Annali di Matematica Pura ed Applicata ( IF 1.0 ) Pub Date : 2021-03-01 , DOI: 10.1007/s10231-021-01077-7 Giovany M. Figueiredo , Vicenţiu D. Rădulescu
In this paper, we are concerned with the problem
$$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$where \(\Omega \subset {\mathbb {R}}^{2}\) is a bounded smooth domain and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.
中文翻译：
具有指数临界增长的规定平均曲率方程的正解
在本文中，我们关注的问题
$$ \ begin {aligned}-\ text {div} \ left（\ displaystyle \ frac {\ nabla u} {\ sqrt {1+ | \ nabla u | ^ 2}} \ right）= f（u）\ \ text {in} \ \ Omega，\ \ u = 0 \ \ text {on} \ \ \ partial \ Omega，\ end {aligned} $$其中\（\ Omega \ subset {\ mathbb {R}} ^ {2} \）是有界平滑域，而\（f：{\ mathbb {R}} \ rightarrow {\ mathbb {R}} \）是一个具有临界指数增长的超线性连续函数。我们首先对规定的平均曲率算符进行截断，并得到一个辅助问题。接下来，我们使用Nehari流形方法显示该辅助问题的正解的存在。最后，我们得出结论，通过使用Moser迭代方法和Stampacchia的估计，辅助问题的解决方案是对原始问题的解决方案。
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