Oscillation Criteria for some Semi-Linear Emden–Fowler ODE

Inspired from earlier works on oscillation criteria for semi-linear elliptic equations, we pinpoint here some straightforward and easy oscillation criteria for Emden–Fowler differential equations. We find out that for α ≥ 0, the equation [ | y ′ | α − 1 y ′ ] ′ + f ( t , y ) = 0 $$\displaystyle{\;[\vert y'\vert ^{\alpha -1}y']' + f(t,y) = 0\;}$$ is oscillatory if for some m, T > 0 and β ∈ [ 1 , α ] ∃ q ∈ C ( [ T , ∞ ) , ( m , ∞ ) ) $$\;\beta \in [1,\;\alpha ]\quad \exists q \in C([T,\;\infty ),\;(m,\;\infty ))\;$$ such that ∀ t > T and ∀ s ∈ ℝ , f ( t , s ) ≥ q ( t ) | s | β − 1 s . $$\displaystyle{\forall t > T\;\text{ and}\;\forall s \in \mathbb{R},\qquad f(t,s) \geq q(t)\vert s\vert ^{\beta -1}s.}$$ The main tools for our investigation are some version of Picone identities Picone identities and comparison methods. We are considering equations of the type ( i ) ϕ ( y ′ ) ′ + Ψ ( t , y , y ′ ) = 0 ( i i ) where ∀ S ∈ ℝ and some α ≥ 0 ϕ ( S ) : = ϕ α ( S ) = | S | α − 1 S ; ( i i i ) Ψ ∈ C ( ℝ 3 , ℝ ) . $$\displaystyle{\left \{\begin{array}{@{}l@{\quad }l@{}} (i)\quad \bigg\{\phi (y')\bigg\}' +\varPsi (t,y,y') = 0 \quad \\ (ii)\quad \mbox{ where $\forall S \in \mathbb{R}$ and some $\alpha \geq 0$}\quad \phi (S):=\phi _{\alpha }(S) = \vert S\vert ^{\alpha -1}S;\quad \\ (iii)\quad \varPsi \in C(\mathbb{R}^{3},\; \mathbb{R}). \quad \\ \quad \end{array} \right.}$$ Usually equations in these contexts have the form a ( t ) ϕ ( y ′ ) ′ + Ψ ( t , y , y ′ ) = 0 $$\displaystyle{\bigg\{a(t)\phi (y')\bigg\}' +\varPsi (t,y,y') = 0}$$ where for some t 0 ≥ 0 , a ∈ C 1 ( [ t 0 , ∞ ) ) $$\;t_{0} \geq 0,\quad \;a \in C^{1}([t_{0},\;\infty ))\;$$ is strictly positive with a′ ≥ 0. Because of these conditions on a, in regard of oscillatory character, that equation is equivalent to ϕ ( y ′ ) ′ + a ′ ( t ) a ( t ) ϕ ( y ′ ) + Ψ ( t , y , y ′ ) a ( t ) = 0 . $$\displaystyle{\bigg\{\phi (y')\bigg\}' + \dfrac{a'(t)} {a(t)}\phi (y') + \dfrac{\varPsi (t,y,y')} {a(t)} = 0.}$$ This is the reason why we take a(t) ≡ 1 in our study and extend the investigation to the equations with damping terms, ϕ(y′), say. We set the following hypotheses: (H): the function Ψ has the form (H1) Ψ(t, u, u′): = f(t, u) where ∀ t ∈ ℝ $$\forall t \in \mathbb{R}\;$$ and u ≠ 0, uf(t, u) > 0; (H2) Ψ ( t , u , u ′ ) : = g ( t , u ′ ) + f ( t , u ) $$\varPsi (t,u,u'):= g(t,u') + f(t,u)\;$$ which f as in (H1) and g ∈ C ( ℝ 2 , ℝ ) $$g \in C(\mathbb{R}^{2},\; \mathbb{R})$$ .