Original scientific paper

https://doi.org/10.3336/gm.48.2.11

Quasilinear elliptic equations with positive exponent on the gradient

Jadranka Kraljević ; Faculty of Economics, University of Zagreb, Kennedyev trg 6, 10000 Zagreb, Croatia
Darko Žubrinić ; Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia

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.

Keywords

Quasilinear elliptic; positive strong solution; ω-solution; critical exponent; existence; nonexistence; weak solution

Hrčak ID:

112215

URI

https://hrcak.srce.hr/112215

Publication date:

16.12.2013.

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