Differential and Integral Equations

Regularity for solutions to nonlinear elliptic equations

Luigi Greco, Gioconda Moscariello, and Gabriella Zecca

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Let $\Omega$ be a domain of ${\mathbb R}^N$, $N>2.$ We establish higher integrability for solutions $u \in W^{1,p}_{\text{loc}}(\Omega)$ of nonlinear PDEs whose prototype~is \begin{equation*} \text{div\,}[|\nabla u|^{p-2}\nabla u +B(x)|u|^{p-2}u]=\text{div\,}(|F|^{p-2}F) \end{equation*} with $1 < p < N$. We assume that the vector field $B\colon\Omega\to{\mathbb R}^N$ belongs to the Marcinkiewicz space $L^{\frac N{p-1},\infty}$. We prove that $F\in L^r\text{loc}(\Omega,{\mathbb R}^N)$, $p < r < N$, implies $u\in L^{r^*} \text{loc}(\Omega).$

Article information

Differential Integral Equations, Volume 26, Number 9/10 (2013), 1105-1113.

First available in Project Euclid: 3 July 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35J20: Variational methods for second-order elliptic equations


Greco, Luigi; Moscariello, Gioconda; Zecca, Gabriella. Regularity for solutions to nonlinear elliptic equations. Differential Integral Equations 26 (2013), no. 9/10, 1105--1113. https://projecteuclid.org/euclid.die/1372858564

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