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On the planar Kirchhoff-type problem involving supercritical exponential growth by Limin Zhang; Xianhua Tang; Peng Chen is a Mathematics article available to read on EtoBox.

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## Abstract This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-M(\Vert \nabla u{\Vert }\_{2}^{2})\Delta u=f\left(x,u),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}\_{1}\left(0),\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}\_{1}\left(0),\end{array}\right. where B 1 ( 0 ) {B}\_{1}\left(0) is the unit ball in R 2 {{\mathbb{R}}}^{2} , M : R + → R + M:{{\mathbb{R}}}^{+}\to {{\mathbb{R}}}^{+} is a Kirchhoff function, and f ( x , t ) f\left(x,t) has supercritical exponential growth on t t , which behaves as exp [ ( β 0 + ∣ x ∣ α ) t 2 ] \exp {[}({\beta }\_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] and exp ( β 0 t 2 + ∣ x ∣ α ) \exp ({\beta }\_{0}{t}^{2+| x{| }^{\alpha }}) with β 0 {\beta }\_{0} , α > 0 \alpha \gt 0 . Based on a deep analysis and some detailed estimate, we obtain Nehari-type ground state solutions for the above problem by variational method. Moreover, we can determine a fine upper bound for the minimax level under weaker assumption on liminf t → ∞ t f ( x , t ) exp [ ( β 0 + ∣ x ∣ α ) t 2 ]

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Author

Limin Zhang; Xianhua Tang; Peng Chen

Publisher

Walter de Gruyter GmbH

Published

2022

Language

EN

Field

Mathematics (Physical Sciences)