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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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