scholarly journals Blow up Points and the Morse Indices of Solutions to the Liouville Equation in Two-Dimension

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Futoshi Takahashi
Keyword(s):  
Bounded Domain ◽  
Liouville Equation ◽  
Morse Index ◽  
Blow Up ◽  
Smooth Bounded Domain ◽  
Two Dimension ◽  
Morse Indices

AbstractWe consider the Liouville equation−Δu = λeu in Ω, u =0 on∂Ω,on a smooth bounded domain Ω in ℝfor m ∈ℕ. We prove that the number of blow up points m is less than or equal to the Morse index of uAs a corollary, we show that if a solution u

scholarly journals Single blow-up solutions for a slightly subcritical biharmonic equation

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Khalil El Mehdi
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Critical Exponent ◽  
Critical Point ◽  
Bounded Domain ◽  
Biharmonic Equation ◽  
Blow Up ◽  
Asymptotic Behavior Of Solutions ◽  
Smooth Bounded Domain ◽  
Nondegenerate Critical Point

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε):∆2u=u9−ε,u>0inΩandu=∆u=0on∂Ω, whereΩis a smooth bounded domain inℝ5,ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient asεgoes to zero. We show that such solutions concentrate around a pointx0∈Ωasε→0, moreoverx0is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical pointx0of the Robin's function, there exist solutions of (Pε) concentrating aroundx0asε→0.

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.

scholarly journals Blow-up for sign-changing solutions of the critical heat equation in domains with a small hole

2016 ◽  
Vol 18 (01) ◽  
pp. 1550017 ◽  
Author(s):  
Isabella Ianni ◽  
Monica Musso ◽  
Angela Pistoia
Keyword(s):  
Heat Equation ◽  
Stationary Solution ◽  
Bounded Domain ◽  
Blow Up ◽  
Initial Conditions ◽  
Small Hole ◽  
Initial Value ◽  
Smooth Bounded Domain ◽  
Sign Changing Solutions

We consider the critical heat equation [Formula: see text] in [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] and [Formula: see text] is a ball of [Formula: see text] of center [Formula: see text] and radius [Formula: see text] small. We show that if [Formula: see text] is small enough, then there exists a sign-changing stationary solution [Formula: see text] of (CH) such that the solution of (CH) with initial value [Formula: see text] blows up if [Formula: see text] is sufficiently small. This shows, in particular, that the set of the initial conditions for which the solution of (CH) is global and bounded is not star-shaped.

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}.

Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli

2017 ◽  
Vol 147 (6) ◽  
pp. 1215-1232
Author(s):  
Zongming Guo ◽  
Linfeng Mei ◽  
Zhitao Zhang
Keyword(s):  
Elliptic Equation ◽  
Symmetry Breaking ◽  
Morse Index ◽  
Blow Up ◽  
Semilinear Elliptic Equation ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Image Position

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

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