scholarly journals Existence, Uniqueness, and Regularity Results for Elliptic Equations with Drift Terms in Critical Weak Spaces

2020 ◽  
Vol 52 (2) ◽  
pp. 1146-1191
Author(s):  
Hyunseok Kim ◽  
Tai-Peng Tsai
Keyword(s):  
Elliptic Equations ◽  
Regularity Results ◽  
Drift Terms

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Aziz Bouhlal ◽  
Abderrahmane El Hachimi ◽  
Jaouad Igbida ◽  
El Mostafa Sadek ◽  
Hamad Talibi Alaoui
Keyword(s):  
Elliptic Problem ◽  
Lebesgue Space ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Regularity Of Solutions ◽  
Natural Growth ◽  
Existence And Regularity Results ◽  
Regularity Results

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f  in  Ω,  u=0  on  ∂Ω,}, where 0<q<1, γ>0 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation.

scholarly journals Regularity results for singular elliptic problems

2006 ◽  
Vol 4 (3) ◽  
pp. 243-259 ◽  
Author(s):  
Loredana Caso
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Elliptic Equations ◽  
Global Regularity ◽  
Elliptic Problems ◽  
The Other ◽  
Weighted Spaces ◽  
Regularity Results

Some local and global regularity results for solutions of linear elliptic equations in weighted spaces are proved. Here the leading coefficients are VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function.

scholarly journals Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction

2020 ◽  
Vol 17 (6) ◽  
Author(s):  
Asadollah Aghajani ◽  
Fatemeh Mottaghi ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Extremal Solution ◽  
General Reaction ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Regularity Results ◽  
Increasing Function

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.

scholarly journals Existence of Renormalized Solutions for Some Anisotropic Quasilinear Elliptic Equations

2020 ◽  
Vol 44 (4) ◽  
pp. 617-637
Author(s):  
T. AHMEDATT ◽  
A. AHMED ◽  
H. HJIAJ ◽  
A. TOUZANI
Keyword(s):  
Dirichlet Problem ◽  
Growth Condition ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Renormalized Solutions ◽  
Carathéodory Function ◽  
Regularity Results ◽  
Quasilinear Elliptic

In this paper, we consider a class of anisotropic quasilinear elliptic equations of the type ( | ∑N { − ∂ia (x, u, ∇u ) + |u|s(x )− 1u = f (x,u ), in Ω, i |( i=1 u = 0 on ∂ Ω, where f(x,s) is a Carathéodory function which satisfies some growth condition. We prove the existence of renormalized solutions for our Dirichlet problem, and some regularity results are concluded.

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