LAYERED SOLUTIONS WITH CONCENTRATION ON LINES IN THREE-DIMENSIONAL DOMAINS

2014 ◽  
Vol 12 (02) ◽  
pp. 161-194 ◽  
Author(s):  
WEIWEI AO ◽  
JUN YANG
Keyword(s):  
Small Parameter ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Singularly Perturbed ◽  
Existence Of A Solution ◽  
Straight Line ◽  
Dimensional Domain ◽  
Exponentially Small

We consider the following singularly perturbed elliptic problem [Formula: see text] where Ω is a bounded domain in ℝ3with smooth boundary, ε is a small parameter, 1 < p < ∞, ν is the outward normal of ∂Ω. We employ techniques already developed in [39] to extend their result to three-dimensional domain. More precisely, let Γ be a straight line intersecting orthogonally with ∂Ω at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution uεconcentrating along a curve [Formula: see text] near Γ, exponentially small in ε at any positive distance from the curve, provided ε is small and away from certain critical numbers. The concentrating curve [Formula: see text] will collapse to Γ as ε → 0.

scholarly journals A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
MirKeysaan Mahshid ◽  
Abdolrahman Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Method ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Singular Term

AbstractHere, we consider the following elliptic problem with variable components: $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , with Dirichlet boundary condition in a bounded domain in $\mathbb{R}^{N}$ R N with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

scholarly journals Existence and Nonexistence Results for Classes of Singular Elliptic Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Peng Zhang ◽  
Jia-Feng Liao
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Existence And Nonexistence

The singular semilinear elliptic problem-Δu+k(x)u-γ=λupinΩ,u>0inΩ,u=0on∂Ω, is considered, whereΩis a bounded domain with smooth boundary inRN,k∈Clocα(Ω)∩C(Ω¯), andγ,p,λare three positive constants. Some existence or nonexistence results are obtained for solutions of this problem by the sub-supersolution method.

Existence of a solution for a nonlocal elliptic system of (p(x),q(x))-Kirchhoff type

2018 ◽  
Vol 9 (3) ◽  
pp. 221-233
Author(s):  
Ali Taghavi ◽  
Horieh Ghorbani
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Variational Methods ◽  
Elliptic System ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Continuous Functions ◽  
Ekeland's Variational Principle ◽  
Existence Of A Solution ◽  
Kirchhoff Type

Abstract In this paper, we consider the system \left\{\begin{aligned} &\displaystyle{-}M_{1}\bigg{(}\int_{\Omega}\frac{\lvert% \nabla u\rvert^{p(x)}+\lvert u\rvert^{p(x)}}{p(x)}\,dx\biggr{)}(\Delta_{p(x)}u% -\lvert u\rvert^{p(x)-2}u)=\lambda a(x)\lvert u\rvert^{r_{1}(x)-2}u-\mu b(x)% \lvert u\rvert^{\alpha(x)-2}u&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M_{2}\bigg{(}\int_{\Omega}\frac{\lvert\nabla v\rvert^{q(x)}+% \lvert v\rvert^{q(x)}}{q(x)}\,dx\biggr{)}(\Delta_{q(x)}v-\lvert v\rvert^{q(x)-% 2}v)=\lambda c(x)\lvert v\rvert^{r_{2}(x)-2}v-\mu d(x)\lvert v\rvert^{\beta(x)% -2}v&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω is a bounded domain in {\mathbb{R}^{N}} ( {N\geq 2} ) with a smooth boundary {\partial\Omega} , {M_{1}(t),M_{2}(t)} are continuous functions and {\lambda,\mu>0} . We prove that for any {\mu>0} there exists {\lambda_{*}} sufficiently small such that for any {\lambda\in(0,\lambda_{*})} the above system has a nontrivial weak solution. The proof relies on some variational arguments based on Ekeland’s variational principle, and some adequate variational methods.

