Quasilinear and Hessian type equations with exponential reaction and measure data

Abstract

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