Concentration Phenomena in a Biharmonic Equation Involving the Critical Sobolev Exponent

2006 ◽  
Vol 6 (4) ◽  
Author(s):  
Abdelbaki Selmi
Keyword(s):  
Biharmonic Equation ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Concentration Phenomena ◽  
Sobolev Exponent

AbstractIn this paper, we consider the problemΔin Ω, u = Δu = 0 on ∂Ω, where Ω is a bounded and smooth domain in ℝ

Construction of dipole type singular solutions for a biharmonic equation with critical Sobolev exponent

2002 ◽  
Vol 2 (4) ◽  
Author(s):  
Sarni Baraket
Keyword(s):  
Fourth Order ◽  
Weak Solutions ◽  
Biharmonic Equation ◽  
Critical Sobolev Exponent ◽  
Singular Solutions ◽  
Invariant Equation ◽  
Conformally Invariant ◽  
Sobolev Exponent

AbstractIn this paper, we construct positive weak solutions of a fourth order conformally invariant equation on S

scholarly journals Existence of Solutions for a Modified Nonlinear Schrödinger System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yujuan Jiao ◽  
Yanli Wang
Keyword(s):  
Perturbation Method ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Schrödinger System

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  x∈Ω,  u=0,  v=0,  x∈∂Ω, whereα>2,  β>2,  α+β<2·2*,  2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN  (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

scholarly journals Towers of Nodal Bubbles for the Bahri–Coron Problem in Punctured Domains

2018 ◽  
Vol 2019 (19) ◽  
pp. 5953-5974
Author(s):  
Mónica Clapp ◽  
Jorge Faya ◽  
Filomena Pacella
Keyword(s):  
Blow Up ◽  
Critical Sobolev Exponent ◽  
Limit Problem ◽  
Smooth Domain ◽  
Critical Problem ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Sign Changing Solutions

Abstract Let Ω be a bounded smooth domain in $\mathbb {R}^{N}$ which contains a ball of radius R centered at the origin, N ≥ 3. Under suitable symmetry assumptions, for each δ ∈ (0, R), we establish the existence of a sequence (um, δ) of nodal solutions to the critical problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u\text{ in }\Omega_{\delta}:=\{x\in\Omega :\left\vert x\right\vert>\delta\},\quad u=0\text{ on }\partial \Omega_{\delta},\nonumber\end{align*}$$ where $2^{\ast }:=\frac {2N}{N-2}$ is the critical Sobolev exponent. We show that, if Ω is strictly star-shaped then, for each $m\in \mathbb {N},$ the solutions um, δ concentrate and blow up at 0, as $\delta \rightarrow 0,$ and their limit profile is a tower of nodal bubbles, that is, it is a sum of rescaled nonradial sign-changing solutions to the limit problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u, \quad u\in D^{1,2}(\mathbb{R}^{N}),\nonumber\end{align*}$$ centered at the origin.

scholarly journals On Critical p-Laplacian Systems

2017 ◽  
Vol 17 (4) ◽  
pp. 641-659
Author(s):  
Zhenyu Guo ◽  
Kanishka Perera ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Least Energy Solution ◽  
Existence And Nonexistence ◽  
Alpha Beta ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Least Energy

AbstractWe consider the critical p-Laplacian system\left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right.where {\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian operator defined onD^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}},endowed with the norm {{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, {N\geq 3}, {1<p<N}, {\lambda,\mu_{1},\mu_{2}\geq 0}, {\gamma\neq 0}, {a,b,\alpha,\beta>1} satisfy {a+b=p}, {\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is {\mathbb{R}^{N}} or a bounded domain in {\mathbb{R}^{N}} and {D_{0}^{1,p}(\Omega)} is the closure of {C_{0}^{\infty}(\Omega)} in {D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

Sign in / Sign up

Export Citation Format

Share Document