Calculus of Variations and Geometric Measure Theory

Q. H. Nguyen - L. VĂ©ron

Quasilinear and Hessian type equations with exponential reaction and measure data

created by nguyen on 22 Jul 2018


Published Paper

Inserted: 22 jul 2018
Last Updated: 22 jul 2018

Journal: Archive for Rational Mechanics and Analysis
Pages: 33
Year: 2014


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 $0<R\leq \infty$ and $f$ is a positive integrable function.