Existence of Positive Solutions for Kirchhoff Type Problem with Singular and Logarithmic Nonlinearity

Abstract

In this paper, we consider the following Kirchhoff type problem with singular and logarithmic nonlinearity

where $Ω ⊂ \mathbb{R}^3$ is a bounded domain with smooth boundary, $0 < γ < 1,$ $4 < p < 6,$ $λ$ is positive constant. By using variational method and the critical point theory for nonsmooth functional, we obtain the existence of two positive solutions. In particular, we propose innovative techniques to tackle the challenges arising from the sign-changing property of $u^{p−1} {\rm ln} |u|,$ which violates both the monotonicity condition and the Ambrosetti-Rabinowitz condition. Moreover, unlike the scenario where $a = 1$ and $b = 0,$ the inclusion of the nonlocal term $(\int_Ω |∇u|^2dx)∆u$ brings about extra complexities. A key issue here is the lack of weak continuity in the functional

for any $v ∈ H_0^1(Ω),$ a factor that substantially complicates the proof that the limit of a $(PS)_c$ sequence corresponds to a nontrivial solution of the problem.