Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth

Abstract

In this article, we study two classes of Kirchhoff-type equations as follows: −a+b∫R3∣∇u∣2dxΔu+V(x)u=(Iα∗∣u∣p)∣u∣p−2u+f(u),inR3,u∈H1(R3),\left\{\begin{array}{l}-\left(a+b\underset{{{\mathbb{R}}}^{3}}{\overset{}{\int }}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+V\left(x)u=({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u+f\left(u),\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{3}),\end{array}\right.\hspace{1.85em} and −a+b∫R3∣∇u∣2dxΔu+V(x)u=(Iα∗∣u∣p)∣u∣p−2u+m∣u∣l−2u,inR3,u∈H1(R3),\left\{\begin{array}{l}-\left(a+b\underset{{{\mathbb{R}}}^{3}}{\overset{}{\int }}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+V\left(x)u=({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u+m| u{| }^{l-2}u,\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{3}),\end{array}\right. where a>0a\gt 0, b≥0b\ge 0, α∈(0,3)\alpha \in \left(0,3), (3+α)/30m\gt 0, V:R3→RV:{{\mathbb{R}}}^{3}\to {\mathbb{R}} is a potential function and Iα{I}_{\alpha } is a Riesz potential whose order is α∈(0,3)\alpha \in \left(0,3). Under some assumptions on V(x)V\left(x) and f(u)f\left(u), we can prove that the equations have ground state solutions by variational methods.

Keywords

• kirchhoff equation
• ground state solutions
• pohozaev identity
• nehari manifold
• 35j60
• 35j35
• 35a15