Advances in Differential Equations

Multiplicity of positive radial solutions of a quasilinear elliptic problem in a ball

Franic Ikechukwu Njoku, Pierpaolo Omari, and Fabio Zanolin

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We prove the existence of infinitely many positive, radially symmetric and decreasing solutions of the quasilinear elliptic problem $$ \begin{cases} -{\hbox{div} (a(|\nabla u|^2) \, \nabla u )} = f(|\hbox{x}|,u), & \hbox{ in }\, B_R,\\ \quad u = 0, & \hbox{ on }\, \partial B_R, \end{cases} $$ where $B_R$ is a ball in ${\Bbb R}^N$. The nonlinear differential operator is rather general and includes, as a particular case, the $m$--Laplacian, with $m>1$. The assumptions on $f$ are placed only at $+\infty$ and require a sort of oscillatory behaviour of the primitive $\int_0^s f(|\hbox{x}|,\xi)\, d\xi$. The proof is based on time--mapping estimates, the upper and lower solutions method and a maximum--like principle. The results are new even in the case of the Laplacian $\Delta$. A counterexample is also constructed in order to show that the assumptions cannot be relaxed.

Article information

Adv. Differential Equations Volume 5, Number 10-12 (2000), 1545-1570.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35J65: Nonlinear boundary value problems for linear elliptic equations


Njoku, Franic Ikechukwu; Omari, Pierpaolo; Zanolin, Fabio. Multiplicity of positive radial solutions of a quasilinear elliptic problem in a ball. Adv. Differential Equations 5 (2000), no. 10-12, 1545--1570.

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