Advances in Nonlinear Analysis (May 2022)
On the planar Kirchhoff-type problem involving supercritical exponential growth
Abstract
This article is concerned with the following nonlinear supercritical elliptic problem: −M(‖∇u‖22)Δu=f(x,u),inB1(0),u=0,on∂B1(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 B1(0){B}_{1}\left(0) is the unit ball in R2{{\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 tt, which behaves as exp[(β0+∣x∣α)t2]\exp {[}({\beta }_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] and exp(β0t2+∣x∣α)\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }}) with β0{\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 liminft→∞tf(x,t)exp[(β0+∣x∣α)t2]{\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em}x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} and liminft→∞tf(x,t)exp(β0t2+∣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)=1M(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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