scholarly journals Regularity of the Extremal Solutions in a Gelfand System Problem

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Craig Cowan
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Extremal Solutions ◽  
Smooth Bounded Domain

AbstractWe examine the elliptic system given bywhere λ, γ are positive parameters and where Ω is a smooth bounded domain in ℝ

scholarly journals Regularity of the extremal solutions associated with elliptic systems

2018 ◽  
Vol 149 (04) ◽  
pp. 1037-1046
Author(s):  
A. Aghajani ◽  
C. Cowan
Keyword(s):  
Convex Function ◽  
Elliptic System ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Extremal Solution ◽  
Second Order ◽  
Extremal Solutions ◽  
Smooth Bounded Domain ◽  
Order Polynomial ◽  
Image Position

AbstractWe examine the elliptic system given by$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill &amp; {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝNandfis aC2positive, nondecreasing and convex function in [0, ∞) such thatf(t)/t→ ∞ ast→ ∞. Assuming$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$we show that the extremal solution (u*,v*) associated with the above system is smooth provided thatN&lt; (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α*&gt; 1 denotes the largest root of the second-order polynomial$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$As a consequence,u*,v* ∈L∞(Ω) forN&lt; 5. Moreover, if τ−= τ+, thenu*,v* ∈L∞(Ω) forN&lt; 10.

Analysis of an elliptic system with infinitely many solutions

2017 ◽  
Vol 6 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Carmen Cortázar ◽  
Manuel Elgueta ◽  
Jorge García-Melián
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Infinitely Many Solutions ◽  
Smooth Bounded Domain ◽  
Content Type ◽  
Outward Unit ◽  
Unit Normal

AbstractWe consider the elliptic system ${\Delta u\hskip-0.284528pt=\hskip-0.284528ptu^{p}v^{q}}$, ${\Delta v\hskip-0.284528pt=\hskip-0.284528ptu^{r}v^{s}}$ in Ω with the boundary conditions ${{\partial u/\partial\eta}=\lambda u}$, ${{\partial v/\partial\eta}=\mu v}$ on ${\partial\Omega}$, where Ω is a smooth bounded domain of ${\mathbb{R}^{N}}$, ${p,s>1}$, ${q,r>0}$, ${\lambda,\mu>0}$ and η stands for the outward unit normal. Assuming the “criticality” hypothesis ${(p-1)(s-1)=qr}$, we completely analyze the values of ${\lambda,\mu}$ for which there exist positive solutions and give a detailed description of the set of solutions.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

scholarly journals On the extremal solutions of superlinear Helmholtz problems

2022 ◽  
Vol 40 ◽  
pp. 1-8
Author(s):  
Makkia Dammak ◽  
Majdi El Ghord ◽  
Saber Ali Kharrati
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Weak Sense ◽  
Critical Value ◽  
Extremal Solutions ◽  
Regular Solutions ◽  
Smooth Bounded Domain ◽  
Bifurcation Phenomena ◽  
Stable Solutions

Abstract: In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

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 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 Regularity of Entropy Solutions of Quasilinear Elliptic Problems Related to Hardy-Sobolev Inequalities

2006 ◽  
Vol 6 (4) ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Eduardo Colorado ◽  
Manel Sanchón
Keyword(s):  
Bounded Domain ◽  
Entropy Solution ◽  
Elliptic Problems ◽  
Entropy Solutions ◽  
Sobolev Inequalities ◽  
Smooth Bounded Domain ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic

AbstractThis article is concerned with the regularity of the entropy solution ofwhere Ω is a smooth bounded domain Ω of ℝ

scholarly journals Non-degeneracy of least-energy solutions for an elliptic problem with nearly critical nonlinearity

2010 ◽  
Vol 140 (1) ◽  
pp. 203-222
Author(s):  
Futoshi Takahashi
Keyword(s):  
Recent Result ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Positive Function ◽  
Coefficient Function ◽  
Critical Nonlinearity ◽  
Smooth Bounded Domain ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Smooth Positive Function

We consider the problem −Δu = c0K(x)upε, u > 0 in Ω, u = 0 on δΩ, where Ω is a smooth, bounded domain in ℝN, N ≥ 3, c0 = N(N − 2), pε = (N + 2)/(N − 2) − ε and K is a smooth, positive function on . We prove that least-energy solutions of the above problem are non-degenerate for small ε > 0 under some assumptions on the coefficient function K. This is a generalization of the recent result by Grossi for K ≡ 1, and needs precise estimates and a new argument.

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