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 ] {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em}x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} and liminf t → ∞ t f ( x , t ) exp ( β 0 t 2 + ∣ x ∣ α ) {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }})} , respectively. Our results generalize and improve the ones in G. M. Figueiredo and U. B. Severo (Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), no. 1, 23–39.) and Q. A. Ngó and V. H. Nguyen (Supercritical Moser-Trudinger inequalities and related elliptic problems, Calc. Var. Partial Differ. Equ. 59 (2020), no. 2, Paper No. 69, 30.) for M ( t ) = 1 M(t)=1 . In particular, if the weighted term ∣ x ∣ α | x\hspace{-0.25em}{| }^{\alpha } is vanishing, we can obtain the ones in S. T. Chen, X. H. Tang, and J. Y. Wei (2021) (Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), no. 1, Paper No. 38, Theorem 1.3 and Theorem 1.4) immediately.

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关于涉及超临界指数增长的平面基尔霍夫型问题

本文关注以下非线性超临界椭圆问题： - 米 ( ‖ ∇ 你 ‖ 2 2 ) Δ 你 = F ( X , 你 ) , 在 乙 1 ( 0 ) , 你 = 0 , 上 ∂ 乙 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. 在哪里 乙 1 ( 0 ) {B}_{1}\left(0) 是单位球在 R 2 {{\mathbb{R}}}^{2} , 米 ： R + → R + M:{{\mathbb{R}}}^{+}\to {{\mathbb{R}}}^{+} 是基尔霍夫函数，并且 F ( X , 吨 ) f\左(x,t) 有超临界指数增长 吨 吨 , 表现为 经验 [ ( β 0 + ∣ X ∣ α ) 吨 2 ] \exp {[}({\beta }_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] 和 经验 ( β 0 吨 2 + ∣ X ∣ α ) \exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }}) 和 β 0 {\beta }_{0} , α > 0 \alpha \gt 0 . 基于深入的分析和一些详细的估计，我们通过变分方法获得了上述问题的 Nehari 型基态解。此外，我们可以在较弱的假设下确定极小极大水平的精细上限 限制 吨 → ∞ 吨 F ( X , 吨 ) 经验 [ ( β 0 + ∣ X ∣ α ) 吨 2 ] {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em }x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} 和 限制 吨 → ∞ 吨 F ( X , 吨 ) 经验 ( β 0 吨 2 + ∣ X ∣ α ) {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }})} ， 分别。我们的结果概括并改进了 GM Figueiredo 和 UB Severo 中的结果（具有指数临界增长的基尔霍夫问题的基态解, 米兰 J. 数学。84（2016），第 1, 23–39.) 和 QA Ngó 和 VH Nguyen (超临界 Moser-Trudinger 不等式和相关的椭圆问题, 计算。变量。部分不同。等。59（2020），没有。2，论文 No. 69, 30.) 为 米 ( 吨 ) = 1 M(t)=1 . 特别是，如果加权项 ∣ X ∣ α | x\hspace{-0.25em}{| }^{\alpha } 正在消失，我们可以在 ST Chen, XH Tang, and JY Wei (2021) 中获得（具有临界指数增长的平面基尔霍夫型椭圆问题的改进结果，Z.安吉。数学。物理。72（2021），不。1, Paper No. 38, Theorem 1.3 and Theorem 1.4) 立即。