INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH

2014 ◽  
Vol 32 (1_2) ◽  
pp. 137-152
Author(s):  
Chenxing Zhou ◽  
Sihua Liang
Keyword(s):  
Critical Growth ◽  
Laplacian Operator

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

On the system of p-Laplacian equations with critical growth

2018 ◽  
Vol 29 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Xiangqing Liu ◽  
Junfang Zhao ◽  
Jiaquan Liu
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Growth ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Laplacian Equations

In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals Existence and Concentration Behavior of Solutions of the Critical Schrödinger–Poisson Equation in R3

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 464
Author(s):  
Jichao Wang ◽  
Ting Yu
Keyword(s):  
Poisson Equation ◽  
Existence Result ◽  
Critical Growth ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Number Of Solutions ◽  
Concentration Behavior ◽  
The Relationship ◽  
Concentration Phenomenon

In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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