Regularizing effect for some p-Laplacian systems

Abstract

We study existence and regularity of weak solutions for the following $p$-Laplacian system \begin{cases} -Δ_p u+Aφ^{θ+1}|u|^{r-2}u=f, \ &u\in W_0^{1,p}(Ω),\\-Δ_p φ=|u|^rφ^θ, \ &φ\in W_0^{1,p}(Ω), \end{cases} where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ $(N\ge 2)$, ${\Delta }_{p}v:=\mathrm{div}(|\nabla v{|}^{p-2}\nabla v)$ is the $p$-Laplacian operator, for $1<p<N$, $A>0$, $r>1$, $0\le \theta <p-1$ and $f$ belongs to a suitable Lebesgue space. In particular, we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions.

15 pages

Cited by in corpus (3)

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