Analysis of a Navier–Stokes–Darcy coupling problem

2016 ◽  
Vol 7 (3) ◽  
Author(s):  
Yassine Mabrouki ◽  
Jamil Satouri
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Standard Condition ◽  
Navier Stokes ◽  
Existence Of A Solution ◽  
Navier Stokes Equations ◽  
Coupling Problem ◽  
Coupled Problem ◽  
Darcy Equations ◽  
Dimensional Domain

AbstractThe aim of this work is to present a model for coupling the Darcy equations in a porous medium with the Navier–Stokes equations in the cracks. We consider a two- or three-dimensional domain with non-standard condition at the interface, namely the continuity of the pressure. We propose a mixed formulation and establish the existence of a solution for the coupled problem.

Blow-up rates of large solutions for a ϕ-Laplacian problem with gradient term

2014 ◽  
Vol 144 (4) ◽  
pp. 669-689 ◽  
Author(s):  
W. Arriagada ◽  
J. Huentutripay
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Blow Up ◽  
Smooth Boundary ◽  
Gradient Term ◽  
Regularly Varying Functions ◽  
Large Solutions ◽  
Regularly Varying ◽  
Increasing Function

We study the behaviour of solutions of a boundary blow-up elliptic problem on a bounded domain Ω with smooth boundary in ℝN. The data of the problem consist of an increasing function f : ℝ+ → ℝ+ and two real regularly varying functions ϕ and g.

Very singular problems with critical nonlinearities in two dimensions

2017 ◽  
Vol 20 (02) ◽  
pp. 1650067 ◽  
Author(s):  
S. Prashanth ◽  
Sweta Tiwari ◽  
K. Sreenadh
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Two Dimensions ◽  
Bifurcation Parameter ◽  
Singular Problems ◽  
Exponential Nonlinearity ◽  
Critical Nonlinearities ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this paper, we consider the following singular elliptic problem involving an exponential nonlinearity in two dimensions: [Formula: see text] [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show the existence and multiplicity of positive solutions globally with respect to the bifurcation parameter [Formula: see text].

Nonlinear elliptic problems on singularly perturbed domains

2001 ◽  
Vol 131 (5) ◽  
pp. 1023-1037 ◽  
Author(s):  
Jaeyoung Byeon
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Singularly Perturbed ◽  
Bounded Domains ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic

We consider how the shape of a domain affects the number of positive solutions of a nonlinear elliptic problem. In fact, we show that if a bounded domain Ω is sufficiently close to a union of disjoint bounded domains Ω1,…, Ωm, the number of positive solutions of a nonlinear elliptic problem on Ω is at least 2m −1.

scholarly journals Implicit Equations Involving the p-Laplace Operator

2021 ◽  
Vol 18 (2) ◽  
Author(s):  
Greta Marino ◽  
Andrea Paratore
Keyword(s):  
Bounded Domain ◽  
Laplace Operator ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Smooth Boundary ◽  
Implicit Equations

AbstractIn this work we study the existence of solutions $$u \in W^{1,p}_0(\Omega )$$ u ∈ W 0 1 , p ( Ω ) to the implicit elliptic problem $$ f(x, u, \nabla u, \Delta _p u)= 0$$ f ( x , u , ∇ u , Δ p u ) = 0 in $$ \Omega $$ Ω , where $$ \Omega $$ Ω is a bounded domain in $$ {\mathbb {R}}^N $$ R N , $$ N \ge 2 $$ N ≥ 2 , with smooth boundary $$ \partial \Omega $$ ∂ Ω , $$ 1< p< \infty $$ 1 < p < ∞ , and $$ f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : Ω × R × R N × R → R . We choose the particular case when the function f can be expressed in the form $$ f(x, z, w, y)= \varphi (x, z, w)- \psi (y) $$ f ( x , z , w , y ) = φ ( x , z , w ) - ψ ( y ) , where the function $$ \psi $$ ψ depends only on the p-Laplacian $$ \Delta _p u $$ Δ p u . We also present some applications of our results.

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