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

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
e-mail: jkraljevic@efzg.hr

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.


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DOI: 10.3336/gm.48.2.11


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