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On regions of existence and nonexistence of solutions for a system of $$p-q$$-Laplacians. (English) Zbl 1157.35350
Summary: We give a new region of existence of solutions to the superhomogeneous Dirichlet problem \begin{aligned} -\Delta_{p}u&=v^{\delta},\quad v>0 \,{\text{in}}\, B,\\-\Delta_{q}v&=u^{\mu},\quad u>0\, {\text{in}}\, B,\\u=v&=0 \quad {\text{on}}\, \partial B \end{aligned} where $$B$$ is the ball of radius $$R>0$$ centered at the origin in $$\mathbb{R}^N$$. Here $$\delta, \mu >0$$ and $$\Delta_{m} u = {\text{div}}(|\nabla u|^{m-2} \nabla u)$$ is the $$m$$-Laplacian operator for $$m>1$$.

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35J60 Nonlinear elliptic equations
##### Keywords:
$$m$$-Laplacian; energy identities