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On Regions of Existence and Nonexistence of solutions for a System of $p$-$q$-Laplacians

Philippe Clement·Marta Garcia-Huidobro·Ignacio Guerra·Raul Manasevich
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Abstract

We give a new region of existence of solutions to the superhomogeneous Dirichlet problem $$ \quad \begin{array}{l} -Δ_{p} u= v^δ\quad v>0\quad {in}\quad B,\cr -Δ_{q} v = u^μ\quad u>0\quad {in}\quad B, \cr u=v=0 \quad {on}\quad \partial B, \end{array}\leqno{(S_R)} $$ where $B$ is the ball of radius $R>0$ centered at the origin in $\RR^N.$ Here $δ, μ>0$ and $ Δ_{m} u={\rm div}(|\nabla u|^{m-2}\nabla u) $ is the $m-$Laplacian operator for $m>1$.

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Philippe Clement, Marta Garcia-Huidobro, Ignacio Guerra, Raul Manasevich. 2006-05-11. On Regions of Existence and Nonexistence of solutions for a System of $p$-$q$-Laplacians. https://arxiv.org/abs/math/0605284

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