An elliptic problem with arbitrarily small positive solutions

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 $max_{Omega} u_n$ decreasing to zero. We assume that $displaystyle{liminf_{so0^+}frac{F(s)}{s^p} = 0}$ and that $displaystyle{limsup_{so 0^+}frac{F(s)}{s^p} = +infty}$, where $F'=f$. We stress that no condition on the sign of $f$ is imposed.

Keywords

• Quasilinear elliptic equation
• positive solution
• upper and lower solutions
• time-mapping estimates.