Positive solutions of the prescribed mean curvature equation with exponential critical growth

Abstract

AbstractIn this paper, we are concerned with the problem $$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ - div ∇ u 1 + | ∇ u | 2 = f ( u ) in Ω , u = 0 on ∂ Ω , where $$\Omega \subset {\mathbb {R}}^{2}$$ Ω ⊂ R 2 is a bounded smooth domain and $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → 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.