Boundary Value Problems
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Published By Springer (Biomed Central Ltd.)

1687-2770, 1687-2762

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Qilin Xie ◽  
Huafeng Xiao

AbstractIn the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.


2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Debao Yan

AbstractWe concentrate on a category of singular boundary value problems of fractional differential equations with integral boundary conditions, in which the nonlinear function f is singular at $t=0$ t = 0 , 1. We use Banach’s fixed-point theorem and Hölder’s inequality to verify the existence and uniqueness of a solution. Moreover, also we prove the existence of solutions by Krasnoselskii’s and Schaefer’s fixed point theorems.


2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Imed Bachar ◽  
Entesar Aljarallah

AbstractWe consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in  D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.


2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Yoshiaki Kusaka

AbstractWe investigate the free-boundary problem of a steadily advancing meniscus in a circular capillary tube. The problem is described using the “interface formation model,” which was originally introduced with the aim of avoiding the singularities that arise when classical hydrodynamics is applied to problems with a moving contact line. We prove the existence of an axially symmetric solution in weighted Hölder spaces for low meniscus speeds.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Q-Heung Choi ◽  
Tacksun Jung

AbstractWe investigate the multiplicity of solutions for problems involving the fractional N-Laplacian. We obtain three theorems depending on the source terms in which the nonlinearities cross some eigenvalues. We obtain these results by direct computations with the eigenvalues and the corresponding eigenfunctions for the fractional N-Laplacian eigenvalue problem in the fractional Orlicz–Sobolev spaces, the contraction mapping principle on the fractional Orlicz–Sobolev spaces and Leray–Schauder degree theory.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang ◽  
Qiongfen Zhang

AbstractIn this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ { − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + λ | u | p − 2 u , x ∈ R N ; u ∈ H 1 ( R N ) , where $\lambda >0$ λ > 0 is a parameter, $\alpha \in (0,N)$ α ∈ ( 0 , N ) , $N\ge 3$ N ≥ 3 , $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$ I α : R N → R is the Riesz potential. As usual, $\alpha /N+1$ α / N + 1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$ λ > λ ∗ for some given number $\lambda _{*}$ λ ∗ in three cases: (i) $2< p<\frac{4}{N}+2$ 2 < p < 4 N + 2 , (ii) $p=\frac{4}{N}+2$ p = 4 N + 2 , and (iii) $\frac{4}{N}+2< p<2^{*}$ 4 N + 2 < p < 2 ∗ . Our result improves the previous related ones in the literature.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohamed Abdelwahed ◽  
Nejmeddine Chorfi ◽  
Henda Ouertani

AbstractThe objective of the article is to improve the algorithms for the resolution of the spectral discretization of the vorticity–velocity–pressure formulation of the Navier–Stokes problem in two and three domains. Two algorithms are proposed. The first one is based on the Uzawa method. In the second one we consider a modified global resolution. The two algorithms are implemented and their results are compared.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xiaowei Li ◽  
Feizhi Wang

AbstractIn this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space $\mathbb{R}^{N}$ R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhongyi Zhang ◽  
Yueqiang Song

AbstractIn the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wei-Chuan Wang

AbstractThe conformable fractional eigenvalue problem $$\begin{aligned} -D_{x}^{\alpha }D_{x}^{\alpha }y+q(x)y=\lambda \rho (x)y \end{aligned}$$ − D x α D x α y + q ( x ) y = λ ρ ( x ) y is considered. We employ an easy and efficient method to derive its eigenvalue asymptotic expansion. On the basis of this result, we also investigate Ambarzumyan problems related to this eigenvalue problem as an application.


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