Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell gauged O(3) sigma model

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Abstract

This article is devoted to the study of the following semilinear equation with measure data which originates in the gravitational Maxwell gauged O(3) sigma model,(E)Δu+A0(j=1k|xpj|2nj)aeu(1+eu)1+a=4πj=1knjδpj4πj=1lmjδqjinR2. In this equation the {δpj}j=1k (resp. {δqj}j=1l) are Dirac masses concentrated at the points {pj}j=1k, (resp. {qj}j=1l), nj and mj are positive integers, and a is a nonnegative real number. We set N=j=1knj and M=j=1lmj.

In previous works [7], [24], some qualitative properties of solutions of (E) with a=0 have been established. Our aim in this article is to study the more general case where a>0. The additional difficulties of this case come from the fact that the nonlinearity is no longer monotone and the data are signed measures. As a consequence we cannot anymore construct directly the solutions by the monotonicity method combined with the supersolutions and subsolutions technique. Instead we develop a new and self-contained approach which enables us to emphasize the role played by the gravitation in the gauged O(3) sigma model. Without the gravitational term, i.e. if a=0, problem (E) has a layer's structure of solutions {uβ}β(2(NM),2], where uβ is the unique non-topological solution such that uβ=βln|x|+O(1) for 2(NM)<β<2 and u2=2ln|x|2lnln|x|+O(1) at infinity respectively. On the contrary, when a>0, the set of solutions to problem (E) has a much richer structure: besides the topological solutions, there exists a sequence of non-topological solutions in type I, i.e. such that u tends to −∞ at infinity, and of non-topological solutions of type II, which tend to ∞ at infinity. The existence of these types of solutions depends on the values of the parameters N,M,β and on the gravitational interaction associated to a.

Introduction

In this paper our goal is to classify the solutions of the following equation with measure dataΔu+A0(j=1k|xpj|2nj)aeu(1+eu)1+a=4πj=1knjδpj4πj=1lmjδqjinR2, where {δpj}j=1k (resp. {δqj}j=1l) are Dirac masses concentrated at the points {pj}j=1k, (resp. {qj}j=1l), pjpj for jj, the related coefficients nj and mj are positive integers, A0>0 is a given constant, a=16πG with G being the Newton's gravitational constant (or more precisely a dimensionless rescaling factor of the gravitational constant [27]) which is of the order of 1030. This means that physically speaking the exponent a is very small. SetP(x)=A0(j=1k|xpj|2nj)a. Since21amin{eu,eau}eu(1+eu)1+amin{eu,eau}, we define the notion of weak solution as follows:

Definition 1.1

A function uLloc1(R2) such that Pmin{eu,eau}Lloc1(R2) is called a weak solution of (E), if for any ξCc(R2),R2u(Δ)ξdx+R2Peu(1+eu)1+aξdx=4πj=1knjξ(pj)4πj=1lmjξ(qj).

This definition means that the following equation holds in the sense of distributions in R2,Δu+Peu(1+eu)1+a=4πj=1knjδpj4πj=1lmjδqj.

We denote by Σ:={p1,,pk,q1,,ql} the set of the supports of the measures in the right-hand side of (1.1). Since the nonlinearity in (1.4) is locally bounded in R2Σ, a weak solution of (1.4) belongs to C2(R2Σ) and is a strong solution ofΔu+Peu(1+eu)1+a=0in R2Σ. The nonlinear term is not monotone, actually the function ueu(1+eu)1+a is increasing on (,lna), and decreasing on (lna,). This makes the structure of solutions of our problem much more complicated than the case where a=0.

Equation (1.1) comes from the Maxwell gauged O(3) sigma model. When a=0, it governs the self-dual O(3) gauged sigma model developed from Heisenberg ferromagnet, see references [1], [2], [18], [21]. When the sigma model for Heisenberg ferromagnet with magnetic field is two-dimensional, it can be expressed by a local U(1)-invariant action density [24, p. 43-49]:L=14FμνFμν+12DμϕDμϕ12(1nϕ)2, where n=(0,0,1), ϕ=(ϕ1,ϕ2,ϕ3) is a spin vector defined over the (2+1)-dimensional Minkowski spacetime R2,1, with value in the unit sphere S2, i.e. |ϕ|=1, Dμ are gauge-covariant derivatives on ϕ, defined byDμϕ=μϕ+Aμ(n×ϕ)where μ=0,1,2 and Fμν=μAννAμ is the electromagnetic curvature induced from the 3-vector connection Aν, ν=0,1,2 as detailed in [26, p. 177-189]. When the time gauge A0 is zero, that is in the static situation, the functional of total energy can be expressed by the following expressionsE(ϕ,A)=12R2((D1ϕ)2+(D2ϕ)2+(1nϕ)2+F122)dx=4π|deg(ϕ)|+12R2((D1ϕ±ϕ×D2ϕ)2+(F12(1nϕ))2)dx, where deg(ϕ) denotes the Brouwer's degree of ϕ. The related Bogomol'nyi equation is obtained by using the stereographic projection ϕϕ˜ from the south pole S=(0,0,1) of S2{S} onto R2 (see e.g. [5], [26] for details). Then the function u=ln|ϕ˜|2 satisfiesΔu+4eu1+eu=4πj=1knjδpj4πj=1lmjδqjinR2. It is pointed out in [24] that the points pj (j=1,,k), which are the poles of ϕ˜ can be viewed as magnetic monopoles and the points qj (j=1,,l), which are the zeros of ϕ˜ as antimonopoles (see [26, p. 55]). They are also called magnetic vortices and anti-vortices respectively.

