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Large solutions for an elliptic system of quasilinear equations. (English) Zbl 1169.35021
Summary: We consider the quasilinear elliptic system Δ p u=u a v b , Δ p v=u c v e in a smooth bounded domain Ω N , with the boundary conditions u=v=+ on Ω. The operator Δ p stands for the p-Laplacian defined by Δ p u=div(|u| p-2 u), p>1, and the exponents verify a,e>p-1, b,c>0 and (a-p+1)(e-p+1)bc. We analyze positive solutions in both components, providing necessary and sufficient conditions for existence. We also prove uniqueness of positive solutions in the case (a-p+1)(e-p+1)>bc and obtain the exact blow-up rate near the boundary of the solution. In the case (a-p+1)(e-p+1)=bc, infinitely many positive solutions are constructed.
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35J60Nonlinear elliptic equations
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