A note on a positive solution of a null mass nonlinear field equation in exterior domains

2019 ◽  
Vol 150 (2) ◽  
pp. 841-870
Author(s):  
Alireza Khatib ◽  
Liliane A. Maia
Keyword(s):  
Field Equation ◽  
Positive Solution ◽  
Ground State ◽  
Bound State ◽  
Nonlinear Field ◽  
Regular Domain ◽  
Exterior Domains ◽  
Bound State Solution ◽  
Image Position ◽  
Bounded Regular Domain

AbstractWe consider the Null Mass nonlinear field equation (𝒫)$$\left\{ {\matrix{ {-\Delta u = f(u){\rm in}\;\;\Omega } \hfill \hfill \hfill \hfill \cr {u > 0} \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \cr {u \vert_{\partial \Omega } = 0} \cr } } \right.$$ where ${\open R}^N \setminus \Omega $ is a bounded regular domain. The existence of a bound state solution is established in situations where this problem does not have a ground state.

scholarly journals Existence of a Positive Solution to a Nonlinear Scalar Field Equation with Zero Mass at Infinity

2018 ◽  
Vol 18 (4) ◽  
pp. 745-762 ◽  
Author(s):  
Mónica Clapp ◽  
Liliane A. Maia
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Positive Solution ◽  
Ground State ◽  
Bound State ◽  
Zero Mass ◽  
Scalar Field Equation ◽  
Nonlinear Scalar Field ◽  
Decay Assumption

AbstractWe establish the existence of a positive solution to the problem-\Delta u+V(x)u=f(u),\quad u\in D^{1,2}(\mathbb{R}^{N}),for {N\geq 3}, when the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.

scholarly journals Existence and Concentration of Positive Solutions for Nonlinear Kirchhoff-Type Problems with a General Critical Nonlinearity

2018 ◽  
Vol 61 (4) ◽  
pp. 1023-1040 ◽  
Author(s):  
Jianjun Zhang ◽  
David G. Costa ◽  
João Marcos do Ó
Keyword(s):  
Positive Solutions ◽  
Local Minimum ◽  
Type Equation ◽  
Bound State ◽  
State Solution ◽  
Critical Nonlinearity ◽  
Bound State Solution ◽  
Kirchhoff Type ◽  
Image Position ◽  
Kirchhoff Type Problems

AbstractWe are concerned with the following Kirchhoff-type equation$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$whereM ∈ C(ℝ+, ℝ+),V ∈ C(ℝN, ℝ+) andf(s) is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum ofVasε → 0 under certain conditions onf(s),MandV. In particular, the monotonicity off(s)/sand the Ambrosetti–Rabinowitz condition are not required.

Influence of the Hardy potential in a semilinear heat equation

2009 ◽  
Vol 139 (5) ◽  
pp. 897-926 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Ireneo Peral ◽  
Ana Primo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Critical Power ◽  
Regular Domain ◽  
Hardy Potential ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Image Position ◽  
Class Of Functions ◽  
Bounded Regular Domain

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problemwhere Ω⊂ℝN, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u0≥0, f≥0 are in a suitable class of functions.There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.The Cauchy problem, Ω=ℝN, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).

EXISTENCE OF POSITIVE SOLUTION FOR INDEFINITE KIRCHHOFF EQUATION IN EXTERIOR DOMAINS WITH SUBCRITICAL OR CRITICAL GROWTH

2016 ◽  
Vol 103 (3) ◽  
pp. 329-340 ◽  
Author(s):  
G. M. FIGUEIREDO ◽  
D. C. DE MORAIS FILHO
Keyword(s):  
Variational Methods ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Exterior Domain ◽  
Critical Case ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Exterior Domains ◽  
Subcritical Case ◽  
Image Position

Using variational methods and depending on a parameter $\unicode[STIX]{x1D706}$ we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ for the subcritical case ($\unicode[STIX]{x1D6FE}=0$) and also for the critical case ($\unicode[STIX]{x1D6FE}=1$).

