00526-016-1032-5.

A. Fiscella and P. Pucci, On certain Hardy–Sobolev critical elliptic drichlet problems, Adv. Differential Equations 21 (2016), 571–599.

A. Fiscella and P. Pucci, Kirchhoff–Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud. 17 (2017), 429–456.

J. Garcia and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 941–957.

N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Adv. Nonlinear Stud. 15 (2015), 527–555.

X. He and H. Zou, Infinitely many solutions for a singular elliptic equation involving critical Sobolev–Hardy exponents in RN . Acta Math. Sci. 30 (2010), 830–840.

V. Maz’ya and T. Shaposhnikova, On the Bourgain, Brézis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, Rev. Mat. Iberoam. 195 (2002), 230–238.

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational methods for nonlocal fractional equations, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.

T. Mukherjee and K. Sreenadh, On Dirichlet problem for fractional p-Laplacian with singular non-linearity, Adv. Nonlinear Anal. (2016), DOI: 10.1515/anona-2016-0100.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, Amer. Math. Soc., Providence, RI, 1986.

Y. Song and S. Shi, Existence of infinitely many solutions for degenerate p-fractional Kirchhoff equations with critical Sobolev-Hardy nonlinearities, Z. Angew. Math. Phys. (2017), DOI 10.1007/s00033-017-0867-8.

X. Wang and J. Yang, Singular critical elliptic problems with fractional Laplacian, Electron. J. Differential Equations 197 (2015), 1–12.

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäser, Boston, Basel,Berlin, 1996.


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