scholarly journals Positive solutions of higher order quasilinear elliptic equations

2002 ◽  
Vol 7 (8) ◽  
pp. 423-452
Author(s):  
Marcelo Montenegro

The higher order quasilinear elliptic equation−Δ(Δp(Δu))=f(x,u)subject to Dirichlet boundary conditions may have unique and regular positive solution. If the domain is a ball, we obtain a priori estimate to the radial solutions via blowup. Extensions to systems and general domains are also presented. The basic ingredients are the maximum principle, Moser iterative scheme, an eigenvalue problem, a priori estimates by rescalings, sub/supersolutions, and Krasnosel'skiĭ fixed point theorem.

2018 ◽  
Vol 7 (4) ◽  
pp. 425-447 ◽  
Author(s):  
Lorenzo D’Ambrosio ◽  
Enzo Mitidieri

AbstractThe paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of {\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case {\Omega=\mathbb{R}^{N}}, we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.


2002 ◽  
Vol 7 (2) ◽  
pp. 207-216
Author(s):  
N. V. Dzenisenko ◽  
A. P. Matus ◽  
P. P. Matus

In order to approximate a multidimensional quasilinear parabolic equation with unlimited nonlinearity the economical vector‐additive scheme is constructed. It is shown that its solution satisfies the maximum principle and, hence, the scheme is monotone. The proof is based on the equivalence of the vector‐additive scheme and the scheme of summarized approximation (locally one‐dimensional scheme). The a priori estimates of the difference solution in the uniform norm are obtained.


2002 ◽  
Vol 130 (10) ◽  
pp. 3043-3050 ◽  
Author(s):  
A. Alexandrou Himonas ◽  
Gerard Misiołek

Author(s):  
Виктор Николаевич Орлов ◽  
Людмила Витальевна Мустафина

В работе приводится доказательство теоремы существования и единственности аналитического решения класса нелинейных дифференциальных уравнений третьего порядка, правая часть которого представлена полиномом шестой степени, в комплексной области. Расширен класс рассматриваемых уравнений за счет новой замены переменных. Получена априорная оценка аналитического приближенного решения. Представлен вариант численного эксперимента оптимизации априорных оценок с помощью апостериорных. The article presents a proof of the theorem of the existence and uniqueness of the analytical solution of the class of nonlinear differential equations of the third order, with a polynomial right-hand side of the sixth degree, in the complex domain. The class of the considered equations has been extended by means of a new change of variables. An a priori estimate of the analytical approximate solution is obtained. A variant of the numerical experiment of optimizing a priori estimates using a posteriori estimates is presented.


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