Existence of non-trivial solutions for systems of n fourth order partial differential equations

Abstract

ABSTRACT. In this paper, employing a very recent local minimum theorem for differentiable functionals due to Bonanno, the existence of at least one nontrivial solution for a class of systems of n fourth order partial differential equations coupled with Navier boundary conditions is established.

Existence of one weak solution for a Steklov problem involving the weighted $p(\cdot)$-Laplacian

“…Electrorheological fluids also have functions in robotics and space technology. Moreover, the p(x)-Laplacian operator was studied extensively; see [10,13,15,16,19,23,25] and the references therein.…”

One solution for nonlocal fourth order equations

“…For classical results obtained on elastic beam equations we refer to [6,12,14,15,38,44], in particular [44] is one of the pioneering works on extensible beams), while [12] settles the existence and multiplicity question for (P f ) in the physical situation p = 2, ρ = 0 and M of the form M (s) = as + b. Recently, the existence of solutions to fourth-order boundary value problems have been studied in many papers and we refer the reader to the papers [8,9,20,22,26,28,29,30,31,36] and the references therein. For example, Candito and Livrea in [9] by using critical point theory, established the existence of infinitely many weak solutions for a class of elliptic Navier boundary value problems depending on two parameters and involving the p-biharmonic operator.…”

Existence results for a class of second order differential equations with nonlinear derivative dependence

“…For existence results for multiple solutions for second order differential equations with nonlinear derivative dependence, we refer to [13,14,15,16] and the references therein. For more details on Theorem 2.1, we refer the readers to [17,18], where the theorem has already been applied to nonlinear secondorder differential equations problems. We also refer to [19,20,21] in which, Theorems 2.1 and 2.2 along with the mountain pass theorem have been successfully employed to ensure multiple solutions for some boundary value problems.…”