Explicit criteria for the qualitative properties of differential equations with p-Laplacian-like operator

The aim of this work is to study qualitative properties of solutions for a fourth-order neutral nonlinear differential equation, driven by a p-Laplace differential operator. Some oscillation criteria for the equation under study have been obtained by comparison theory. The obtained results improve the well-known oscillation results present in the literature. Some examples are provided to show the applicability of the obtained results.


Introduction
Differential equations of fourth-order appear in models concerning biological, physical, and chemical phenomena, optimization, mathematics of networks, dynamical systems, see [1].
We study the oscillatory behavior of the fourth-order neutral nonlinear differential equation of the form ⎧ ⎨ ⎩ (r(x)|w (x)| p 1 -2 w (x)) + j i=1 q i (x)|u (ϑ i (x))| p 2 -2 u (ϑ i (x)) = 0, j ≥ 1, where w(x) := u(x) + a(x)u(τ (x)) and the first term means the p-Laplace-type operator (1 < p i < ∞, i = 1, 2). The main results are obtained under the following conditions: The p-Laplace equations have some significant applications in elasticity theory and continuum mechanics, see [2] (power-law fluids), and in general in nonlinear phenomena, see [3] (capillary phenomena). For some results concerning the oscillatory behavior of equations driven by a p-Laplace differential operator, we mention the papers [4][5][6]. In [7], the authors used a classical variational approach based on the critical points theory to prove the existence of at least one nontrivial weak solution of a double-phase Dirichlet problem. Here the differential operator of the problem is the sum of two p-Laplaciantype operators with variable exponents. This fact could provide new ideas for further investigations. The authors characterized the continuous spectrum of double-phase equations (to improve the regularity theory for such a kind of operators and classify solutions).
Nastasi [8] established an existence result of a nontrivial weak solution to (p, q)-Laplacian problem on a noncompact Riemannian manifold. The special setting led the author to develop the Maz'ya's approach, by working with isocapacitary inequalities to characterize the compact embeddings.

Mathematical background-hypotheses
In this section we collect some relevant facts and auxiliary results from the existing literature. Also, we fix the notation.
As we already mentioned in the Introduction, our aim here is to provide complementary results to [28,29,31]. For this purpose we briefly discuss these results.

Definition 2.1 Define sequences of functions {δ
We see by induction that Now, we are ready to introduce the precise hypotheses on the data of (1): (H1) u is an eventually positive solution of (1). where (H3) For some μ ∈ (0, 1), there are positive constants M 1 , M 2 such that and lim inf where and lim sup for some n. and

Main results
Next, we mention some important lemmas: for all sufficiently large x.
The rest of the proof is the same as that for the case (G 2 ). Theorem 3.3 is proved. Proof Proceeding as in the proof of Theorem 3.2, in the case (G 1 ), from (23) we obtain ω(x) ≥ δ 0 (x). By induction we can also see that ω(x) ≥ δ n (x) for x ≥ x 0 , n > 1. Since the sequence {δ n (x)} ∞ n=0 is monotone increasing and bounded above, it converges to δ(x). Using Lebesgue's monotone convergence theorem, we find Since δ n (x) ≤ δ(x), it follows from (32) that Hence, we get This implies which contradicts (9). The proof of the case where (G 2 ) holds is the same as that of (G 1 ). Corollary 3.1 is proved.

Conclusions
Our aim of this article was to study the qualitative behavior of a fourth-order neutral nonlinear differential equation, driven by a p-Laplace differential operator. The obtained oscillation theorems complement the well-known oscillation results present in the literature. In this line of work, one can investigate oscillatory conditions for a fourth-order equation of the type: ⎧ ⎨ ⎩ (r(x)|y (x)| p 1 -2 y (x)) + a(x)f (y (x)) + j i=1 q i (x)|y(σ i (x))| p 2 -2 y(σ i (x)) = 0, which is of interest to the authors, in particular, the case of p 2 > p 1 .