Advances in Nonlinear Analysis (Dec 2024)
On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
Abstract
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 , 00\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.
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