Multiplicity and Existence Results for a Nonlinear Elliptic Equation With Sobolev Exponent

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Zakaria Bouchech ◽  
Hichem Chtioui
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent

AbstractIn this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)u

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

scholarly journals A Regularization Method for the Elliptic Equation with Inhomogeneous Source

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tuan H. Nguyen ◽  
Binh Thanh Tran
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Regularization Method ◽  
Dirichlet Boundary ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Ill Posed ◽  
Sharp Error

We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x=0 and x=π. The problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.

INFINITELY MANY SOLUTIONS FOR THE DIRICHLET PROBLEM ON THE SIERPINSKI GASKET

2011 ◽  
Vol 09 (03) ◽  
pp. 235-248 ◽  
Author(s):  
BRIGITTE E. BRECKNER ◽  
VICENŢIU D. RĂDULESCU ◽  
CSABA VARGA
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Sierpinski Gasket ◽  
Infinitely Many Solutions ◽  
Sierpiński Gasket ◽  
Nonlinear Elliptic ◽  
Fractal Domains

We study the nonlinear elliptic equation Δu(x) + a(x)u(x) = g(x)f(u(x)) on the Sierpinski gasket and with zero Dirichlet boundary condition. By extending a method introduced by Faraci and Kristály in the framework of Sobolev spaces to the case of function spaces on fractal domains, we establish the existence of infinitely many weak solutions.

scholarly journals Weak Solutions for ap-Laplacian Impulsive Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Wei Dong ◽  
Jiafa Xu ◽  
Xiaoyan Zhang
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Boundary Conditions ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Impulsive Differential Equation ◽  
Dirichlet Boundary ◽  
Point Theory ◽  
Existence Results ◽  
Dirichlet Boundary Conditions

By the virtue of variational method and critical point theory, we give some existence results of weak solutions for ap-Laplacian impulsive differential equation with Dirichlet boundary conditions.

scholarly journals Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

2004 ◽  
Vol 2004 (46) ◽  
pp. 2453-2472
Author(s):  
Abdallah El Hamidi
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Riemannian Manifolds ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equation ◽  
Existence Results ◽  
Real Parameter ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Characteristic Values

This paper deals with an elliptic equation involving Paneitz type operators on compact Riemannian manifolds with concave-convex nonlinearities and a real parameter. Nonlocal and multiple existence results are established. Characteristic values of the real parameter are introduced and their role in the change of the energy sign and the existence of positive solutions are highlighted.

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