scholarly journals A finite-dimensional reduction method for slightly supercritical elliptic problems

2004 ◽  
Vol 2004 (8) ◽  
pp. 683-689 ◽  
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Sobolev Space ◽  
Dimensional Reduction ◽  
Reduction Method ◽  
Elliptic Problems ◽  
Subcritical Case ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Dimensional Reduction Method ◽  
Usual Sobolev Space ◽  
Large Level

We describe a finite-dimensional reduction method to find solutions for a class of slightly supercritical elliptic problems. A suitable truncation argument allows us to work in the usual Sobolev space even in the presence of supercritical nonlinearities: we modify the supercritical term in such a way to have subcritical approximating problems; for these problems, the finite-dimensional reduction can be obtained applying the methods already developed in the subcritical case; finally, we show that, if the truncation is realized at a sufficiently large level, then the solutions of the approximating problems, given by these methods, also solve the supercritical problems when the parameter is small enough.

Large number of bubble solutions for a fractional elliptic equation with almost critical exponents

2020 ◽  
pp. 1-40
Author(s):  
Chunhua Wang ◽  
Suting Wei
Keyword(s):  
Critical Exponents ◽  
Fractional Laplacian ◽  
Reduction Method ◽  
Laplacian Operator ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Dimensional Reduction Method ◽  
Non Linear Equation ◽  
Image Position ◽  
Fractional Laplacian Operator

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: \[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \] where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$ .

Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

2020 ◽  
Vol 20 (1) ◽  
pp. 53-75 ◽  
Author(s):  
Billel Gheraibia ◽  
Chunhua Wang
Keyword(s):  
Positive Integer ◽  
Local Minimum ◽  
Reduction Method ◽  
Type System ◽  
Finite Dimensional Reduction ◽  
Nonlinear Schrödinger ◽  
Finite Dimensional ◽  
Local Minimum Point ◽  
Dimensional Reduction Method ◽  
Continuous Potential

AbstractIn this paper, we study the following nonlinear Schrödinger–Newton type system:\left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right.where {\epsilon>0} and {Q(x)} is a positive bounded continuous potential on {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point {x_{0}} of {Q(x)} in {\mathbb{R}^{3}}, provided that {\epsilon>0} is sufficiently small.

Wavelet-like collocation method for finite-dimensional reduction of distributed systems

2000 ◽  
Vol 24 (12) ◽  
pp. 2687-2703 ◽  
Author(s):  
A. Adrover ◽  
G. Continillo ◽  
S. Crescitelli ◽  
M. Giona ◽  
L. Russo
Keyword(s):  
Distributed Systems ◽  
Collocation Method ◽  
Dimensional Reduction ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional
Sign in / Sign up

Export Citation Format

Share Document