Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

Abstract

In this paper we study the existence, multiplicity, and regularity of positive weak solutions for the following Kirchhoff–Choquard problem: M()∬ℝ2Nfalse|ufalse(xfalse)−ufalse(yfalse)false|2false|x−yfalse|N+2s0.1emdxdyfalse(−normalΔfalse)su=λuγ+()∫normalΩfalse|ufalse(yfalse)false|2μ,s∗false|x−yfalse|μ0.1emdyfalse|ufalse|2μ,s∗−2u0.51emin0.51emnormalΩ,3emu=00.51emin0.51emℝN\normalΩ, where Ω is open bounded domain of ℝN with C2 boundary, N > 2s and s ∈ (0, 1). M models Kirchhoff‐type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. (− Δ)s is fractional Laplace operator, λ > 0 is a real parameter, γ ∈ (0, 1) and 2μ,s∗=2N−μN−2s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Hölder regularity of order s. Furthermore, using the variational methods and truncation arguments, we prove the existence of two positive solutions.

High Energy Solutions for p-Kirchhoff Elliptic Problems with Hardy–Littlewood–Sobolev Nonlinearity

“…For this class of problems, extensive literature exists on the existence and multiplicity of results using variational methods. For instance, one can see [1,8,12,14,24,[27][28][29]33] and the references therein. For the energy functional associated to (1.1), Kirchhoff term is homogeneous with degree 4, so to obtain the geometric structure and the boundedness of Palais-Smale sequences for the energy functional, one usually assumes one of the following conditions:…”