Right-indefinite half-linear Sturm–Liouville problems

We study the regular half-linear Sturm–Liouville equation where r(u)=ur−1u, r>0, , and p>0 a.e. on J. Let N(λ) denote the number of zeros in J of a nontrivial solution of the equation. Asymptotic formulas are found for N(λ) when w≥0 a.e. and w 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 w 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.