Nonlocal $p$-Kirchhoff equations with singular and critical nonlinearity terms

Abstract

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{σ-1}(-Δ)^s_p u = \fracλ{u^γ}+u^{ p_s^{*}-1 }\quad \text{in }Ω,\\ u>0,\;\;\;\;\quad \text{in }Ω,\\ u=0,\;\;\;\;\quad \text{in }\mathbb{R}^{N}\setminus Ω,\end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with the smooth boundary $\partial \Omega$, $0<s<1<p<\infty$, $N>sp$, $1<\sigma <p_s^{\ast }/p,$ with $p_s^{\ast }=\frac{Np}{N-ps},$ $(-\Delta {)}_p^s$ is the nonlocal $p$-Laplace operator and $[u{]}_{s,p}$ is the Gagliardo $p$-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.

References in corpus (1)

• Bifurcation and multiplicity results for critical fractional p-Laplacian problems