A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

作者：Fiscella, Alessio^*

来源：Advances in Nonlinear Analysis, 2019, 8(1): 645-660.

DOI：10.1515/anona-2017-0075

• Kirchhoff-type problems
• fractional Laplacian
• singularities
• critical nonlinearities
• perturbation methods

In this paper, we consider the following critical nonlocal problem: @@@ { M( integral(R)integral(2N) vertical bar u(x) - u(y)vertical bar(2)/vertical bar x - y vertical bar(N+2s) dxdy)(-Delta)(s)u = lambda/u(gamma) + u(2s)*(-1) in Omega, @@@ u > 0 in Omega, @@@ u = 0 in R-N \ Omega, @@@ where Omega is an open bounded subset of R-N with continuous boundary, dimension N > 2s with parameter s is an element of (0, 1), 2(s)* = 2N/(N - 2s) is the fractional critical Sobolev exponent, lambda > 0 is a real parameter, gamma is an element of (0, 1) and M models a Kirchhoff-type coefficient, while (-Delta)(s) is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

• 出版日期2019-1