Oscillation constants for Euler type differential equations involving the \(p(t)\)-Laplacian
Oscillation problems are studied for the half-linear differential equation \[ (|x'|^{p(t)-2}x')'+\frac{\lambda}{t^{p(t)}}|x|^{p(t)-2}x=0 \] with \(p(t) > 1\). The main results are under the assumption that \(p(t)\) is non-decreasing and there exist \(p_\ast> 1\) and \(M > 0\) such that \(t^{|p_\ast-p(t)|} < M\) (i.e., \(\lim_{t\to\infty} p(t)=p_\ast\)). Using the notation \(\gamma(p)=((p-1)/p)^p\), the authors show that (a) If \(0<M<e\) and \(\lambda>\gamma(p_\ast)\), then all solutions of the equation are oscillatory; (b) If \(\lambda<\gamma(p_\ast)\), then the equation has a nonoscillatory solution. Further results are obtained for the case that \(\lim_{t\to\infty} p(t)=\infty\).
scientific article; zbMATH DE number 6991328
scientific article