Zero Mass Case Research Articles

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

Local-Nonlocal Schrödinger Equation with Critical Exponent: The Zero Mass Case

Local-Nonlocal Schrödinger Equation with Critical Exponent: The Zero Mass Case

Generalized Chern–Simons–Schrödinger system with critical exponential growth: The zero-mass case

We consider the existence of ground state solutions for a class of zero-mass Chern–Simons–Schrödinger systems − Δ u + A 0 u + ∑ j = 1 2 A j 2 u = f ( u ) − a ( x ) | u | p − 2 u , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 | u | 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 | u | 2 , ∂ 2 A 0 = − A 1 | u | 2 , where a : R 2 → R + is an external potential, p ∈ ( 1 , 2 ) and f ∈ C ( R ) denotes the nonlinearity that fulfills the critical exponential growth in the Trudinger–Moser sense at infinity. By introducing an improvement of the version of Trudinger–Moser inequality approached in (J. Differential Equations 393 (2024) 204–237), we are able to investigate the existence of positive ground state solutions for the given system using variational method.

Massive Klein tunneling in topological photonic crystals

Klein’s paradox refers to the transmission of a relativistic particle through a high potential barrier. Although it has a simple resolution in terms of particle-to-antiparticle tunneling (Klein tunneling), debates on its physical meaning seem lasting partially due to the lack of direct experimental verification. In this article, we point out that honeycomb-type photonic crystals (PhCs) provide an ideal platform to investigate the nature of Klein tunneling, where the effective Dirac mass can be tuned in a relatively easy way from a positive value (trivial PhC) to a negative value (topological PhC) via a zero-mass case (PhC graphene). Specifically, we show that analysis of the transmission between domains with opposite Dirac masses—a case hardly be treated within the scheme available so far—sheds new light on the understanding of the Klein tunneling.

Choquard equations with critical exponential nonlinearities in the zero mass case

<p>We investigate Choquard equations in $ \mathbb R^N $ driven by a weighted $ N $-Laplace operator with polynomial kernel and zero mass. Since the setting is limiting for the Sobolev embedding, we work with nonlinearities which may grow up to the critical exponential. We establish the existence of a positive solution by variational methods, complementing the analysis in <sup>[<xref ref-type="bibr" rid="b32">32</xref>]</sup>, where the case of a logarithmic kernel was considered.</p>

Born-Infeld problem with general nonlinearity

In this paper, using variational methods, we look for non-trivial solutions to the following problem{−div(a(|∇u|2)∇u)=g(u),in RN,N≥3,u(x)→0,as |x|→+∞, under general assumptions on the continuous nonlinearity g. We assume growth conditions of g at 0 and, in the zero mass case, growth conditions at infinity are imposed. If a(s)=(1−s)−1/2, we obtain the well-known Born-Infeld operator, but we are able to study also a general class of a such that a(s)→+∞ as s→1−. We find a radial solution to the problem with finite energy.

Axially symmetric solutions for planar Schrödinger–Poisson systems with critical exponential growth and non-negative potential

This paper is concerned with the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R,R) has critical exponential growth in the sense of Trudinger–Moser. This system can be converted into the integro-differential equation with logarithmic convolution potential. We prove the existence of axially symmetric solutions by some new useful estimates on logarithmic convolution potential. Our result not only improves the ones of Chen–Tang (2020), but covers the zero mass case that V(x)≡0.

Quasilinear equation with critical exponential growth in the zero mass case

In this work we study the existence of solutions for the following class of quasilinear elliptic equations −divA(|x|)|∇u|N−2∇u=Q(|x|)f(u),inRN,where N≥2 and f has critical exponential growth. We establish conditions on the non-homogeneous weights A and Q to introduce a suitable function space where we are able to apply Variational Methods to obtain weak solutions. The main key is a Hardy type inequality for radial functions. Our approach is based on a new Trudinger–Moser type inequality, a version of the Symmetric Criticality Principle and Mountain Pass Theorem.

Radial and non-radial multiple solutions to a general mixed dispersion NLS equation

We study the following nonlinear Schrödinger equation with a fourth-order dispersion term Δ2u−βΔu=g(u)in RN in the positive and zero mass regimes: in the former, and , where m > 0 depends on g; in the latter, and β > 0. In either regimes, we find an infinite sequence of solutions under rather generic assumptions about g; if N = 2 in the positive mass case, or N = 4 in the zero mass case, we need to strengthen such assumptions. Our approach is variational.

Kirchhoff–Boussinesq-type problems with positive and zero mass

Using variational methods we show the existence of solutions for the following class of elliptic Kirchhoff–Boussinesq-type problems given by Δ 2 u − Δ p u + u = h ( u ) , in R N and Δ 2 u − Δ p u = f ( u ) , in R N , where 2 < p ≤ 2 N N − 2 for N ≥ 3 and 2 ∗ ∗ = ∞ for N = 3, N = 4, 2 ∗ ∗ = 2 N N − 4 for N ≥ 5 and h and f are continuous functions that satisfy hypotheses considered by Berestycki and Lions [Nonlinear scalar field. Arch Rational Mech Anal. 1983;82:313–345]. More precisely, the problem with the nonlinearity h is related to the Positive mass case and the problem with the nonlinearity f is related to the Zero mass case. The main argument is to find a Palais–Smale sequence satisfying a property related to Pohozaev identity, as in Hirata et al. [Nonlinear scalar field equations in RN: mountain pass and symmetric mountain pass approaches. Topol Methods Nonlinear Anal. 2010;35:253–276], which was used for the first time by Jeanjean [On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer- type problem set on R N . Proc R Soc Edinb Sect A. 1999;129:787–809].

Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).

Existence and Regularity of Solutions for a Choquard Equation with Zero Mass

This paper concerns with the existence and regularity of solutions for the following Choquard type equation, $$-\Delta_u = \big(I_{\mu} * F(u)\big) f(u) {\rm in} \mathbb{R}^3, \quad \quad (P)$$ where \({I_\mu = \frac{1}{|x|^\mu}, 0 < \mu < 3}\), is the Riesz potential, \({F(s)}\) is the primitive of the continuous function f(s), and \({I_{\mu} * F(u)}\) denotes the convolution of \({I_{\mu}}\) and F(u). By using the variational method, we prove that problem (P), in the zero mass case, possesses at least a nontrivial solution under certain conditions on f.

Orbital stability of solitary waves for derivative nonlinear Schrödinger equation

In this paper, we show the orbital stability of solitons arising in the cubic derivative nonlinear Schrodinger equations. We consider the zero mass case that is not covered by earlier works. As this case enjoys L2 scaling invariance, we expect orbital stability (up to scaling symmetry) in addition to spatial and phase translations. We also show a self-similar type blow up criterion of solutions with the critical mass 4π.

Ground state solutions for a class of Choquard equations with potential vanishing at infinity

This paper is dedicated to studying the following nonlinear Choquard equation−△u+V(x)u=(∫RNQ(y)F(u(y))|x−y|μdy)Q(x)f(u),u∈D1,2(RN), where N≥3, μ∈(0,N), V∈C(RN,[0,∞)), Q∈C(RN,(0,∞)), f∈C(R,R) and F(t)=∫0tf(s)ds. By combining the non-Nehari manifold approach with some new inequalities, we prove that the above equation has a ground state solution in the case when V(x)→0 as |x|→∞, where the strict monotonicity condition on f is not required. Moreover, by using perturbation method, we obtain the existence of a least energy solution in the zero mass case, i.e. V=0, where f satisfies the condition that F(t0)≠0 for some t0∈R instead of the usual Ambrosetti–Rabinowitz type condition. These results extend the ones in Alves, Figueiredo and Yang (2015) [1] and some related literature.

The zero mass limit of Kerr and Kerr-(anti-)de-Sitter space-times: exact solutions and wormholes

Heun-type exact solutions emerge for both the radial and the angular equations for the case of a scalar particle coupled to the zero mass limit of both the Kerr and Kerr-(anti)de-Sitter spacetime. Since any type D metric has Heun-type solutions, it is interesting that this property is retained in the zero mass case. This work further refutes the claims that $M$ going to zero limit of the Kerr metric is both locally and globally the same as the Minkowski metric.

Some quasilinear elliptic equations involving multiple $p$-Laplacians

This paper is devoted to the study, with variational technique, of (p,q)-Laplacian equations in presence of general nonlinearities. Especially we obtain the existence result for the zero mass case, which includes a large class of pure power nonlinearities. More general quasilinear problems of Born-Infeld type are also considered.

Renormalizability, van Dam-Veltman-Zakharov discontinuity, and Newtonian singularity in higher-derivative gravity

It was proposed that if a higher-derivative gravity is renormalizable, it implies necessarily a finite Newtonian potential at the origin, but the reverse of this statement is not true. Here we show that the reverse is true when taking into account the vDVZ discontinuity which states that the theory obtained from massive one by taking zero mass limit is not equivalent to the theory obtained in the zero mass case. The surviving degree of freedom in the zero mass limit is an extra scalar which does not affect the light bending angle, but affects the Newtonian potential. This asserts that in order to make the singularity cancellation, the number of massive ghost and healthy tensors matches with that of massive ghost and healthy scalars.

Tight Constraint on Photon Mass from Pulsar Spindown

Abstract Pulsars are magnetized rotating compact objects. They spin down due to magnetic dipole radiation and wind emission. If a photon has nonzero mass, the spin-down rate will be lower than in the zero-mass case. We show that an upper limit of the photon mass, i.e., , may be placed if a pulsar with period P is observed to spin down. Recently, a white dwarf (WD)–M dwarf binary, AR Scorpii, was discovered to emit pulsed broadband emission. The spin-down luminosity of the WD can comfortably power non-thermal radiation from the system. Applying our results to the WD pulsar with P = 117 s, we obtain a stringent upper limit of the photon mass between , assuming a vacuum dipole spindown, and , assuming spindown due to a fully developed pulsar wind.

A positive mass theorem in three dimensional Cauchy–Riemann geometry

We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral formula for the p-mass, and we reduce its positivity to a solution of Kohn's equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive CR Yamabe invariant and with non-negative CR Paneitz operator. Under these assumptions, we also characterize the zero mass case as the standard three dimensional CR sphere. We then show the existence of (non-embeddable) CR 3-manifolds having nonpositive Paneitz operator or negative p-mass through a second variation formula. Finally, we apply our main result to find solutions of the CR Yamabe problem with minimal energy.

Exact results for N $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

We provide a contour integral formula for the exact partition function of ${\cal N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)$ ${\cal N}=2$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the ${\cal N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.