scholarly journals Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

2020 ◽  
Vol 150 (5) ◽  
pp. 2682-2718 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Antonio J. Fernández
Keyword(s):  
Bounded Domain ◽  
Lebesgue Space ◽  
Fractional Laplacian ◽  
Existence Result ◽  
Existence Results ◽  
Real Parameter ◽  
Nonlocal Diffusion ◽  
Smooth Bounded Domain ◽  
Image Position ◽  
Regularity Results

AbstractLet$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form $$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$ where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, $\mu \in L^{\infty}$ and $\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by $$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$ Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

scholarly journals Existence of positive solutions to the fractional Laplacian with positive Dirichlet data

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1795-1807
Author(s):  
Lijuan Liu
Keyword(s):  
Classical Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Fractional Laplacian ◽  
Nonnegative Function ◽  
Existence Results ◽  
Smooth Bounded Domain ◽  
Dirichlet Data ◽  
Existence Of Positive Solutions

We consider the fractional Laplacian with positive Dirichlet data { (-?)?/2 u = ?up in ?, u > 0 in ?, u = ? in Rn\?, where p > 1,0 < ? < min{2,n}, ? ? Rn is a smooth bounded domain, ? is a nonnegative function, positive somewhere and satisfying some other conditions. We prove that there exists ?* > 0 such that for any 0 < ? < ?*, the problem admits at least one positive classical solution; for ? > ?*, the problem admits no classical solution. Moreover, for 1 < p ? n+?/n-?, there exists 0 < ?? ? ?* such that for any 0 < ? < ??, the problem admits a second positive classical solution. From the results obtained, we can see that the existence results of the fractional Laplacian with positive Dirichlet data are quite different from the fractional Laplacian with zero Dirichlet data.

On a critical nonlinear problem involving the fractional Laplacian

2021 ◽  
pp. 2150066
Author(s):  
Azeb Alghanemi ◽  
Hichem Chtioui
Keyword(s):  
Global Existence ◽  
Bounded Domain ◽  
Nonlinear Problem ◽  
Fractional Laplacian ◽  
Existence Result ◽  
Counting Formula ◽  
Global Existence Result

Fractional Yamabe-type equations of the form [Formula: see text] in [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded domain of [Formula: see text], [Formula: see text] is a given function on [Formula: see text] and [Formula: see text], is the fractional Laplacian are considered. Bahri’s estimates in the fractional setting will be proved and used to establish a global existence result through an index-counting formula.

scholarly journals Multiple Solutions for Superlinear Fractional Problems via Theorems of Mixed Type

2018 ◽  
Vol 18 (4) ◽  
pp. 799-817
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Mixed Type ◽  
Fractional Laplacian ◽  
Linking Theorem ◽  
Smooth Bounded Domain

AbstractIn this paper, we investigate the existence of multiple solutions for the following two fractional problems:\left\{\begin{aligned} \displaystyle(-\Delta_{\Omega})^{s}u-\lambda u&% \displaystyle=f(x,u)&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{in }\partial\Omega\end{% aligned}\right.\qquad\text{and}\qquad\left\{\begin{aligned} \displaystyle(-% \Delta_{\mathbb{R}^{N}})^{s}u-\lambda u&\displaystyle=f(x,u)&&\displaystyle% \text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{in }\mathbb{R}^{N}% \setminus\Omega,\end{aligned}\right.where{s\in(0,1)},{N>2s}, Ω is a smooth bounded domain of{\mathbb{R}^{N}}, and{f:\bar{\Omega}\times\mathbb{R}\to\mathbb{R}}is a superlinear continuous function which does not satisfy the well-known Ambrosetti–Rabinowitz condition. Here{(-\Delta_{\Omega})^{s}}is the spectral Laplacian and{(-\Delta_{\mathbb{R}^{N}})^{s}}is the fractional Laplacian in{\mathbb{R}^{N}}. By applying variational theorems of mixed type due to Marino and Saccon and the Linking Theorem, we prove the existence of multiple solutions for the above problems.

scholarly journals ASYMPTOTICALLY LINEAR ELLIPTIC SYSTEMS WITH PARAMETERS

2010 ◽  
Vol 52 (2) ◽  
pp. 383-389 ◽  
Author(s):  
CHAOQUAN PENG
Keyword(s):  
Bounded Domain ◽  
Trivial Solution ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Asymptotically Linear ◽  
Smooth Bounded Domain ◽  
Negative Numbers ◽  
Solution Pair ◽  
Image Position

AbstractIn this paper, we show that the semi-linear elliptic systems of the form (0.1) possess at least one non-trivial solution pair (u, v) ∈ H01(Ω) × H01(Ω), where Ω is a smooth bounded domain in ℝN, λ and μ are non-negative numbers, f(x, t) and g(x, t) are continuous functions on Ω × ℝ and asymptotically linear at infinity.

