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Positive radial solutions for a quasilinear system. (English) Zbl 1100.34018
The authors use the Schauder fixed-point theorem to show the existence of positive radial solutions for the general quasilinear system
\[ -\text{div } \big ( | Du| ^{p-2}Du \big ) = a(x) f(u, v), x \in B_1, \]
\[ -\text{div } \big ( | Dv| ^{q-2}Dv \big ) = b(x) g(u, v), x \in B_1, \]
\[ u(x) = v(x) = 0, x \in \partial B_1, \]
where \(p,q > 1\) and \(B_1\) is the unit ball in \(\mathbb{R}^n\), \(B_1 = \{ x \in {\mathbb{R}}^n : | x| < 1 \}\). Their main assumptions on the functions \(a\) and \(b\) are that these functions are integrable over \([0, 1]\) and that their integrals are positive. There are no assumptions made on the functions \(f\) and \(g\) at either zero or infinity. This is a well written and easy to read paper that will appeal to anyone studying quasilinear equations or systems of quasilinear equations.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
35J20 Variational methods for second-order elliptic equations
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References:
[1] DOI: 10.1080/03605309308821005 · Zbl 0802.35044
[2] DOI: 10.1080/03605309208820912 · Zbl 0813.35020
[3] DOI: 10.1088/0951-7715/9/4/007 · Zbl 0896.35048
[4] DOI: 10.1016/S0362-546X(97)00506-3 · Zbl 0932.35097
[5] DOI: 10.1007/BF01221125 · Zbl 0425.35020
[6] DOI: 10.1080/00036819408840222 · Zbl 0841.35008
[7] DOI: 10.1016/S0362-546X(99)00269-2 · Zbl 0992.34012
[8] DOI: 10.1016/S0022-0396(02)00094-3 · Zbl 1016.35020
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