Boundary value problems on the half-line for a generalised Painlevé II equation
Consider the differential equations \[ p''=a|p|^\gamma p+b |p|^\delta pp'\pm xp+c \tag{\((1)^\pm\)} \] with constants \(a>0\), \(\gamma\geq 0\), \(\delta\geq 0\), \(b\in\mathbb{R}\) \(c\in \mathbb{R}_*\) for \(0\leq x<\infty\). The authors prove the existence of at least one solution of \((1)^+\) satisfying the boundary conditions \[ p(0)=p_0,\;\lim_{x\to+ \infty}p(x)=0, \] and at least one solution of \((1)^-\) satisfying \[ p(0)=p_0,\;\lim_{x\to+ \infty}\frac{p(x)}{x}=0 \] in case \(\gamma>\max\{1,\delta\}\). The proofs are based on lower and upper solutions technique.
scientific article; zbMATH DE number 5563018