Abstract
Let $\Omega$ be an open subset of $\mathbb{R}^{n}$, we establish regularity results for solutions of some degenerate nonhomogeneous equations of the type $${\rm div}\langle {A}(x)Du,Du\rangle^{\frac{p-2}{2}}{ A}(x)Du={\rm div} F{\rm in} \Omega$$ where $p\ge 2$. The nonnegative function $\mathcal K(x)$, which measures the degree of degeneracy of ellipticity bounds, lies in the exponential class, i.e. $\mathrm {exp}(\lambda \mathcal K(x))$ is integrable for some $\lambda>0$. Under this assumption, the gradient of a finite energy solution of (1) lies in the Orlicz-Zygmund class $L^{p}\log^{-1}L(\Omega)$. Our results states that the gradient of such solution is more regular provided $\lambda$ is sufficiently large and the datum $F=F(x)$ belongs to a suitable Orlicz-Zygmund class.
Citation
Luigi D'Onofrio. Gioconda Moscariello. "On finite energy solutions for nonhomogeneous $p$-harmonic equations." Funct. Approx. Comment. Math. 40 (1) 139 - 150, March 2009. https://doi.org/10.7169/facm/1238418804
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