An important quantity for the gauged sigma model is the total magnetic flux. It is customary [20] to identity it to the integral of the curvature as follows:M(ϕ)=R2F12. Using the variable u its value coincides with R2Δudx (the Laplacian being taken a.e.). Thus, for the sake of simplicity, we identify M(ϕ) and M(u), an expression which will be called the total flux in the sequel. Here and in what follows, we denoteN=j=1knjandM=j=1lmj.

When the gravitation constant G is replaced by zero, a layer's structure of solutions of (1.1) has been determined in the following result:

Theorem 1.1

[7], [24] (i) If M=N1, then problem (1.6) has no solution.

(ii) If M<N1, then for any β[2,2(NM)) problem (1.6) has a unique solution uβ verifyingM(uβ)=2π(2(NM)+β), with the following behavior as |x|,uβ(x)={βln|x|+O(1)ifβ(2,2(NM)),2ln|x|2lnln|x|+O(1)ifβ=2. Furthermore the correspondence βuβ is decreasing.

(iii) If M<N1 and u is a non-topological solution of (1.6) with finite total magnetic flux, i.e. M(u)<, then there exists a unique β[2,2(NM)) such that u=uβ.

These equations have been studied extensively, motivated by a large range of many applications in physics such as the gauged sigma models with broken symmetry [25], the gravitational Maxwell gauged O(3) sigma model [5], [8], [9], [20], [21], the self-dual Chern-Simons-Higgs model [6], [13], magnetic vortices [15], Toda system [14], [17], Liouville equation [11] and the references therein. It is also motivated by important questions in the theory of nonlinear partial differential equations [4], [22], [23], which has its own features in two dimensional space.

When a=16πG, equation (1.1) governs the gravitational Maxwell gauged O(3) sigma model restricted to a plane. Because of the gravitational interaction between particles, the Lagrangian density becomesL=14gμμgννFμνFμν+12DμϕDμϕ12(1nϕ)2 with stress energy tensorTμν=gμνFμνFμν+DμϕDμϕgμνL. We simplify the Einstein equationRμν12Rgμν=8πGTμν, where Rμν is the Ricci tensor and R is a scalar tensor of the metric in considering a metric conformal to the (2 + 1)-dimensional Minkowski onegμν=(1000eη000eη). Then12eηΔη=8πGT00, whereT00=12(eηF12±(1nϕ))2±eηF12(1nϕ))±eηϕ(D1ϕ×D2ϕ)+12(D1ϕ±ϕ×D2ϕ)2. The minimum of the energy is achieved if and only if (ϕ,A) satisfies the self-dual equations (the Bogomol'nyi equations)D1ϕ=ϕ×D2ϕ,F12=±eη(1nϕ). Furthermore, a standard analysis yields equation (1.1). In particular, Yang in [26] studied equation (1.1) when there is only one concentrated pole, i.e. k=1 and l=0. For multiple poles, Chae showed in [5] that problem (1.1) has a sequence of non-topological solutions uβ such thatuβ(x)=βln|x|+O(1)when|x| for β(min{6,2(NM)},2), whenaN<1andNM2. Under the assumption (1.8), the existence of solutions has been improved up to the range β(2(NM),2) by Song in [21]. However, these existence results do not show the role of the gravitation played in the gauged sigma model and the features of the interaction of the diffusion and the non-monotone nonlinearity of equation (1.1) in the whole two dimensional space.

Note that if we take into account the gravitation, the total magnetic flux turns out to beM(u)=R2P(x)eu(1+eu)1+adx, which, due to the potential and the decay to zero for et(1+et)1+a as t, allows the existence of solutions with very wild behaviors at infinity. In fact, the following three types of solutions are considered in this paper{ a solutionuof(1.1)is topological iflim|x|+u(x)=R, a solutionuof(1.1)is non-topological of type I iflim|x|+u(x)=, a solutionuof(1.1)is non-topological of type II iflim|x|+u(x)=+.

