Periodic solutions of singular second order equations at resonance
The paper deals with the \(2\pi\)-periodic problem for the equation \[ x''+\lambda x+h(x)=p(t), \] where \(\lambda\) is an eigenvalue of the corresponding Dirichlet problem (i.e., \(\lambda=n^2/4\) for some \(n\in\mathbb{N}\)), \(p:\mathbb{R}\to\mathbb{R}\) is continuous and \(2\pi\)-periodic, \(h:(0,\infty)\to\mathbb{R}\) is locally Lipschitz continuous and has a strong repulsive singularity in the origin (i.e., \(h(0+)=-\infty\) and \(\lim_{x\to 0+}H(x)=+\infty,\) where \(H(x)=\int_1^xh(s)ds\)). The authors show that if, moreover, \(\lim_{x\to\infty}h(x)=\infty\), \(\lim_{x\to\infty}h(x)/x=0\) and \(\lim_{x\to 0+} H(x)/h(x)=0\), then the given equation has at least one positive \(2\pi\)-periodic solution. The main tools are a thorough qualitative analysis and topological degree theory. In addition, an analogous result is obtained for a related radially symmetric \(k\)-dimensional system \[ x''+(\lambda\,|x|+h(|x|)-p(t))\,\frac{x}{|x|}=0. \]
repulsive singularity, strong singularity
scientific article; zbMATH DE number 6497395
scientific article