Advances in Differential Equations

Singular solutions to a quasilinear {ODE}

Francesca Dalbono and M. García-Huidobro

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In this paper, we prove the existence of infinitely many radial solutions having a singular behaviour at the origin for a superlinear problem of the form $-\Delta_pu=|u|^{\delta-1}u$ in $B(0,1)\setminus\{0\}\subset\mathbb R^N$,\, $u=0$ for $|x|=1$, where $N>p>1$ and $\delta>p-1$. Solutions are characterized by their nodal properties. The case $\delta+1 <\frac{Np}{N-p}$ is treated. The study of the singularity is based on some energy considerations and takes into account the classification of the behaviour of the possible solutions available in the literature. By following a shooting approach, we are able to deduce the main multiplicity result from some estimates on the rotation numbers associated to the solutions.

Article information

Adv. Differential Equations Volume 10, Number 7 (2005), 747-765.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B16: Singular nonlinear boundary value problems
Secondary: 34B15: Nonlinear boundary value problems 35J60: Nonlinear elliptic equations


Dalbono, Francesca; García-Huidobro, M. Singular solutions to a quasilinear {ODE}. Adv. Differential Equations 10 (2005), no. 7, 747--765.

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