The first result of this paper deals with non-topological solutions of type I for (1.1). For such a task we introduce two important quantities:β#=max{2(NM),22aNa}andβ=min{0,2aN2,α2(NM)}, whereα:=12πR2P(x)dx.

Notice that α= if anj1 for some j or aN1, otherwise α is finite, in this case, a free parameter A0 should be taken into account. If aN1, we have that β=2aN20.

Theorem 1.2

Let a=16πG, anj<1forj=1,,k and M be the total magnetic flux given in (1.9).

(i) IfaN1andM<(1+a)N1, then for any β(2(NM),β), problem (1.1) possesses a minimal solution uβ,min satisfyinguβ,min(x)=βln|x|+O(1)as|x|+. Moreover, for some real number C,uβ,min(x)=βln|x|+C+O(|x|aNβ2aNβ1)as|x|+, and the total magnetic flux of the solution uβ,min is equal to 2π[2(NM)+β], i.e.M(uβ,min)=2π[2(NM)+β]. (ii) IfaN>1andM<N, then β#<0 and for any β(β#,0), problem (1.1) possesses a sequence of non-topological solutions uβ,i of type I satisfyinguβ,i(x)=βln|x|+Ci+O(|x|2aN2β22aN2β1)as|x|, whereCi<Ci+1asi+. Moreover, the total magnetic flux of the solutions {uβ,i}i is equal to 2π[2(NM)+β].

Note that our assumption (1.12) is much weaker than (1.8) and Theorem 1.2 provides a larger range of β for existence of solutions uβ verifying uβ=βln|x|+o(1) at infinity. Furthermore we obtain a minimal solution and not just a finite energy solution as in [21, Theorem 1.3]. Note also that the assumption M<(1+a)N1 implies that β>2(NM), and our second interest is to consider this extremal case β=β, which is 2aN2 under the assumption (1.12).

Theorem 1.3

Assume that a=16πG, anj<1forj=1,,k, the magnetic flux M is given by (1.9) and let (1.12) hold.

Then problem (1.1) possesses a minimal non-topological solution uβ,min satisfyinguβ,min(x)=βln|x|2lnln|x|+O(1)as|x|+, and the total magnetic flux of uβ,min is equal to 2π[2(NM)+β].

The existence of non-topological states of type II to (1.1) states as follows.

Theorem 1.4

Assume that a=16πG, anj<1forj=1,,k and β# is given by (1.10), then for anyβ>β+#=max{0,β#}, problem (1.1) possesses a sequence of non-topological solutions {uβ,i}i such thatuβ,i(x)=βln|x|+Ci+O(|x|2aN2β22aN2β1)as|x|+, whereCi<Ci+1+asi+. Moreover, the total magnetic flux of the solutions {uβ,i}i is equal to 2π[2(NM)+β].

Our existence statements of solutions of (1.1) are summarized in Table 1, Table 2, Table 3.

Concerning topological solutions of (1.1), we have following result,

Theorem 1.5

Let a=16πG, anj<1forj=1,,k and (1.15) hold true.

Then problem (1.1) possesses infinitely many topological solutions u0,i satisfyingu0,i(x)=Ci+O(|x|2aN22aN1)as|x|, whereCi<Ci+1asi. Moreover, the total magnetic flux of the solutions {u0,i}i is equal to 4π(NM).

Note that Theorem 1.4 and Theorem 1.5 provide respectively infinitely many non-topological solutions of Type II and topological solutions. Furthermore, there is no upper bound for these solutions, this is due to the failure of the Keller-Osserman condition for the nonlinearity 4eu(1+eu)1+a, see [12], [16]. More precisely equation (1.1) admits no solution with boundary blow-up in a bounded domain. The existence of these solutions illustrates that the gravitation plays an important role in the Maxwell gauged O(3) sigma model:

(i) the set of solutions is extended to topological and two types of non-topological solutions;

(ii) the uniqueness fails for the solution under the given condition uβ(x)=βln|x|+O(1) at infinity;

(iii) the numbers (counted with multiplicity) of magnetic poles N,M do no longer verify M<N+1. In fact, for the non-topological solution of type I, it becomes M<(1+a)N+1, but for the non-topological solution of type II, there is no restriction on N and M, if β>0 is large enough.

The biggest difference with the case that a=0 is that the nonlinearity is no longer monotone, which makes more difficult to construct super and sub solutions to (1.1). Our main idea is to approximate the solution by monotone iterative schemes for some related equations with an increasing nonlinearity.

Finally, we concentrate on the nonexistence of solutions uβ for (1.1) with the behavior βln|x|+O(1) at infinity for some β.

