Positive solutions for prescribed mean curvature equations with critical growth in the whole R2

Abstract

In this paper, we study the problem$-\mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u{|}^2}}\right)+V\left(x\right)u=\lambda u^{p-1}+f\left(u\right),u\in H^1\left(R^2\right),u>0,$ where $p>2$, $\lambda >0$, $V:R^2\to R$ is a continuous potential bounded away from zero and $f:R\to R$ is continuous and has exponential critical growth without the assumption of monotonicity on $s\mapsto f\left(s\right)/s$. Firstly, we truncate the prescribed mean curvature operator and obtain a nonzero solution for an auxiliary problem. Next, we use the Moser iteration technique to get some uniform estimates of this solution. We finalize by proving that the solution of the auxiliary problem is positive and actually is a solution of the original problem when λ is large.