Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

Abstract

In this paper, we prove the following nonlocal Kirchhoff problem of the type {left leftarray−a+b∫ℝ2|∇u|2dxΔu+V(x)u=f(x,u),arrayx∈ℝ2;arrayu∈H1(ℝ2)array$$ \left\{\begin{array}{ll}-\left(a+b\underset{{\mathbb{R}}^2}{\int }{\left|\nabla u\right|}^2\mathrm{d}x\right)\Delta u+V(x)u=f\left(x,u\right),& x\in {\mathbb{R}}^2;\\ {}u\in {H}^1\left({\mathbb{R}}^2\right)& \end{array}\right. $$ has a Nehari‐type ground state solution when Vfalse(xfalse)$$ V(x) $$ and ffalse(x,ufalse)$$ f\left(x,u\right) $$ are periodic on x$$ x $$ and ffalse(x,ufalse)$$ f\left(x,u\right) $$ has critical exponential growth in the sense of Trudinger–Moser inequality on u$$ u $$. We develop some new approaches to estimate precisely the minimax level of the energy functional and to recover the compactness of Cerami sequences of the associated Euler–Lagrange functional. With this in hand, we can overcome difficulties arising from the appearance of the nonlocal term ∫ℝ2false|∇ufalse|2normaldx$$ {\int}_{{\mathbb{R}}^2}{\left|\nabla u\right|}^2\mathrm{d}x $$ and the nonlinearity which is of critical growth of Trudinger–Moser type in whole Euclidean space ℝ2$$ {\mathbb{R}}^2 $$.