Theorem 1.6

Assume that a=16πG and anj<1forj=1,,k.

(i) If aN<1 and β<β<2aNa, then problem (1.1) has no solution uβ with the aymptotic behavioruβ(x)=βln|x|+o(ln|x|)as|x|. (ii) If aN=1, then problem (1.1) has no topological solution.

The remaining of this paper is organized as follows. In Section 2, we present some decompositions of solutions of (1.1), some important estimates are provided and related forms of equations are considered. We prove that problem (1.1) has a minimal non-topological solution of Type I and minimal solutions in Section 3. Existence of infinitely many non-topological solutions of Type II is obtained in Section 4. Infinitely many topological solutions and minimal topological solution are constructed in Section 5. Finally, Section 6 deals with the classification of general non-topological solutions of (1.1) with infinite total magnetic flux.

Section snippets

Regularity

We begin our analysis by considering the regularity of weak solutions of (1.1). Let ζ be a smooth and increasing function defined in (0,) and such thatζ(t)={lntfor0<t1/2,0fort1. Setν1(x)=2i=1kniζ(|xpi|σ)andν2(x)=2j=1lmjζ(|xqj|σ), where σ(0,1) is chosen such that any two balls of the set{Bσ(pi),Bσ(qj):i=1,k,j=1,l} do not intersect. We fix a positive number r0ee large enough such that Bσ(pi),Bσ(qj)Br0(0) for i=1,,k and j=1,,l, and we denoteΣ1={p1,,pk},Σ2={q1,,ql}andΣ=Σ1Σ2. If u

Minimal solution

In order to consider solutions w of (2.3) with asymptotic behavior βln|x|+O(1), we look for w under the form w=βln|x|+v where v is a bounded function satisfying some related equation. In particular, we look for non-topological solution uβ of problem (1.1) under the formuβ=ν1+ν2+βlnλ+vβorwβ=vβ+βlnλ, where λ is given by (2.30) and vβ is a bounded classical solution ofΔv+Vλβev(eν1ν2+λβev)1+a=gβinR2, with V being defined in (2.4) and where gβ is defined in (2.34). Here and in what follows,

Non-topological solutions

If N>M and aN<1, we recall that by Theorem 2.1, for any β(2(NM),β), with β=2(aN1)<0, there exists a unique bounded solution vβ to equationΔv+Vea(ν1ν2)λβeveν1ν2+λβev=gβinR2, and by Theorem 2.2, there exists a unique bounded solution vβ toΔv+Vea(ν1ν2)λβΛ2eveν1ν2+λβΛ2ev=gβinR2. For β(2(NM),β), we first setwβ=vβ+βlnλ. Then wβ is the unique solution ofΔw+Vea(ν1ν2)eweν1ν2+ew=f1f2inR2 such that wβlnλ is bounded in R2.

When β=β, we setwβ=vβ+βlnλ2lnΛ, thus wβ is the

Non-topological solutions

Let β0 and uβ be a solution of problem (1.1) with the asymptotic behavioruβ(x)=βln|x|+O(1)as |x|+. Then uβ can be written under the formuβ=ν1+ν2+βlnλ+vβ, where vβ is a bounded solution of the following equation equivalent to (3.1)Δv+Wβev(eν1ν2λβ+ev)1+a=gβinR2 with Wβ=Vλaβ, and where gβ is expressed bygβ=f1f2+βΔlnλ. Note that it is a smooth function with compact support in Br0(0) and it verifiesR2gβdx=2π[2(NM)+β]. As for Wβ it satisfieslimxpjWβ(x)=A0(ij|pjpi|2ni)a,limxqjWβ(x

Nonexistence

Lemma 6.1

Let aN<1. Then

(i) Problem (1.1) has no solution uβ verifyinguβ(x)βln|x|=o(ln|x|)as|x|+ for β<β0.

(ii) Problem (1.1) has no solution uβ verifying (6.1) if 0β<2aNa.

(iii) Problem (1.1) has no topological solution.

Proof

We recall that a solution verifying (6.1) with β<0 (resp. β>0) is called non-topological of type II (resp. type I). Given a function, we denote by w the circular average of w, i.e.w(r)=12πrBr(0)w(ξ)dθ(ξ)=12π02πw(r,θ)dθ. For |x|r0, there exists c45>0 such thatP(x)c45|x|aN,

Acknowledgements

The authors are grateful to the referees for careful checking of the manuscript and relevant observations concerning some theoretical aspects of the physical theory.

H. Chen is supported by NSFC (No: 12071189), by the Jiangxi Provincial Natural Science Foundation (No: 20212ACB211005 and No: 20202ACBL20101), by the Science and Technology Research Project of Jiangxi Provincial Department of Education (No: GJJ200307 and No: GJJ 200325).

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