The Nehari manifold for a fractional $p$-Kirchhoff system involving sign-changing weight function and concave-convex nonlinearities
Summary: In this paper, we are concerned with the following fractional $p$-Kirchhoff system with sign-changing nonlinearities: \[\mathcal{M}\Biggl(\int_{\mathbb{R}^{2n}} (|u(x)-u(y)|^p/|[x-y]|^{n+ps})\,dx\,dy)(-\Delta)^s_pu= \lambda a(x)|u|^{q-2} u+(\alpha/(\alpha+\beta)) f(x)|u|^{\alpha-2} u|v|^\beta\text{ in }\Omega,\] \[\mathcal{M}\Biggl(\int_{\mathbb{R}^{2n}} )|v(x)-v(y)|^p/|x-y|^{n+ps})\,dx\,dy\Biggr)(-\Delta)^s_pv= \mu b(x)|v|^{q-2} v+(\beta/(\alpha+\beta)) f(x)|u|^\alpha|v|^{\beta-2}\text{ in }\Omega,\] and $u=v=0$, in $\mathbb{R}^n\setminus\Omega$, where $\Omega$ is a smooth bounded domain in $\mathcal{R}^n$, $n\ge ps$, $s\in(0, 1)$, $\lambda$, $\mu$ are two real parameters, $1< q< p< p(h +1)< \alpha+\beta< p^*_s= np/(n-ps)$, $\mathcal{M}$ is a continuous function, given by $\mathcal{M}(t)= k+lt^h$, $k>0$, $l\ge 0$, $h\ge 1$, $a(x), b(x)\in L^{(\alpha+\beta)/(\alpha+\beta-q)}(\Omega)$ are sign changing and either $a^{\pm}= \max\{\pm a,0\}\not\equiv 0$ or $b^{\pm}= \max\{\pm b, 0\}\not\equiv 0$, $f\in L(\overline\Omega)$ with $\Vert f\Vert_\infty= 1$, and $f\ge 0$.\par Using Nehari manifold method, we prove that the system has at least two solutions with respect to the pair of parameters $(\lambda,\mu)$.
scientific article; zbMATH DE number 7040453
scientific article