The cone Moser-Trudinger inequalities and their applications

TL;DR

This paper establishes cone Moser-Trudinger inequalities on bounded and unbounded domains, and applies these results to prove the existence of solutions for a nonlinear PDE involving a Fuchsian type Laplace operator with exponential growth nonlinearities.

Contribution

It introduces and analyzes cone Moser-Trudinger inequalities and applies them to solve a class of nonlinear PDEs with exponential growth terms involving a degenerate Laplace operator.

Findings

Established cone Moser-Trudinger inequalities with best exponents.

Proved existence of weak solutions for nonlinear PDEs with exponential growth.

Analyzed PDEs with Fuchsian type Laplace operators on bounded domains.

Abstract

In this article, we firstly study the cone Moser-Trudinger inequalities and their best exponents $\alpha_2$ on both bounded and unbounded domains $\mathbb{R}^2_{+}$. Then, using the cone Moser-Trudinger inequalities, we study the existence of weak solutions to the nonlinear equation \begin{equation*} \left\{\begin{array}{ll} -\Delta_{\mathbb{B}} u=f(x, u), &\mbox{in}\ x\in \mbox{int} (\mathbb{B}), \\ u= 0, &\mbox{on}\ \partial\mathbb{B}, \end{array} \right. \end{equation*} where $\Delta_{\mathbb{B}}$ is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary $x_1 =0$, and the nonlinearity $f$ has the subcritical exponential growth or the critical exponential growth.