Right-indefinite half-linear Sturm–Liouville problems

We study the regular half-linear Sturm–Liouville equation −(pϕr(y′))′+qϕr(y)=λwϕr(y)on J=(a,b), where ϕr(u)=|u|r−1uϕr(u)=|u|r−1u, r>0r>0, p−1r,q,w∈L(a,b), and p>0p>0 a.e. on JJ. Let N(λ)N(λ) denote the number of zeros in JJ of a nontrivial solution of the equation. Asymptotic formulas are found for N(λ)N(λ) when w≥0w≥0 a.e. and ww changes sign, respectively. As a consequence, the existence and asymptotics of real eigenvalues are established for the half-linear Sturm–Liouville problem consisting of the above equation and a separated boundary condition when ww changes sign. Our results cover the work of Atkinson and Mingarelli on second-order linear equations as a special case. The generalized Prüfer transformation plays a key role in the proofs.