Solutions for a Class of Singular Elliptic Equation

2013 ◽  
Vol 694-697 ◽  
pp. 2910-2913
Author(s):  
Zong Hu Xiu
Keyword(s):  
Elliptic Equation ◽  
Variational Method ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Singular Elliptic Equation

In this paper, we consider a class of singular elliptic equation on bounded domain. By the variational method, we prove the existence of solutions.

scholarly journals A uniqueness result for a singular elliptic equation with gradient term

2018 ◽  
Vol 148 (5) ◽  
pp. 983-994 ◽  
Author(s):  
José Carmona ◽  
Tommaso Leonori
Keyword(s):  
Elliptic Equation ◽  
Bounded Domain ◽  
Nonlinear Term ◽  
Uniqueness Result ◽  
Gradient Term ◽  
Singular Elliptic Equation ◽  
Uniqueness Results ◽  
Image Position ◽  
Uniqueness Of A Solution

We prove the uniqueness of a solution for a problem whose simplest model iswith k ≥ 1, 0 f ∈ L∞(Ω) and Ω is a bounded domain of ℝN, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

Existence of Solutions for Elliptic Equation with Critical Sobolev Exponent

2013 ◽  
Vol 300-301 ◽  
pp. 1205-1208
Author(s):  
Zong Hu Xiu
Keyword(s):  
Elliptic Equation ◽  
Variational Method ◽  
Unbounded Domain ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Sobolev Exponent

In this paper, we consider a class of elliptic equation on unbounded domain. By the variational method, we prove the existence of solutions.

scholarly journals Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

2020 ◽  
Vol 20 (2) ◽  
pp. 373-384
Author(s):  
Quoc-Hung Nguyen ◽  
Nguyen Cong Phuc
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Linear Range ◽  
Measure Data ◽  
Regularity Conditions ◽  
Compactly Supported ◽  
Singular Data ◽  
Quasilinear Elliptic

AbstractWe characterize the existence of solutions to the quasilinear Riccati-type equation\left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.with a distributional or measure datum σ. Here {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ({p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that {p>1} and {q>p}. For measure data, we assume that they are compactly supported in Ω, {p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range {p-1<q<1}. We also assume more regularity conditions on {\mathcal{A}} and on {\partial\Omega\Omega} in this case.

Existence of Solutions for a Problem of Resonance Using Variational Method

2012 ◽  
Vol 20 (2) ◽  
pp. 161-178
Author(s):  
Antonio Ronaldo G. Garcia ◽  
Adrião D. D. Neto ◽  
Moisés D. Santos
Keyword(s):  
Variational Method ◽  
Existence Of Solutions

Nonlinear Elliptic Equations in Strip-Like Domains

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
Jaeyoung Byeon ◽  
Kazunaga Tanaka
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic Equations ◽  
Nonlinear Elliptic

AbstractWe study the existence of a positive solution of a nonlinear elliptic equationwhere k ≥ 2 and D is a bounded domain domain in R

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

Free boundary solutions to a log-singular elliptic equation

2013 ◽  
Vol 82 (1-2) ◽  
pp. 91-107 ◽  
Author(s):  
Marcelo Montenegro ◽  
Sebastián Lorca
Keyword(s):  
Elliptic Equation ◽  
Free Boundary ◽  
Singular Elliptic Equation ◽  
Boundary Solutions
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