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.
An elliptic problem with arbitrarily small positive solutions
OMARI, PIERPAOLO;
2000-01-01
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.File in questo prodotto:
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