Nonlinear Differential Equations,
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 301-308.

An elliptic problem with arbitrarily small positive solutions

Pierpaolo Omari & Fabio Zanolin

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 13, 2000.
Math Subject Classifications: 35J65, 34B15, 34C25, 47H15.
Key Words: Quasilinear elliptic equation, positive solution, upper and lower solutions, time-mapping estimates.

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Pierpaolo Omari
Dipartimento di Scienze Matematiche
Universita, Piazzale Europa 1
I-34127 Trieste, Italy

Fabio Zanolin
Dipartimento di Matematica e Informatica
Universita, Via delle Scienze 206
I--33100 Udine, Italy

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