Ground state solutions of Pohožaev type for Kirchhoff‐type problems with general convolution nonlinearity and variable potential

Abstract

This paper is devoted to dealing with the following nonlinear Kirchhoff‐type problem with general convolution nonlinearity and variable potential: {left leftarray−(a+b∫ℝ3|∇u|2dx)Δu+V(x)u=(Iα∗F(u))f(u),inℝ3,arrayu∈H1(ℝ3),$$ \left\{\begin{array}{l}-\left(a+b{\int}_{{\mathbb{R}}^3}{\left|\nabla u\right|}^2 dx\right)\Delta u+V(x)u=\left({I}_{\alpha}\ast F(u)\right)f(u),\kern0.30em \mathrm{in}\kern0.50em {\mathbb{R}}^3,\\ {}u\in {H}^1\left({\mathbb{R}}^3\right),\end{array}\right. $$ where a>0$$ a>0 $$, b≥0$$ b\ge 0 $$ are constants; V∈C1false(ℝ3,false[0,+∞false)false)$$ V\in {C}^1\left({\mathbb{R}}^3,\right[0,+\infty \left)\right) $$; f∈Cfalse(ℝ,ℝfalse)$$ f\in C\left(\mathbb{R},\mathbb{R}\right) $$, Ffalse(tfalse)=∫0tffalse(sfalse)ds$$ F(t)={\int}_0^tf(s) ds $$; and Iα:ℝ3→ℝ$$ {I}_{\alpha }:{\mathbb{R}}^3\to \mathbb{R} $$ is the Riesz potential, α∈false(0,3false)$$ \alpha \in \left(0,3\right) $$. By applying some new analytical tricks introduced by Tang and Chen, the existence results of ground state solutions of Pohožaev type for the above Kirchhoff type problem are obtained under some mild assumptions on V$$ V $$ and the general “Berestycki‐Lions assumptions” on the nonlinearity f$$ f $$. Our results generalize and improve the ones obtained by Chen and Liu and other related results in the literature.

High Energy Solutions for p-Kirchhoff Elliptic Problems with Hardy–Littlewood–Sobolev Nonlinearity

“…But using Nehari-Pohožaev manifold with some strong assumptions on the potential V (x), Chen & Liu [9], obtained a ground-state solution in R 3 for the complete range, q ∈ (1+α/3, 3+α). For more recent results on ground-state solutions, refer to [10,35].…”