scholarly journals Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
T. Godoy ◽  
U. Kaufmann
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Bounded Domains ◽  
Smooth Bounded Domain ◽  
Unbounded Function

AbstractLet Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k

Nonlinear elliptic problems on singularly perturbed domains

2001 ◽  
Vol 131 (5) ◽  
pp. 1023-1037 ◽  
Author(s):  
Jaeyoung Byeon
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Singularly Perturbed ◽  
Bounded Domains ◽  
Nonlinear Elliptic Problem ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic

We consider how the shape of a domain affects the number of positive solutions of a nonlinear elliptic problem. In fact, we show that if a bounded domain Ω is sufficiently close to a union of disjoint bounded domains Ω1,…, Ωm, the number of positive solutions of a nonlinear elliptic problem on Ω is at least 2m −1.

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 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.

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.

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 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.

scholarly journals Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Chunmei Yuan ◽  
Shujuan Guo ◽  
Kaiyu Tong
Keyword(s):  
Variational Method ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Real Parameter ◽  
Existence Of Positive Solutions

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities-Δu=λa(x)up+b(x)uqinΩ,u>0inΩ,u=0on∂Ω, whereλ>0is a real parameter,0<p<1<q.Ωis a bounded domain inRN  (N≥3), anda(x)andb(x)are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

Minimum action solutions of non-linear elliptic equations in unbounded domains containing a plane

1986 ◽  
Vol 102 (3-4) ◽  
pp. 327-343 ◽  
Author(s):  
Otared Kavian
Keyword(s):  
Elliptic Equation ◽  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Necessary And Sufficient Conditions ◽  
Smooth Bounded Domain ◽  
Non Linear ◽  
Necessary And Sufficient

SynopsisLet d ≧ 1 be an integer and ω ⊂ℝd a smooth bounded domain and consider the elliptic equation − Δu = g(u) on Ω = ℝ2 × ω. We prove that under (almost) necessary and sufficient conditions on the continuous function g: ℝm→ ℝm the above equation has a minimum-action solution.

L∞-Bounds for Solutions of Supercritical Elliptic Problems with Finite Morse Index

2010 ◽  
Vol 10 (4) ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Salem Rebhi
Keyword(s):  
Critical Exponent ◽  
Half Space ◽  
Bounded Domain ◽  
Morse Index ◽  
Elliptic Problems ◽  
Entire Space ◽  
Smooth Bounded Domain ◽  
Sobolev Critical Exponent

AbstractWe study here finite Morse index solutions of -∆u = f(u) on the entire space or half space and their application to smooth bounded domain problems when the growth of the non-linearity is faster than the usual Sobolev critical exponent.

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