Existence of Solutions with a Given Number of Zeros to a Higher-Order Regular Nonlinear Emden–Fowler Equation

Abstract

We consider the nonlinear Emden–Fowler equation

\({y^{(n)}} + p(t,y,y\prime, \ldots ,{y^{(n - 1)}})|y{|^k}{\rm{sgn }}y = 0,\)

, where n ∈ ℕ, n ≥ 2, k ∈ ℝ, k > 1, and the function p(t, ξ₁,…, ξ_n) is jointly continuous in all the variables, satisfies the Lipschitz condition with respect to the variables ξ₁,…, ξ_n, and obeys the inequalities m ≤ p(t, ξ₁,…, ξ_n) ≤ M with some positive constants M and m. For this equation, we prove the existence of solutions that are defined on an arbitrary given interval or half-interval and have a prescribed number of zeros.