Multiplicity and asymptotic behavior of solutions to a class of Kirchhoff-type equations involving the fractional p-Laplacian

Abstract

Abstract The present study is concerned with the following fractional p-Laplacian equation involving a critical Sobolev exponent of Kirchhoff type: [a+b(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)θ−1](−Δ)psu=|u|ps∗−2u+λf(x)|u|q−2uin RN, $$\biggl[a+b \biggl( \int_{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy \biggr)^{\theta-1} \biggr](-\Delta)_{p}^{s}u =|u|^{p_{s}^{*}-2}u+\lambda f(x)|u|^{q-2}u \quad\text{in } \mathbb {R}^{N}, $$ where a,b>0 $a,b>0$, θ=(N−ps/2)/(N−ps) $\theta=(N-ps/2)/(N-ps)$ and q∈(1,p) $q\in(1,p)$ are constants, and (−Δ)ps $(-\Delta)_{p}^{s}$ is the fractional p-Laplacian operator with 00 $a,b>0$. Moreover, we regard a>0 $a>0$ and b>0 $b>0$ as parameters to obtain convergent properties of solutions for the given problem as a↘0+ $a\searrow0^{+}$ and b↘0+ $b\searrow0^{+}$, respectively.

Keywords

• Critical Sobolev exponent
• Fractional p-Laplacian
• Kirchhoff
• Multiplicity
• Asymptotic behavior