Electronic Journal of Differential Equations,
Conference 05 (2000), pp. 301-308.
 
Title: An elliptic problem with arbitrarily small positive solutions
 
Authors:  Pierpaolo Omari (Univ., Piazzale Europa 1, Trieste, Italy)
          Fabio Zanolin (Univ., Via delle Scienze 206, Trieste, Italy)
  
Abstract:
 We show that for each $\lambda > 0$, the problem
 $-\Delta_p u = \lambda f(u)$ in $Omega$,
 $u = 0$ on $\partial \Omega$
 has a sequence of positive solutions $(u_n)_n$ 
 with $\max_{\bar\Omega} u_n$ decreasing to zero.
 We assume that $\displaystyle{\liminf_{s\to0^+}\frac{F(s)}{s^p} = 0}$ 
 and that 
 $\displaystyle{\limsup_{s\to 0^+}\frac{F(s)}{s^p} = +\infty}$, 
 where $F'=f$. We stress that no condition on the sign of $f$ is imposed.

Published October 24, 2000.
Math Subject Classifications: 35J65, 34B15, 34C25, 47H15.
Key Words: Quasilinear elliptic equation; positive solution; 
upper and lower solutions; time-mapping estimates.