A nonoscillation theorem for superlinear Emden–Fowler equations with near-critical coefficients

We are interested in the oscillatory behavior of solutions of the Emden–Fowler equation y″+a(x)|y|γ−1y=0, γ>1, where a(x) is a positive continuous function on (0,∞). In the special case when the coefficient a(x) is a power of x, i.e. a(x)=xα for some constant α, the value α∗=−(γ+3)/2 plays a critical role: The equation has both oscillatory and nonoscillatory solutions if α>α∗, while all solutions are nonoscillatory if α<α∗. When a(x) is close to the critical exponent, one of the known results is that if a(x)=x−(γ+3)/2log−σ(x), where σ>0, then all solutions are nonoscillatory. In this paper, this result is further extended to include a class of coefficients in which the above condition with log(x) can be replaced by loglog(x), or logloglog(x) and so on.