Asymptotic oscillations of continua of positive solutions of a semilinear Sturm-Liouville problem
The semilinear Sturm-Liouville boundary value problem \(-u''=\lambda u+f(u)\) in \((0,\pi)\), \(u(0)=u(\pi)=0\), is considered, where \(f:[0,\infty)\to \mathbb R\) is Lipschitz continuous. A solution \((\lambda ,u)\in \mathbb R\times C^2[0,2\pi]\) satisfying \(u>0\) on \((0,2\pi)\) is called a positive solution. A closed connected set of positive solutions \(\mathcal C\) is said to oscillate over the interval \(I=[\lambda ^-,\lambda ^+]\) if there are two sequences of positive numbers \(\{\xi_n^\pm\}\) such that \(\xi_n^\pm\to \infty \) as \(n\to \infty \), any \((\lambda ,u)\in \mathcal C\) with \(\max\{|u(x)|:x\in [0,\pi]\}=\xi_n^\pm\) must satisfy \(\lambda ^-<\lambda <\lambda ^+\), and such solutions do exist for sufficiently large \(n\); if \(\lambda ^-=\lambda ^+=\lambda \), \(\mathcal C\) is said to oscillate over the point \(\lambda \). Putting \(f=0\), \(\mu_0=1\) is the principal eigenvalue. The general assumption is that \(f(s)=O(s^2)\). Redefining \(f(s)\) and \(\lambda \) to be \(f(s)-\gamma s\) and \(\lambda +\gamma \) for a suitable \(\gamma \), it follows that \(\beta =\lim_{s\to\infty }G(s)/s^2\). The main results are: If \(\beta =0\), then \(\mathcal C\) can oscillate at most over the point \(\mu_0\), and under an additional condition, \(\mathcal C\) oscillates over the point \(\mu_0\). If \(\beta >0\), then there exists a closed interval \(I\), containing \(\mu_0\) in its interior, such that \(\mathcal C\) oscillates over \(I\).
scientific article; zbMATH DE number 1563083
scientific article