Ground state solutions for elliptic Kirchhoff–Boussinesq type problems with supercritical exponential growth

Abstract

Abstract In this work, we are interested in studying the existence of a ground state solution for the problem Δ ( w β ( x ) Δ u ) ± div ( w β ( x ) | ∇ u | p − 2 ∇ u ) = f ( x , u ) in B , and u = ∂ u ∂ ν = 0 on ∂ B , $$ \Delta (w_{\beta}(x)\Delta u ) \pm \text{div}(w_{\beta} (x) | \nabla u |^{p-2} \nabla u)= f(x,u) \text{ in }B,\quad \text{and} \ u= \dfrac{\partial u}{\partial \nu}=0 \ \text{on}\ \partial B, $$ where B is the unit ball in R 4 $\mathbb{R}^{4}$ , w β ( x ) = ( log e | x | ) β $w_{\beta}(x)=\big(\log \frac{e}{|x|}\big)^{\beta}$ or w β ( x ) = ( log ( 1 | x | ) β $w_{\beta}(x)=\big(\log (\frac{1}{|x|}\big)^{\beta}$ for β ∈ ( 0 , 1 ) $\beta \in (0,1)$ , 2 < p < 4 $2< p < 4$ and f : R → R $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous class function with supercritical exponential growth. We utilize the Nehari manifold method to establish the existence result.

Keywords

• Biharmonic operator
• p-Laplacian
• Ground state
• Logarithmic weight
• Supercritical exponential growth