TWO SOLUTIONS OF THE BAHRI–CORON PROBLEM IN PUNCTURED DOMAINS VIA THE FIXED POINT TRANSFER

2008 ◽  
Vol 10 (01) ◽  
pp. 81-101 ◽  
Author(s):  
MÓNICA CLAPP ◽  
TOBIAS WETH
Keyword(s):  
Fixed Point ◽  
Positive Solution ◽  
Energy Estimate ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
The Nehari Manifold

We consider the problem [Formula: see text] where Ω is a bounded smooth domain in ℝN, N ≥ 3, and [Formula: see text] is the critical Sobolev exponent. We assume that Ω is annular shaped, i.e. there are constants R2 > R1 > 0 such that {x ∈ ℝN : R1 < |x| < R2} ⊂ Ω and {x ∈ ℝN : |x| < R1}\Ω ≠ ∅. Coron [7] showed that there is one positive solution to this problem if R2/R1 is large enough. We establish the existence of at least two pairs of nontrivial solutions in this case. The proof combines a deformation argument on the Nehari manifold with cohomological information derived from Dold's fixed point transfer. To deal with the lack of compactness, an energy estimate recently proved by one of the authors is used.

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 Multiplicity of Positive Solutions for ap-q-Laplacian Type Equation with Critical Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Positive Solutions ◽  
Type Equation ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Smooth Domain ◽  
Laplacian Equation ◽  
Critical Nonlinearities ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions

We study the effect of the coefficientf(x)of the critical nonlinearity on the number of positive solutions for ap-q-Laplacian equation. Under suitable assumptions forf(x)andg(x), we should prove that for sufficiently smallλ>0, there exist at leastkpositive solutions of the followingp-q-Laplacian equation,-Δpu-Δqu=fxu|p*-2u+λgxu|r-2u  in  Ω,u=0  on  ∂Ω,whereΩ⊂RNis a bounded smooth domain,N>p,1<q<N(p-1)/(N-1)<p≤max⁡{p,p^*-q/(p-1)}<r<p^*,p^*=Np/(N-p)is the critical Sobolev exponent, andΔsu=div(|∇u|s-2∇uis thes-Laplacian ofu.

On inhomogeneous biharmonic equations involving critical exponents

1999 ◽  
Vol 129 (5) ◽  
pp. 925-946 ◽  
Author(s):  
Yinbin Deng ◽  
Gengsheng Wang
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Mountain Pass Lemma ◽  
Biharmonic Equations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Image Position

In this paper, we consider the existence of multiple solutions of biharmonic equations boundary value problemwhere Ω is a bounded smooth domain in ℝN, N ≥ 5; λ ∈ ℝ1 is a given constant; p = 2N/(N − 4) is the critical Sobolev exponent for the embedding ; Δ2 = ΔΔ denotes iterated N-dimensional Laplacian; f(x) is a given function. Some results on the existence and non-existence of multiple solutions for the above problem have been obtained by Ekeland's variational principle and the mountain-pass lemma under some assumptions on f(x) and N.

Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent

2014 ◽  
Vol 144 (4) ◽  
pp. 691-709 ◽  
Author(s):  
Ching-yu Chen ◽  
Tsung-fang Wu
Keyword(s):  
Positive Solutions ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problems ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold ◽  
Indefinite Elliptic Problem

In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

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.

Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

2016 ◽  
Vol 146 (6) ◽  
pp. 1167-1193 ◽  
Author(s):  
Wenjing Chen ◽  
Shengbing Deng
Keyword(s):  
Elliptic System ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Dirichlet Boundary Conditions ◽  
Bounded Set ◽  
Sobolev Exponent ◽  
Image Position ◽  
Two Parameters ◽  
The Nehari Manifold

We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:where Ω is a smooth bounded set in ℝn, n > 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ > 0 are two parameters; the exponent n/(n – 2s) ⩽ q < 2; α > 1, β > 1 satisfy is the fractional critical Sobolev exponent.

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

2021 ◽  
pp. 1-19
Author(s):  
Jing Zhang ◽  
Lin Li
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Hardy Potential ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Positive and nodal solutions for an elliptic equation with critical growth

2016 ◽  
Vol 18 (02) ◽  
pp. 1550021 ◽  
Author(s):  
Marcelo F. Furtado ◽  
Bruno N. Souza
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Critical Growth ◽  
Smooth Domain ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Continuous Potential ◽  
Bounded Smooth

We consider the problem [Formula: see text] where [Formula: see text] is a bounded smooth domain, [Formula: see text], [Formula: see text], [Formula: see text]. Under some suitable conditions on the continuous potential [Formula: see text] and on the parameter [Formula: see text], we obtain one nodal solution for [Formula: see text] and one positive solution for [Formula: see text].

Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

2014 ◽  
Vol 526 ◽  
pp. 177-181
Author(s):  
Yuan Li ◽  
Ai Hui Sheng
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Perturbation Theorem ◽  
Existence And Multiplicity ◽  
Manifold Theory ◽  
The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

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