Topological Methods in Nonlinear Analysis

Existence of positive solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

Somayeh Rastegarzadeh and Nemat Nyamoradi

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In this paper, we have used variational methods to study existence of solutions for the following critical nonlocal fractional Hardy elliptic equation \begin{equation*} (- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{|u|^{2_s^*(b) - 2} u}{|x|^{b}} + \lambda f (x, u ), \quad \text{in } \mathbb{R}^N, \end{equation*} where $N > 2 s $, $ 0< s< 1 $, $ \gamma, \lambda $ are real parameters, $(- \Delta)^s$ is the fractional Laplace operator, $2_s^*(b) = {2 (N - b)}/(N - 2s)$ is a critical Hardy-Sobolev exponent with $b \in [0, 2s)$ and $ f \in C(\mathbb R^{N} \times \mathbb{R}, \mathbb{R})$.

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Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 731-746.

First available in Project Euclid: 11 May 2019

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Rastegarzadeh, Somayeh; Nyamoradi, Nemat. Existence of positive solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 731--746. doi:10.12775/TMNA.2019.021.

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  • C.O. Alves and M.A.S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations 254 (2013), 1977–1991.
  • X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.
  • M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl. 195 (2016), no. 6, 2099–2129.
  • G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A. 112 (1978), 332–336. (Italian)
  • G. Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl. 124 (1980), 161–179.
  • E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker'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 55 (2016), 29 pp.
  • S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche 68 (2013), 201–216.
  • P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.
  • G.M. Figueiredo, G. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskiĭ's genus, Asymptot. Anal. 94 (2015), 347–361.
  • A. Fiscella and P. Pucci, On certain nonlocal Hardy–Sobolev critical elliptic Dirichlet problems, Adv. Differential Equations 21 (2016), no. 5–6, 571–599.
  • A. Fiscella and P. Pucci, $p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. 35 (2017), 350–378.
  • A. Fiscella, P. Pucci and S. Saldi, Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators, Nonlinear Anal. 158 (2017), 109–131.
  • A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170.
  • R. L.Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities J. Funct. Anal. 255 (2008), 3407–3430.
  • Q. Li, K. Teng and X. Wu, Existence of positive solutions for a class of critical fractional Schrödinger equations with potential vanishing at infinity, Mediterr. J. Math. (2017), DOI: 10.1007/s00009-017-0846-5. (preprint)
  • V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), 230–238.
  • X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity 29 (2016), 357–374.
  • G. Molica Bisci and V. Radulescu, Ground state solutions of scalar field fractional Schroedinger equations, Calc. Var. Partial Differential Equations 54 (2015), 2985–3008.
  • G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and its Applications, 162 Cambride University Press, ISBN: 9781107111943, 2016.
  • G. Molica Bisci and D. Repov\us, On doubly nonlocal fractional elliptic equations, Rend. Lincei Mat. Appl. 26 (2015), 161–176.
  • G. Molica Bisci and F. Tulone, An existence result for fractional Kirchhoff-type equations, Z. Anal. Anwend. 35 (2016), 181–197.
  • G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math. 19 (2017), no. 1, 1550088, 23 pp.
  • P. Piersanti and P. Pucci, Entire solutions for critical $p$-fractional Hardy Schrödinger–Kirchhoff equations, Publ. Mat. 61 (2017), 26 pp.
  • P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam. 32 (2016), no. 1, 1–22.
  • P. Pucci, M.Q. Xiang and B.L. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785–2806.
  • S. Secchi, Ground state solutions for the fractional Schrödinger in $\mathbb{R}^N$, J. Math. Phys. 54 (2013), 031501.
  • R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.
  • R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.
  • L. Wang, B. Zhang and H. Zhang, Fractional Laplacian system involving doubly critical nonlinearities in $\mathbb{R}^N$, Electron. J. Qual. Theory Differ. Equ. Monogr. Ser. 57 (2017), 1–17.
  • B. Zhang, G. Molica Bisci and M. Xiang, Multiplicity results for nonlocal fractional $p$-Kirchhoff equations via Morse theory, Topol. Methods Nonlinear Anal. 49 (2017), no. 2, 445–461.