Glasnik Matematicki, Vol. 48, No. 2 (2013), 391402.
QUASILINEAR ELLIPTIC EQUATIONS WITH POSITIVE EXPONENT ON THE GRADIENT
Jadranka Kraljević and Darko Žubrinić
Faculty of Economics,
University of Zagreb,
Kennedyev trg 6, 10000 Zagreb,
Croatia
email: jkraljevic@efzg.hr
Department of Applied Mathematics,
Faculty of Electrical Engineering and Computing,
University of Zagreb,
Unska 3, 10000 Zagreb,
Croatia
email: darko.zubrinic@fer.hr
Abstract. We study the existence and nonexistence of positive, spherically symmetric solutions
of a quasilinear elliptic equation (1.1) involving pLaplace operator, with an arbitrary positive
growth rate e_{0} on the gradient on the righthand side. We show
that e_{0}=p1 is the critical exponent: for e_{0}< p1 there exists a
strong solution for any choice of the coefficients, which is a known result, while for
e_{0}>p1 we have existencenonexistence splitting of the
coefficients and . The elliptic problem is studied by relating it to the corresponding singular ODE of the first order. We give sufficient conditions for a strong radial solution to be the weak solution. We also examined when ωsolutions of (1.1), defined in Definition 2.3, are weak solutions. We found conditions under which strong solutions are weak solutions in the critical case of e_{0}=p1.
2010 Mathematics Subject Classification.
35J92, 35D35, 35D30, 35B33.
Key words and phrases. Quasilinear elliptic,
positive strong solution, ωsolution, critical exponent, existence, nonexistence, weak solution.
Full text (PDF) (access from subscribing institutions only)
DOI: 10.3336/gm.48.2.11
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