A Neumann problem at resonance with the nonlinearity restricted in one direction
The Neumann boundary value problem \[ x''+f(t,x)+h(t,x)=0, \qquad x'(0)=x'(1)=0, \tag{1} \] is considered. Here, \(f,h: (0,1)\times \mathbb{R}\to \mathbb{R}\) are \(L^1\)-Carathéodory functions and moreover \(f(t,x)\geq 0\) for \(x\in \mathbb{R}\) with \(|x|\geq r>0\) a.e. in \(t\in (0,1)\) and \(|h(t,x)|\leq\overline{h}(t)\) for \(x\in \mathbb{R}\) a.e. in \(t\in (0,1)\) for some \(\overline{h}\in L^1(0,1)\). As the authors mention, this problem was initially studied by \textit{G. D. Figueiredo} and \textit{B. Ruf} [J. Reine Angew. Math. 421, 1-22 (1991; Zbl 0732.34024)] under the conditions \[ \lim_{x\to+\infty} f(t,x)/x=\lambda\in (0, \pi^2/4) ,\qquad \lim_{x\to-\infty} f(t,x)/x=+\infty, \] and some other supplementary conditions. The problem was latter studied by \textit{S. Villegas} [J. Differ. Equations 145, 145-155 (1998; Zbl 0910.34035)] and it was found that \[ \lim_{s\to-\infty} (f(x,s)/s) < {\pi^2 \over 4}, \quad\text{and}\quad \lim_{s\to- \infty} f(x,s)<-\int^1_0 h(x) dx <\lim_{s\to \infty} f(x,s) \] uniformly in \(x\in [0,1]\) is sufficient for (1) to have a solution. A question was proposed whether the number \(\pi^2 \over 4\) in the above-stated sufficient condition can be increased. The author gives a negative answer by means of a counterexample. However, the sufficient condition is improved. Some related results concerning periodic boundary conditions are discussed, too.
scientific article; zbMATH DE number 1836612
scientific article