On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity - Yang Baoling; Zhang Deli; Liang Sihua (Advances in Nonlinear Analysis)
This work focuses on the Kirchhoff-Schrödinger-Poisson-type system with singular term and critical Sobolev nonlinearity as follows: −a+b∫Ω∣∇u∣pdxΔpu+ϕ∣u∣q−2u=λu−γ+∣u∣p∗−2uinΩ,−Δϕ=∣u∣qinΩ,u=ϕ=0on∂Ω,\left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{p}{\rm{d}}x\right){\Delta }_{p}u+\phi {| u| }^{q-2}u=\lambda {u}^{-\gamma }+{| u| }^{{p}^{\ast }-2}u\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -\Delta \phi ={| u| }^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω\Omega is a bounded domain in RN{{\mathbb{R}}}^{N} with Lipschitz boundary ∂Ω\partial \Omega , 0<γ<10\lt \gamma \lt 1,Δpu=div(∣∇u∣p−2∇u){\Delta }_{p}u={\rm{div}}\left({| \nabla u| }^{p-2}\nabla u), 1<p<q<p*21\lt p\lt q\lt \frac{{p}^{* }}{2}, p*=Np⁄N−p{p}^{* }=Np/N-p is the critical Sobolev exponent, and λ>0\lambda \gt 0. With the Nehari manifold approach, the above problem is discovered to have at least one weak solution. Furthermore, the singular term and critical nonlinearity arise concurrently, which is the main innovation and difficulty of this article. To some extent, we generalize the previous results.