Existence and multiplicity of positive solutions for a fourth-order elliptic equation

2019 ◽  
Vol 150 (2) ◽  
pp. 1053-1069
Author(s):  
Giovany M. Figueiredo ◽  
Marcelo F. Furtado ◽  
João Pablo P. da Silva
Keyword(s):  
Critical Sobolev Exponent ◽  
Order Elliptic Equation ◽  
Regular Domain ◽  
Lebesgue Spaces ◽  
Existence And Multiplicity ◽  
Fourth Order Elliptic Equation ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Bounded Regular Domain

AbstractWe prove existence and multiplicity of solutions for the problem$$\left\{ {\matrix{ {\Delta ^2u + \lambda \Delta u = \vert u \vert ^{2*-2u},{\rm in }\Omega ,} \hfill \hfill \hfill \hfill \cr {u,-\Delta u > 0,\quad {\rm in}\;\Omega ,\quad u = \Delta u = 0,\quad {\rm on}\;\partial \Omega ,} \cr } } \right.$$where$\Omega \subset {\open R}^N$,$N \ges 5$, is a bounded regular domain,$\lambda >0$and$2^*=2N/(N-4)$is the critical Sobolev exponent for the embedding of$W^{2,2}(\Omega )$into the Lebesgue spaces.

scholarly journals Detecting dark photons from atomic rearrangement in the galaxy

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Eiger ◽  
Michael Geller
Keyword(s):  
Dark Matter ◽  
Ground State ◽  
Bound State ◽  
Current Data ◽  
Dark Sector ◽  
Atomic Process ◽  
The Galaxy ◽  
Atomic States ◽  
Atomic Rearrangement

Abstract We study a new dark sector signature for an atomic process of “rearrangement” in the galaxy. In this process, a hydrogen-like atomic dark matter state together with its anti-particle can rearrange to form a highly-excited bound state. This bound state will then de-excite into the ground state emitting a large number of dark photons that can be measured in experiments on Earth through their kinetic mixing with the photon. We find that for DM masses in the GeV range, the dark photons have enough energy to pass the thresholds of neutrino observatories such as Borexino and Super-Kamiokande that can probe for our scenario even when our atomic states constitute a small fraction of the total DM abundance. We study the corresponding bounds on the parameters of our model from current data as well as the prospects for future detectors.

GROUND STATE CHARGE OF FERMION SOLITON SYSTEM IN 1+1 AND 3+1 DIMENSIONS

1993 ◽  
Vol 08 (04) ◽  
pp. 705-721
Author(s):  
M. RAVENDRANADHAN ◽  
M. SABIR
Keyword(s):  
Ground State ◽  
Energy State ◽  
Flow Analysis ◽  
Background Field ◽  
Bound State ◽  
Fermion Mass ◽  
Spectral Flow ◽  
Bound State Spectrum ◽  
Spectral Asymmetry ◽  
State Charge

Ground state charge of some fermion soliton system without C-invariance is calculated in 1+1 and 3+1 dimensions by a combination of adiabatic method and spectral flow analysis. Induced charge is calculated by evolving adiabatically the fields from a vacuum having a background field which has a zero energy state and spectral symmetry. The spectral flow is calculated by an analysis of the bound state spectrum. In 1+1 dimension our calculations are in agreement with the results already found in the literature. In 3+1 dimension we study the interaction of fermions with monopoles and dyons. In the case of monopoles, even though there is spectral asymmetry, ground state charge is found to be ±1/2. It is shown that ground state charge gets contribution only from the lowest angular momentum states and is discontinuous at the fermion mass.

Approximate Solution of a Nonlinear Field Equation

1963 ◽  
Vol 4 (3) ◽  
pp. 334-338 ◽  
Author(s):  
D. D. Betts ◽  
H. Schiff ◽  
W. B. Strickfaden
Keyword(s):  
Approximate Solution ◽  
Field Equation ◽  
Nonlinear Field

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

Sign in / Sign up

Export Citation Format

Share Document