MR Giovanni ANELLO Giovanni ANELLO Funkcialaj Ekvacioj. Serio Internacia 59 2016 113--122 http://fe.math.kobe-u.ac.jp/FE/FullPapers/59-1/59_113.pdf http://www.ams.org/mathscinet-getitem?mr=MR Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with smooth boundary. Let $f:[0,+\infty[\rightarrow[0,+\infty[$, with $f(0)=0$, be a continuous function such that, for some $a>0$, the function $\xi\i ]0,+\infty[\rightarrow\xi^{-2}\cdot\int_0^\xi f(t)dt$ is non increasing in $]0,a[$. Finally, let $\alpha :\overline{\Omega}\rightarrow[0,+\infty[$ be a continuous function with $\alpha(x)>0$, for all $x\in \Omega$. We establish a necessary and sufficient condition for the existence of solutions to the following problem $-\Delta u=\lambda\alpha(x)f(u)$ in $\Omega$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\lambda$ is a positive parameter. Our result extends to higher dimension a similar characterization very recently established by Ricceri in the one dimensional case. Elliptic equation, Boundary value problem, Positive solution, Variational method. 35J20, 35J25. 59-113 2016 Giovanni ANELLO Giovanni ANELLO

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