scholarly journals Existence and Multiplicity of Solutions for a Robin Problem Involving the p(x)-Laplace Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Najib Tsouli ◽  
Omar Chakrone ◽  
Omar Darhouche ◽  
Mostafa Rahmani
Keyword(s):  
Continuous Function ◽  
Variational Method ◽  
Smooth Boundary ◽  
Normal Derivative ◽  
Robin Problem ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Robin Boundary ◽  
Robin Boundary Value Problem ◽  
Outer Unit

We study the following nonlinear Robin boundary-value problem −Δp(x)u=λf(x,u) in Ω, |∇u|p(x)-2(∂u/∂v)+β(x)|u|p(x)−2u=0 on ∂Ω, where Ω⊂ℝN is a bounded domain with smooth boundary ∂Ω, ∂u/∂v is the outer unit normal derivative on ∂Ω, λ>0 is a real number, p is a continuous function on Ω¯ with infx∈Ω¯p(x)>1, β∈L∞(∂Ω) with β−:=infx∈∂Ωβ(x)>0, and f:Ω×ℝ→ℝ is a continuous function. Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.

scholarly journals Existence of Solutions for a Robin Problem Involving thep(x)-Laplace Operator

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Mostafa Allaoui
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Laplace Operator ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Robin Problem ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

In this article we study the nonlinear Robin boundary-value problem-Δp(x)u=f(x,u)  in  Ω,|∇u|px-2(∂u/∂ν)+β(x)up(x)-2u=0on∂Ω. Using the variational method, under appropriate assumptions onf, we obtain results on existence and multiplicity of solutions.

scholarly journals Multiplicity of solutions for Robin problem involving the p(x)-laplacian

2021 ◽  
Vol 13 (2) ◽  
pp. 321-335
Author(s):  
Hassan Belaouidel ◽  
Anass Ourraoui ◽  
Najib Tsouli
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
Technical Approach ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Robin Boundary ◽  
Laplacian Equations

Abstract This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian equations with Robin boundary condition. Our technical approach is based on variational methods.

Existence and multiplicity of solutions for a class of quasilinear elliptic field equation on ℝN

2018 ◽  
Vol 24 (3) ◽  
pp. 1231-1248
Author(s):  
Claudianor O. Alves ◽  
Alan C.B. dos Santos
Keyword(s):  
Field Equation ◽  
Continuous Function ◽  
Singular Function ◽  
Positive Continuous Function ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic

In this paper, we establish existence and multiplicity of solutions for the following class of quasilinear field equation    −Δu + V(x)u − Δpu + W′(u) = 0,  in  ℝN,    (P) where u = (u1, u2, … , uN+1), p > N ≥ 2, W is a singular function and V is a positive continuous function.

scholarly journals Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

2015 ◽  
Vol 4 (1) ◽  
pp. 37-58 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Continuous Function ◽  
Weight Function ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Smooth Boundary ◽  
Existence Results ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Beta 2 ◽  
Fractional Laplace Operator

AbstractIn this article, we study the following p-fractional Laplacian equation: $ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF THE BOUNDARY VALUE PROBLEM WITH A PARAMETER

2017 ◽  
Vol 21 (6) ◽  
pp. 135-140
Author(s):  
A.V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Laplace Operator ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Simple Eigenvalue ◽  
Robin Problem ◽  
The Laplace Operator ◽  
Robin Boundary

The paper presents the investigation of an eigenvalue problem for the Laplace operator with Robin boundary condition in a bounded domain with smooth boundary. The case of boundary condition containing a real parameter is con- sidered. It is proved that multiplicity of the eigenvalue to the Robin problem for all values of the parameter greater than some number does not exceed the mul- tiplicity of the corresponding eigenvalue to the Dirichlet problem for the Laplace operator. For simple eigenvalue of the Dirichlet problem the convergence of eigen- function of the Robin problem to the eigenfunction of the Dirichlet problem for unlimited increase of the parameter is proved. The formula for derivative on the parameter for eigenvalues of the Robin problem is established. This formula is used to justify the asymptotic expansions of eigenvalues of the Robin problem for large positive values of the parameter.

scholarly journals Existence and multiplicity of solutions for anisotropic elliptic equation

2022 ◽  
Vol 40 ◽  
pp. 1-12
Author(s):  
El Amrouss Abdelrachid ◽  
Ali El Mahraoui
Keyword(s):  
Elliptic Equation ◽  
Variational Method ◽  
Nonlinear Problem ◽  
Multiplicity Of Solutions ◽  
Anisotropic Elliptic Equation ◽  
Existence And Multiplicity

In this article we study the nonlinear problem $$\left\{ \begin{array}{lr} -\sum_{i=1}^{N}\partial_{x_{i}}a_{i}(x,\partial_{x_{i}}u)+ b(x)~|u|^{P_{+}^{+}-2}u =\lambda f(x,u) \quad in \quad \Omega\\ u=0 \qquad on \qquad \partial\Omega \end{array} \right.$$ Using the variational method, under appropriate assumptions on $f$, we obtain a result on existence and multiplicity of solutions.

scholarly journals Computing Robin Problem on Unbounded Simply Connected Domain via an Integral Equation with the Generalized Neumann Kernel

2017 ◽  
Vol 2 (3) ◽  
pp. 120-124
Author(s):  
Shwan H. H. Al-Shatri ◽  
Karzan Wakil ◽  
Munira Ismail
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Connected Domain ◽  
Boundary Value ◽  
Solution Technique ◽  
Robin Problem ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation, and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth.

scholarly journals Existence And Multiplicity Of Solutions For A Robin Problem

2014 ◽  
Vol 10 (03) ◽  
pp. 163-172
Author(s):  
Mostafa Allaoui ◽  
Abdel Rachid El Amrouss ◽  
Fouad Kissi ◽  
Anass Ourraoui
Keyword(s):  
Robin Problem ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

scholarly journals On a Robin type problem involving 𝑝(𝑥)-Laplacian operator

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdesslem Ayoujil ◽  
Anass Ourraoui
Keyword(s):  
Variational Approach ◽  
Growth Conditions ◽  
Laplacian Operator ◽  
Robin Problem ◽  
Type Problem ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Cerami Condition ◽  
Weaker Conditions

Abstract This paper deals with the existence and multiplicity of solutions for the p ⁢ ( x ) p(x) -Laplacian Robin problem without the well-known Ambrosetti–Rabinowitz type growth conditions. By means of the variational approach (with the Cerami condition), existence and multiplicity results of solutions are established under weaker conditions.

The Nehari manifold for a quasilinear polyharmonic equation with exponential nonlinearities and a sign-changing weight function

2015 ◽  
Vol 4 (3) ◽  
pp. 177-200 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Weight Function ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Nehari Manifold ◽  
Polyharmonic Equation ◽  
Unbounded Function ◽  
Multiplicity Of Solutions ◽  
The Real ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

AbstractIn this article, we consider the following quasilinear polyharmonic equation: Δn/mmu = λh(x)|u|q-1u + u|u|pe|u|β in Ω, u = ∇u = ⋯ = ∇m-1u = 0 on ∂Ω, where Ω ⊂ ℝn, n ≥ 2m ≥ 2, is a bounded domain with smooth boundary. The real-valued function h is a sign-changing and unbounded function. The exponents p, q and β satisfy 0 < q < n/(m-1) < p+1, β ∈ (1,n/(n-m)] and λ > 0. Using the Nehari manifold and fibering maps, we show the existence and multiplicity of solutions.

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