Qualitative properties of solutions to semilinear elliptic equations from the gravitational Maxwell gauged O(3) sigma model
Introduction
In this paper our goal is to classify the solutions of the following equation with measure data where (resp. ) are Dirac masses concentrated at the points , (resp. ), for , the related coefficients and are positive integers, is a given constant, 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 . This means that physically speaking the exponent a is very small. Set Since we define the notion of weak solution as follows: Definition 1.1 A function such that is called a weak solution of (E), if for any ,
We denote by 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 , a weak solution of (1.4) belongs to and is a strong solution of The nonlinear term is not monotone, actually the function is increasing on , and decreasing on . This makes the structure of solutions of our problem much more complicated than the case where .
Equation (1.1) comes from the Maxwell gauged sigma model. When , it governs the self-dual 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 -invariant action density [24, p. 43-49]: where , is a spin vector defined over the -dimensional Minkowski spacetime , with value in the unit sphere , i.e. , are gauge-covariant derivatives on ϕ, defined by and is the electromagnetic curvature induced from the 3-vector connection , as detailed in [26, p. 177-189]. When the time gauge is zero, that is in the static situation, the functional of total energy can be expressed by the following expressions where denotes the Brouwer's degree of ϕ. The related Bogomol'nyi equation is obtained by using the stereographic projection from the south pole of onto (see e.g. [5], [26] for details). Then the function satisfies It is pointed out in [24] that the points (), which are the poles of can be viewed as magnetic monopoles and the points (), which are the zeros of 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: Using the variable u its value coincides with (the Laplacian being taken a.e.). Thus, for the sake of simplicity, we identify and , an expression which will be called the total flux in the sequel. Here and in what follows, we denote
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 , then problem (1.6) has no solution. If , then for any problem (1.6) has a unique solution verifying with the following behavior as , Furthermore the correspondence is decreasing. If and u is a non-topological solution of (1.6) with finite total magnetic flux, i.e. , then there exists a unique such that .
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 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 , equation (1.1) governs the gravitational Maxwell gauged sigma model restricted to a plane. Because of the gravitational interaction between particles, the Lagrangian density becomes with stress energy tensor We simplify the Einstein equation where is the Ricci tensor and R is a scalar tensor of the metric in considering a metric conformal to the (2 + 1)-dimensional Minkowski one Then where The minimum of the energy is achieved if and only if satisfies the self-dual equations (the Bogomol'nyi equations) 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. and . For multiple poles, Chae showed in [5] that problem (1.1) has a sequence of non-topological solutions such that for , when Under the assumption (1.8), the existence of solutions has been improved up to the range 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 be which, due to the potential and the decay to zero for as , allows the existence of solutions with very wild behaviors at infinity. In fact, the following three types of solutions are considered in this paper
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: where
Notice that if for some j or , otherwise is finite, in this case, a free parameter should be taken into account. If , we have that .
Theorem 1.2 Let , and be the total magnetic flux given in (1.9). (i) If then for any , problem (1.1) possesses a minimal solution satisfying Moreover, for some real number , and the total magnetic flux of the solution is equal to , i.e. If then and for any , problem (1.1) possesses a sequence of non-topological solutions of type I satisfying where Moreover, the total magnetic flux of the solutions is equal to .
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 verifying 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 implies that , and our second interest is to consider this extremal case , which is under the assumption (1.12).
Theorem 1.3 Assume that , , the magnetic flux is given by (1.9) and let (1.12) hold. Then problem (1.1) possesses a minimal non-topological solution satisfying and the total magnetic flux of is equal to .
The existence of non-topological states of type II to (1.1) states as follows.
Theorem 1.4 Assume that , and is given by (1.10), then , problem (1.1) possesses a sequence of non-topological solutions such that where Moreover, the total magnetic flux of the solutions is equal to .
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 , and (1.15) hold true. Then problem (1.1) possesses infinitely many topological solutions satisfying where Moreover, the total magnetic flux of the solutions is equal to .
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 , 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 sigma model:
(i) the set of solutions is extended to topological and two types of non-topological solutions;
the uniqueness fails for the solution under the given condition at infinity;
the numbers (counted with multiplicity) of magnetic poles do no longer verify . In fact, for the non-topological solution of type I, it becomes , but for the non-topological solution of type II, there is no restriction on N and M, if is large enough.
The biggest difference with the case that 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 for (1.1) with the behavior at infinity for some β.
Theorem 1.6 Assume that and . (i) If and , then problem (1.1) has no solution with the aymptotic behavior If , 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 and such that Set where is chosen such that any two balls of the set do not intersect. We fix a positive number large enough such that for and , and we denote If u
Minimal solution
In order to consider solutions w of (2.3) with asymptotic behavior , we look for w under the form where v is a bounded function satisfying some related equation. In particular, we look for non-topological solution of problem (1.1) under the form where λ is given by (2.30) and is a bounded classical solution of with V being defined in (2.4) and where is defined in (2.34). Here and in what follows,
Non-topological solutions
If and , we recall that by Theorem 2.1, for any , with , there exists a unique bounded solution to equation and by Theorem 2.2, there exists a unique bounded solution to For , we first set Then is the unique solution of such that is bounded in .
When , we set thus is the
Non-topological solutions
Let and be a solution of problem (1.1) with the asymptotic behavior Then can be written under the form where is a bounded solution of the following equation equivalent to (3.1) with , and where is expressed by Note that it is a smooth function with compact support in and it verifies As for it satisfies
Nonexistence
Lemma 6.1 Let . Then (i) Problem (1.1) has no solution verifying for . Problem (1.1) has no solution verifying (6.1) if . Problem (1.1) has no topological solution. Proof We recall that a solution verifying (6.1) with (resp. ) is called non-topological of type II (resp. type I). Given a function, we denote by the circular average of w, i.e. For , there exists such that
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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