Harnack inequality for solutions of the p(x)-Laplace equation under the precise non-logarithmic Zhikov’s conditions
Copyright policy )Harnack inequality for solutions of the p(x)-Laplace equation under the precise non-logarithmic Zhikov’s conditions
AbstractWe prove continuity and Harnack’s inequality for bounded solutions to the equation $$\begin{aligned} \textrm{div}\big (\mid \nabla u\mid ^{p(x)-2}\,\nabla u \big )&=0, \quad p(x)= {\bar{p}} + L\frac{\log \log \frac{1}{\mid x-x_{0}\mid }}{\log \frac{1}{\mid x-x_{0}\mid }}, \\ {\bar{p}}&>1, \quad L>0, \end{aligned}$$ div ( ∣ ∇ u ∣ p ( x ) - 2 ∇ u ) = 0 , p ( x ) = p ¯ + L log log 1 ∣ x - x 0 ∣ log 1 ∣ x - x 0 ∣ , p ¯ > 1 , L > 0 , under the precise non-logarithmic condition on the function p(x).
\(p\)-Laplacian equation with variable exponent, Zhikov's condition, Asymptotic behavior of solutions to PDEs, Smoothness and regularity of solutions to PDEs, Positive solutions to PDEs, Harnack inequality;continuity, A priori estimates in context of PDEs, 35B09, 35B40, 35B45, 35B65, non-negative bounded solutions, Mathematics - Analysis of PDEs, FOS: Mathematics, Quasilinear elliptic equations with \(p\)-Laplacian, Analysis of PDEs (math.AP)
\(p\)-Laplacian equation with variable exponent, Zhikov's condition, Asymptotic behavior of solutions to PDEs, Smoothness and regularity of solutions to PDEs, Positive solutions to PDEs, Harnack inequality;continuity, A priori estimates in context of PDEs, 35B09, 35B40, 35B45, 35B65, non-negative bounded solutions, Mathematics - Analysis of PDEs, FOS: Mathematics, Quasilinear elliptic equations with \(p\)-Laplacian, Analysis of PDEs (math.AP)
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