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

doi:10.1007/s10473-020-0307-2

摘要/Abstract

摘要： 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 ? R^N (N > α) is a bounded smooth domain containing the origin, α ∈ (0, 2), 0 ≤ s, t < α, 1 ≤ q < 2, λ > 0, 2_α^*(t)=2(N-t)/N -α is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γ_H, and γ_H 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.

关键词: Fractional Laplacian, Hardy potential, multiple positive solutions, critical Sobolev-Hardy exponent

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 ? R^N (N > α) is a bounded smooth domain containing the origin, α ∈ (0, 2), 0 ≤ s, t < α, 1 ≤ q < 2, λ > 0, 2_α^*(t)=2(N-t)/N -α is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γ_H, and γ_H 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

中图分类号:

47G20

张金国, 许清山. MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES[J]. 数学物理学报(英文版), 2020, 40(3): 679-699.

Jinguo ZHANG, Tsing-San HSU. MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES[J]. Acta mathematica scientia,Series B, 2020, 40(3): 679-699.

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