On the planar Kirchhoff-type problem involving supercritical exponential growth

This article is concerned with the following nonlinear supercritical elliptic problem: { -M(parallel to del u parallel to(2)(2))Delta u = f(x, u), in B-1(0), u = 0, on partial derivative B-1(0), where B-1(0) is the unit ball in R-2, M : R+ -> R+ is a Kirchhoff function, and f (x, t) has supercritical exponential growth on t, which behaves as exp[(beta(0) + vertical bar x vertical bar(alpha))t(2)] and exp(beta(0)t(2+vertical bar x vertical bar alpha)) with beta(0), alpha > 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 ->infinity)tf(x, t)/exp[(beta(0) + vertical bar x vertical bar(alpha))t(2)] and liminf(t ->infinity) tf(x, t)/exp(beta(0)t(2+vertical bar x vertical bar alpha)) respectively. Our results gen- eralize 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. Ngo 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. In particular, if the weighted term vertical bar x vertical bar(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.

Automated summary

This article focuses on a nonlinear supercritical elliptic problem and obtains Nehari-type ground state solutions using variational methods. It also determines an upper bound for the minimax level under weaker assumptions on convergence conditions. These results generalize and improve previous research and can immediately give corresponding results under specific conditions.