### Multiplicity results in a ball for $p$-Laplace equation with positive nonlinearity

#### Abstract

We consider the equation $-\Delta_{p}u=u^{\alpha}+u^{q}$ where $0\le q <p-1 <\alpha\le p^{*}-1$ in the ball $B_{R}(0)\subset \mathbb R^{N}, N\ge 2.$ Here, $p^{*}=Np/(N-p)$. We show the existence of at least two positive solutions to the above equation for small enough balls when $\alpha=p^{*}-1$ and $q>0.$ Further if $p\in (1,2)$ and $\alpha\le p^{*}-1$, we show the existence of exactly two positive solutions for small enough balls when $q>0$, and at most two solutions when $q=0$. This we do by the asymptotic analysis of the corresponding Emden-Fowler equation.

#### Article information

Source
Adv. Differential Equations, Volume 7, Number 7 (2002), 877-896.

Dates
First available in Project Euclid: 27 December 2012

Prashanth, S.; Sreenadh, K. Multiplicity results in a ball for $p$-Laplace equation with positive nonlinearity. Adv. Differential Equations 7 (2002), no. 7, 877--896. https://projecteuclid.org/euclid.ade/1356651709