nonlinear elliptic equation
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2021 ◽  
Vol 71 (6) ◽  
pp. 1427-1440
Author(s):  
Houssem Eddine Kadem ◽  
Saida Bendaas

Abstract The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation ε 2△u = f(u). We describe the asymptotic behavior as ε tends to zero of the solutions on a spherical crown C of RN , (N ≥ 2) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics.Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.



2021 ◽  
Vol 55 (2) ◽  
pp. 201-205
Author(s):  
S. Aliev ◽  
T. Gajiev ◽  
Ya. Rustamov ◽  
T. Maharramova

We establish apriori estimate for the solutions of a degenerate non-divergence nonlinear elliptic equation. For this goal we study forcing the system by a drift.





2021 ◽  
Vol 39 (2) ◽  
pp. 39-61
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul ◽  
Badr Lahmi

The main aim of this paper is to prove, by using the topological degree methods, the existence of solutions for nonlinear elliptic equation Au = f where Au  is partial dierential operators of general divergence form.





2020 ◽  
Vol 6 (2) ◽  
pp. 231-242
Author(s):  
Adil Abbassi ◽  
Chakir Allalou ◽  
Abderrazak Kassidi

AbstractIn this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation- div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),where Ω is a bounded smooth domain of 𝕉N.



2020 ◽  
Vol 17 (6) ◽  
Author(s):  
Asadollah Aghajani ◽  
Fatemeh Mottaghi ◽  
Vicenţiu D. Rădulescu

AbstractWe consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.



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