scholarly journals Existence of Weak Solutions for Nonlinear Time-Fractionalp-Laplace Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Meilan Qiu ◽  
Liquan Mei
Keyword(s):  
Fixed Point ◽  
Weak Solutions ◽  
Nehari Manifold ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Fractional Equations ◽  
Compact Imbedding ◽  
The Nehari Manifold ◽  
The Relationship ◽  
Laplace Problem

The existence of weak solution forp-Laplace problem is studied in the paper. By exploiting the relationship between the Nehari manifold and fibering maps and combining the compact imbedding theorem and the behavior of Palais-Smale sequences in the Nehari manifold, the existence of weak solutions is established. By means of the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations of thep-Laplace problem are obtained.

scholarly journals The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition

2007 ◽  
Vol 12 (2) ◽  
pp. 143-155 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. Mahdavi ◽  
Z. Naghizadeh
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Laplacian Equation ◽  
The Nehari Manifold ◽  
The Relationship

The Nehari manifold for the equation −∆pu(x) = λu(x)|u(x)|p−2 + b(x)|u(x)|γ−2u(x) for x ∈ Ω together with Dirichlet boundary condition is investigated in the case where 0 < γ < p. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form of t → J(tu) where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as λ changes, and show how existence results for positive solutions of the equation are linked to the properties of Nehari manifold.

Weak Solutions for Fractional Differential Equations via Henstock–Kurzweil–Pettis Integrals

2020 ◽  
Vol 21 (2) ◽  
pp. 135-145 ◽  
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Weak Solutions ◽  
Measure Of Noncompactness ◽  
Pettis Integral ◽  
Existence Of Weak Solutions ◽  
Fractional Differential

AbstractIn this paper, we used Henstock–Kurzweil–Pettis integral instead of classical integrals. Using fixed point theorem and weak measure of noncompactness, we study the existence of weak solutions of boundary value problem for fractional integro-differential equations in Banach spaces. Our results generalize some known results. Finally, an example is given to demonstrate the feasibility of our conclusions.

scholarly journals Multiple of Solutions for Nonlocal Elliptic Equations with Critical Exponent Driven by the Fractional p-Laplacian of Order s

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
M. Khiddi
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nehari Manifold ◽  
Lusternik Schnirelmann ◽  
The Nehari Manifold

In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional p-Laplacian of order s. We show the above result when λ>0 is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.

scholarly journals Regularity and multiplicity results for fractional (p,q)-Laplacian equations

2019 ◽  
Vol 22 (08) ◽  
pp. 1950065 ◽  
Author(s):  
Divya Goel ◽  
Deepak Kumar ◽  
K. Sreenadh
Keyword(s):  
Weak Solutions ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Subcritical Case ◽  
Existence Of A Solution ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
The Nehari Manifold ◽  
Weak Harnack Inequality ◽  
Laplacian Equations

This paper deals with the study of the following nonlinear doubly nonlocal equation: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are parameters. Here [Formula: see text] and [Formula: see text] are sign-changing functions. We prove [Formula: see text] estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case [Formula: see text] Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of [Formula: see text], we show the existence of a solution.

scholarly journals Existence results of solutions for some fractional neutral functional integro-differential equations with infinite delay

2017 ◽  
Author(s):  
Kazem Nouri ◽  
Marjan Nazari ◽  
Bagher Keramati
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Existence Results ◽  
Banach Fixed Point Theorem ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Equations

In this paper, by means of the Banach fixed point theorem and the Krasnoselskii's fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results.

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 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 Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tianfang Wang ◽  
Wen Zhang
Keyword(s):  
Ground State ◽  
Nehari Manifold ◽  
Existence Results ◽  
Infinitely Many Solutions ◽  
Choquard Equation ◽  
Ground State Solutions ◽  
Existence And Multiplicity ◽  
Variational Tools ◽  
Nonlinear Choquard Equation ◽  
The Nehari Manifold

AbstractIn this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , where $N\geq 3$ N ≥ 3 , $0<\mu <N$ 0 < μ < N , $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ 2 N − μ N ≤ p < 2 N − μ N − 2 , ∗ represents the convolution between two functions. We assume that the potential function $V(x)$ V ( x ) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

scholarly journals Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tuhina Mukherjee ◽  
Patrizia Pucci ◽  
Mingqi Xiang
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Nehari Manifold ◽  
Real Parameter ◽  
Combined Effects ◽  
Smooth Bounded Domain ◽  
Existence Proofs ◽  
The Nehari Manifold ◽  
Kirchhoff Problem

<p style='text-indent:20px;'>In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u&gt;0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula> is a real parameter, <inline-formula><tex-math id="M6">\begin{document}$ \beta &lt;{n/(n-s)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q\in (0,1) $\end{document}</tex-math></inline-formula>.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.</p>

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