Weak Solutions of Quasilinear Biharmonic Problems With Positive, Increasing and Convex Nonlinearities

Abstract

We study the existence of positive weak solutions to a fourth-order semilinear elliptic equation with Navier boundary conditions and a positive, increasing and convex source term. We also prove the uniqueness of extremal solutions. In particular, we generalize results of Mironescu and Rădulescu for the bi-Laplacian operator.

Elliptic Weighted Problem with Indefinite Asymptotically Linear Nonlinearity

“…When f is superlinear and ℓ = (0, +), the problem (1.2) was studied in (Brezis et al, 1996;Martel, 1997) and the references therein and it is generated to the p-Laplace operator in (Filippakis and Papageorgiou, 2006;Sanchón, 2007). The same problem with Bi-Laplace operator has been studied in (Arioli et al, 2005;Abid et al, 2008;Saanouni and Trabelsi, 2016b;Wei, 1996).…”

Schrödinger equation with asymptotically linear nonlinearities

“…The proof of the Theorem 1.1 will be given in many steps. After introducing the Energy I by the formula (6), we have interest to use the norm u σ given by (3). So, the first elementary result is the following.…”

A class of singular perturbation solutions to semilinear equations of fourth order

“…Obviously, the solution Y 00 to the problem (8)- (9) is the solution Y 0 (x) to the reduced problem (4)- (3). And from the problem (10)-(11), we can obtain Y jk (j, k = 0, 1, 2, · · · ) successively.…”

“…During the last decade, many approximate methods have been developed and refined, including the methods of averaging, the boundary layer methods, the methods of matched asymptotic expansion, and the multiple scale methods. Recently, many scholars have done a great deal of work [1][2][3][4][5] . Using the method of differential inequalities and other methods, Mo et al considered a class of nonlinear singularly perturbed boundary value problems for the ordinary differential equations [6] , the reaction diffusion equations [7][8] , a class of activator inhibitor systems [9] , the ecological environment [10] , the shock wave [11] , the soliton [12][13] , the laser pulse [14] , the ocean science [15] , and the problems of atmospheric physics [16] .…”