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An elliptic problem with arbitrarily small positive solutions. (English) Zbl 0959.35059
We show that for each $$\lambda>0$$, the problem $-\Delta_pu=\lambda f(u)\quad\text{in } \Omega$
$u=0\quad\text{on } \partial\Omega$ has a sequence of positive solutions $$(u_n)_N$$ such that $$\max_\Omega u_n$$ decreases to zero. We assume that $$\liminf_{s\to 0^+}\frac{F(s)}{s^p}=0$$ and that $$\limsup_{s\to 0^+}\frac{F(s)}{s^p}=+\infty$$, where $$F^\prime=f$$. We stress that no condition on the sign of $$f$$ is imposed.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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