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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.

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
Full Text: DOI
[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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