Quasilinear and Hessian type equations with exponential reaction and measure data

TL;DR

This paper investigates the existence and estimates of solutions to quasilinear and Hessian equations with exponential nonlinearities and measure data, providing conditions based on potentials and capacities.

Contribution

It introduces new existence criteria and estimates for solutions to complex nonlinear PDEs involving exponential reactions and measure data, extending previous results.

Findings

Existence results under conditions involving fractional maximal potentials.

Two-sided estimates of solutions using Wolff potentials.

Necessary conditions expressed via Orlicz capacities.

Abstract

We prove existence results concerning equations of the type $-\Delta_pu=P(u)+\mu$ for $p>1$ and $F_k[-u]=P(u)+\mu$ with $1\leq k<\frac{N}{2}$ in a bounded domain $\Omega$ or the whole $\mathbb{R}^N$, where $\mu$ is a positive Radon measure and $P(u)\sim e^{au^\beta}$ with $a>0$ and $\beta\geq 1$. Sufficient conditions for existence are expressed in terms of the fractional maximal potential of $\mu$. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of $\mu$. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form $u={\bf W}_{\alpha,p}^R[P(u)]+f$ in $\mathbb{R}^N$, where $0