PERTURBATIONS FROM INDEFINITE SYMMETRIC ELLIPTIC BOUNDARY VALUE PROBLEMS

2017 ◽  
Vol 59 (3) ◽  
pp. 635-648 ◽  
Author(s):  
LIANG ZHANG ◽  
XIANHUA TANG
Keyword(s):  
Boundary Value Problems ◽  
Bounded Domain ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Smooth Bounded Domain ◽  
Multiplicity Of Solutions ◽  
Elliptic Boundary Value ◽  
Formula Group ◽  
Image Position

AbstractIn this paper, we study the multiplicity of solutions for the following problem: $$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$ where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$N, θ is a parameter and g, h ∈ C($\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation.

Existence of a Least Energy Nodal Solution for a Class of p&q-Quasilinear Elliptic Equations

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Sara Barile ◽  
Giovany M. Figueiredo
Keyword(s):  
Bounded Domain ◽  
Elliptic Equations ◽  
Existence Result ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solution ◽  
Smooth Bounded Domain ◽  
Quasilinear Elliptic ◽  
Least Energy

AbstractIn this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given bywhere Ω is a smooth bounded domain in ℝ

scholarly journals Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

2018 ◽  
Vol 20 (03) ◽  
pp. 1750032 ◽  
Author(s):  
Alexander Quaas ◽  
Aliang Xia
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Degree Theory ◽  
Elliptic Systems ◽  
Existence Results ◽  
Nonlinear Elliptic Problem ◽  
Topological Degree Theory ◽  
Scaling Method ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic

In this paper, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: [Formula: see text] where [Formula: see text] denotes the fractional Laplacian and [Formula: see text] is a smooth bounded domain in [Formula: see text]. It shown that under some assumptions on [Formula: see text] and [Formula: see text], the problem has at least one positive solution [Formula: see text]. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.

An indefinite and critical concave—convex-type equation

2013 ◽  
Vol 143 (1) ◽  
pp. 169-184 ◽  
Author(s):  
Humberto Ramos Quoirin ◽  
Pedro Ubilla
Keyword(s):  
Bounded Domain ◽  
Type Equation ◽  
Nehari Manifold ◽  
Real Parameter ◽  
Linear Case ◽  
Energy Estimates ◽  
Multiplicity Result ◽  
Image Position

We analyse the multiplicity of non-negative solutions for the critical concave—convex-type equationwhere Ω is a bounded domain of ℝN, 1 < r < p and λ > 0, and μ is a real parameter. Combining minimization on the underlying Nehari manifold with energy estimates, we show that, under suitable conditions on a and b, three non-negative solutions may exist when λ is positive and sufficiently small and μ is in a right neighbourhood of μ1, the first weighted eigenvalue of the p-Laplacian. To the best of our knowledge, our multiplicity result is new, even in the semi-linear case p = 2.

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>

GEOMETRY OF SOBOLEV SPACES WITH VARIABLE EXPONENT AND A GENERALIZATION OF THE p-LAPLACIAN

2009 ◽  
Vol 07 (04) ◽  
pp. 373-390 ◽  
Author(s):  
GEORGE DINCA ◽  
PAVEL MATEI
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Uniformly Convex ◽  
Smooth Bounded Domain ◽  
Generalized Lebesgue Space

Let Ω ⊂ ℝN, N ≥ 2, be a smooth bounded domain. It is shown that: (a) if [Formula: see text] and ess inf x ∈ y p(x) > 1, then the generalized Lebesgue space (Lp (·)(Ω), ‖·‖p(·)) is smooth; (b) if [Formula: see text] and p(x) > 1, [Formula: see text], then the generalized Sobolev space [Formula: see text] is smooth. In both situations, the formulae for the Gâteaux gradient of the norm corresponding to each of the above spaces are given; (c) if [Formula: see text] and p(x) ≥ 2, [Formula: see text], then [Formula: see text] is uniformly convex and smooth.

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