scholarly journals Existence results for some nonlinear degenerate problems in the anisotropic spaces

2021 ◽  
Vol 39 (6) ◽  
pp. 53-66
Author(s):  
Mohamed Boukhrij ◽  
Benali Aharrouch ◽  
Jaouad Bennouna ◽  
Ahmed Aberqi
Keyword(s):  
Elliptic Problem ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Sign Condition ◽  
Degenerate Elliptic ◽  
Anisotropic Spaces ◽  
Nonlinear Degenerate

Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,where for $i=1,...,N$ $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and $H_i(x,u,\nabla u)$ is a nonlinear term without a sign condition. Under suitable conditions on $a_i$ and $H_i$, we prove the existence of weak solutions.

scholarly journals Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

2019 ◽  
Vol 5 (2) ◽  
pp. 164-178
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Degenerate Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Nonlinear Degenerate

AbstractIn this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations\left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right.in the setting of the weighted Sobolev spaces.

Existence theorems for fractional p-Laplacian problems

2016 ◽  
Vol 15 (05) ◽  
pp. 607-640 ◽  
Author(s):  
Paolo Piersanti ◽  
Patrizia Pucci
Keyword(s):  
Weak Solutions ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Positive Function ◽  
Existence Results ◽  
Laplacian Operator ◽  
Bounded Domains ◽  
Lipschitz Boundary ◽  
Nontrivial Solutions ◽  
Existence Of Weak Solutions

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter [Formula: see text] under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional [Formula: see text]-Laplacian operator. Denoting by [Formula: see text] a sequence of eigenvalues obtained via mini–max methods and linking structures we prove the existence of (weak) solutions both when there exists [Formula: see text] such that [Formula: see text] and when [Formula: see text]. The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at [Formula: see text] is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional [Formula: see text]-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linking methods.

scholarly journals Existence of Solutions for Degenerate Elliptic Problems in Weighted Sobolev Space

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Lili Dai ◽  
Wenjie Gao ◽  
Zhongqing Li
Keyword(s):  
Sobolev Space ◽  
Growth Condition ◽  
Elliptic Problem ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Weighted Sobolev Space ◽  
Elliptic Problems ◽  
Weight Functions ◽  
General Elliptic ◽  
Degenerate Elliptic

This paper is devoted to the study of the existence of solutions to a general elliptic problemAu+g(x,u,∇u)=f-div⁡F, withf∈L1(Ω)andF∈∏i=1NLp'(Ω,ωi*), whereAis a Leray-Lions operator from a weighted Sobolev space into its dual andg(x,s,ξ)is a nonlinear term satisfyinggx,s,ξsgn⁡(s)≥ρ∑i=1Nωi|ξi|p,|s|≥h>0, and a growth condition with respect toξ. Here,ωi,ωi*are weight functions that will be defined in the Preliminaries.

Existence Results for a Class of p(x)-Kirchhoff Problems

2021 ◽  
Vol 58 (1) ◽  
pp. 1-14
Author(s):  
Mostafa Allaoui
Keyword(s):  
Variational Methods ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Boundary Data ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Variable Exponent Sobolev Spaces ◽  
Robin Boundary

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows:Where and satisfies Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Some Existence Results for High Order Fractional Impulsive Differential Equation on Infinite Interval

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zhaocai Hao ◽  
Tian Wang
Keyword(s):  
Differential Equation ◽  
Fixed Points ◽  
Impulsive Differential Equation ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Existence Results ◽  
High Order ◽  
Infinite Interval ◽  
Derivative Operator ◽  
Lower Order Derivative

In this paper, we consider the high order impulsive differential equation on infinite interval D 0 + α u t + f t , u t , J 0 + β u t , D 0 + α − 1 u t = 0 ,   t ∈ 0 , ∞ ∖ t k k = 1 m △ u t k = I k u t k ,   t = t k , k = 1 , … , m u 0 = u ′ 0 = ⋯ = u n − 2 0 = 0 , D 0 + α − 1 u ∞ = u 0 By applying Schauder fixed points and Altman fixed points, we obtain some new results on the existence of solutions. The nonlinear term of the equation contains fractional integral operator J β u t and lower order derivative operator D 0 + α − 1 u t . An example is presented to illustrate our results.

scholarly journals Weighted Steklov problem under nonresonance conditions

2018 ◽  
Vol 36 (4) ◽  
pp. 87-105
Author(s):  
Jonas Doumatè ◽  
Aboubacar Marcos
Keyword(s):  
Boundary Condition ◽  
Nonlinear Problem ◽  
Weak Solutions ◽  
Existence Results ◽  
Smooth Domain ◽  
Existence Of Weak Solutions ◽  
Steklov Problem ◽  
Bounded Smooth Domain ◽  
Carathéodory Function ◽  
Bounded Smooth

We deal with the existence of weak solutions of the nonlinear problem $-\Delta_{p}u+V|u|^{p-2}u$ in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$ which is subject to the boundary condition $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=f(x,u)$. Here $V\in L^{\infty}(\Omega)$ possibly exhibit both signs which leads to an extension  of particular cases in literature and $f$ is a Carathéodory function that satisfies some additional conditions. Finally we prove, under and between nonresonance condtions, existence results for the problem.

Existence Results to Some Damped-Like Fractional Differential Equations

2017 ◽  
Vol 18 (3-4) ◽  
pp. 233-243 ◽  
Author(s):  
Nemat Nyamoradi ◽  
Yong Zhou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Fractional Differential ◽  
New Criteria ◽  
Fractional Boundary Value Problems

Abstract:In this paper, by using critical point theory and variational methods, we prove the existence of weak solutions for damped-like fractional differential equations. We given some new criteria to distinguish that the fractional boundary value problems have at least one solution. Some examples are also given to illustrate the main results.

scholarly journals Existence Results for A Perturbed Fourth-Order Equation

2017 ◽  
Vol 23 (2) ◽  
pp. 55-65
Author(s):  
Mohammad Reza Heidari Tavani
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Results ◽  
Order Equation ◽  
Fourth Order Equation

‎The existence of at least three weak solutions for a class of perturbed‎‎fourth-order problems with a perturbed nonlinear term is investigated‎. ‎Our‎‎approach is based on variational methods and critical point theory‎.

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