Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

2009 ◽  
Vol 70 (2) ◽  
pp. 671-692 ◽  
Author(s):  
Norimichi Hirano
Keyword(s):  
Riemannian Manifold ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic

Multiple existence of solutions for a singularly perturbed nonlinear elliptic problem on a Riemannian manifold

2012 ◽  
Vol 142 (1) ◽  
pp. 81-114
Author(s):  
Norimichi Hirano
Keyword(s):  
Riemannian Manifold ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Singularly Perturbed ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Image Position

Let (M, g) be a compact smooth N-dimensional Riemannian manifold without boundary. We consider the multiple existence of positive solutions of the problemwhere Δg stands for the Laplacian in M and f ε C2(M).

scholarly journals On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Anna Maria Micheletti ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Multiplicity Result ◽  
Nonlinear Elliptic Problem ◽  
Nodal Solutions ◽  
Nonlinear Elliptic ◽  
Sign Changing Solutions

Given thatis a smooth compact and symmetric Riemannian -manifold, , we prove a multiplicity result for antisymmetric sign changing solutions of the problem in . Here if and if .

scholarly journals Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

2021 ◽  
Vol 7 (1) ◽  
pp. 50-65
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Problem ◽  
Topological Degree ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Problem ◽  
Technical Approach ◽  
Nonlinear Elliptic ◽  
Variable Exponent Sobolev Spaces ◽  
Topological Degree Method

AbstractThe aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form\left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right.where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and f ∈ W−1,p′(.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.

scholarly journals Study of a second-order nonlinear elliptic problem generated by a divergence type operator on a compact Riemannian manifold

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4811-4820
Author(s):  
A. Abnoune ◽  
E. Azroul ◽  
M.T.K. Abbassi
Keyword(s):  
Riemannian Manifold ◽  
Elliptic Problem ◽  
Compact Riemannian Manifold ◽  
Second Order ◽  
Type Operator ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic ◽  
Divergence Type

In this paper, we will study a second-order nonlinear elliptic problem generated by an operator of divergence type (or type leray-Lion) : (P1){ A(u) = f in M u = 0 on ? (1) on (M,g) a compact Riemannian manifold et ? its border.

Existence and non-existence of solutions to elliptic equations with a general convection term

2014 ◽  
Vol 144 (2) ◽  
pp. 225-239 ◽  
Author(s):  
Salomón Alarcón ◽  
Jorge García-Melián ◽  
Alexander Quaas
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Classical Solutions ◽  
Convection Term ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Comparison Arguments ◽  
Increasing Function

In this paper we consider the nonlinear elliptic problem −Δu + αu = g(∣∇u∣) + λh(x) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain of ℝN, α ≥ 0, g is an arbitrary C1 increasing function and h ∈ C1() is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter λ. We show that there exist solutions for every λ when α = 0 and the integral 1/g(s)ds = ∞, or when α > 0 and the integral s/g(s)ds = ∞. Conversely, when the respective integrals converge and h is non-trivial on ∂Ω, existence depends on the size of λ. Moreover, non-existence holds for large λ. Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on λ.

scholarly journals Solutions with Polynomial Growth to an Autonomous Nonlinear Elliptic Problem

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Juncheng Wei ◽  
Kelei Wang
Keyword(s):  
Periodic Function ◽  
Elliptic Problem ◽  
Polynomial Growth ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic

AbstractWe study the nonlinear elliptic problem-Δu = Fʹ (u) in ℝwhere F(u) is a periodic function. Moser (1986) showed that for any minimal and nonself-intersecting solution, there exist α ∈ ℝ(*) |u - α · x| ≤ C.He also showed the existence of solutions with any prescribed α ∈ ℝ

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