One solution for nonlocal fourth order equations

A critical point result for differentiable functionals is exploited in order to prove that a suitable class of fourth-order boundary value problem of Kirchhoff-type possesses at least one weak solution under an asymptotical behavior of the nonlinear datum at zero. Some examples to illustrate the results are given.

Some new oscillation criteria of fourth-order quasi-linear differential equations with neutral term

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems.

ON COUPLED SYSTEMS OF LIDSTONE-TYPE BOUNDARY VALUE PROBLEMS

This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.

On the existence of solutions of one-dimensional fourth-order equations

UDC 517.9 Using variational methods and critical point theorems, we prove the existence of nontrivial solutions for one-dimensional fourth-order equations. Multiplicity results are also pointed out.

On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity

We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.