Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (3): 679-699.doi: 10.1007/s10473-020-0307-2

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MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES

Jinguo ZHANG1, Tsing-San HSU2   

  1. 1 School of Mathematics, Jiangxi Normal University, Nanchang 330022, China;
    2 Center for General Education, Chang Gung University, Tao-Yuan, Taiwan, China
  • Received:2018-11-07 Online:2020-06-25 Published:2020-07-17
  • Contact: Tsing-San HSU E-mail:tshsu@mail.cgu.edu.tw

Abstract: In this article, we study the following critical problem involving the fractional Laplacian:\[\left\{ \begin{array}{l} { - \Delta)^{\frac{\alpha }{2}}}u - \gamma \frac{u}{{|x{|^\alpha }}} = \lambda \frac{{|u{|^{q - 2}}}}{{|x{|^s}}} + \frac{{|u{|^{2_\alpha ^*(t) - 2}}u}}{{|x{|^t}}}\quad {\rm{in }}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \Omega,\\ u = 0\quad {\rm{ }}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{in }}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {{\rm{\mathbb{R} }}^N}\backslash \Omega , \end{array} \right.\]where Ω ? RN (N > α) is a bounded smooth domain containing the origin, α ∈ (0, 2), 0 ≤ s, t < α, 1 ≤ q < 2, λ > 0, 2α*(t)=2(N-t)/N -α is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γH, and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.

Key words: Fractional Laplacian, Hardy potential, multiple positive solutions, critical Sobolev-Hardy exponent

CLC Number: 

  • 47G20
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