scholarly journals Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions

2018 ◽  
Vol 265 (9) ◽  
pp. 4133-4157 ◽  
Author(s):  
Haiyun Deng ◽  
Hairong Liu ◽  
Long Tian
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Points ◽  
Quasilinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic

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.

scholarly journals On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

2019 ◽  
Vol 149 (5) ◽  
pp. 1163-1173
Author(s):  
Vladimir Bobkov ◽  
Sergey Kolonitskii
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Nodal Set ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic ◽  
Least Energy

AbstractIn this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

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 Dirichlet impulsive equations

2014 ◽  
Vol 3 (S1) ◽  
pp. s89-s98 ◽  
Author(s):  
Massimiliano Ferrara ◽  
Shapour Heidarkhani ◽  
Pasquale F. Pizzimenti
Keyword(s):  
Banach Space ◽  
Boundary Conditions ◽  
Critical Points ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Reflexive Banach Space ◽  
Dirichlet Boundary Conditions ◽  
Nontrivial Solutions ◽  
Three Solutions ◽  
Impulsive Equations

AbstractIn this paper we are interested to ensure the existence of multiple nontrivial solutions for some classes of problems under Dirichlet boundary conditions with impulsive effects. More precisely, by using a suitable analytical setting, the existence of at least three solutions is proved exploiting a recent three-critical points result for smooth functionals defined in a reflexive Banach space. Our approach generalizes some well-known results in the classical framework.

scholarly journals A uniqueness result for some singular semilinear elliptic equations

2016 ◽  
Vol 18 (06) ◽  
pp. 1550084 ◽  
Author(s):  
Annamaria Canino ◽  
Berardino Sciunzi
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Open Subset ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Uniqueness Result ◽  
Dirichlet Boundary Conditions ◽  
Semilinear Elliptic ◽  
Bounded Open Subset

Given [Formula: see text] a bounded open subset of [Formula: see text], we consider non-negative solutions to the singular semilinear elliptic equation [Formula: see text] in [Formula: see text], under zero Dirichlet boundary conditions. For [Formula: see text] and [Formula: see text], we prove that the solution is unique.

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