Glasnik Matematicki, Vol. 48, No. 2 (2013), 391-402.

QUASILINEAR ELLIPTIC EQUATIONS WITH POSITIVE EXPONENT ON THE GRADIENT

Jadranka Kraljević and Darko Žubrinić

Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
e-mail: darko.zubrinic@fer.hr

Abstract.   We study the existence and nonexistence of positive, spherically symmetric solutions of a quasilinear elliptic equation (1.1) involving p-Laplace operator, with an arbitrary positive growth rate e₀ on the gradient on the right-hand side. We show that e₀=p-1 is the critical exponent: for e₀< p-1 there exists a strong solution for any choice of the coefficients, which is a known result, while for e₀>p-1 we have existence-nonexistence 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 ω-solutions 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 e₀=p-1.

2010 Mathematics Subject Classification.   35J92, 35D35, 35D30, 35B33.

Key words and phrases.   Quasilinear elliptic, positive strong solution, ω-solution, critical exponent, existence, nonexistence, weak solution.

DOI: 10.3336/gm.48.2.11

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