Existence of solution for biharmonic systems with indefinite weights

Abstract

Abstract. In this article we deal with the existence questions to the nonlinear biharmonic systems. Using theory of monotone operators, we show the existence of a unique weak solution to the weighted biharmonic systems. We also show the existence of a positive solution to weighted biharmonic systems in the unit ball in R n , using Leray Schauder fixed point theorem. In this study we allow sign-changing weights.Mathematics subject classification (2010): 35J08, 35J58, 35J60, 35J75, 35J91.

Existence of Positive Solution for a Nonlinear Weighted Bi-Harmonic System of Elliptic Partial Differential Equations via Fixed-Point Argument

“…The established results provide an easy and straightforward technique to cheek the existence and non-existence of solution to nonlinear weighted bi-harmonic system of elliptic PDEs given by (1.1). Furthermore, the results of this research extend the corresponding results of Soltani and Yazidi [28] and Dwivedi [13].…”

“…The nonlinear partial differential equations (PDEs for short) have proved to be valuable tools for the modeling of many physical, chemical and biological phenomena, see for instance [27,32,34] and references therein. In the last few decays, there has been a noticeable interest on the study of existence of solution to nonlinear elliptic systems, especially when the nonlinear term appears as a source in the equation with the Dirichlet's or Neumann's boundary conditions, see for instance [13,28,32] and references therein. Nonlinear systems are divided into two broad classes, first one is with a variational structure, namely Hamiltonian or gradients systems, see for instance [1,16,18] and second one is the class of non-variational problems, which can be maintained through the topological methods (fixed-point arguments), see for instance [2,4,9,10].…”

“…After Dalmasso [14], several authors studied the BVP (1.3) using different techniques and theorems for different values of ( ) ( ) , , 1, a x b x  = see for instance Chen [12] and their references. In 2014 Dwivedi [13] studied the existence of positive solutions to BVP (1.1) by using Leray-Schauder fixed-point theorem. Recently, Soltani and Yazidi [4] established the existence and non-existence of positive solutions to the following system of bi-Laplacian equations…”