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In this paper, we consider curvature equations 0.1 Δ u + e u = 16 π δ 0 on E τ , and 0.2 Δ u + e u = 8 π (δ 0 + δ p + δ q) on E τ , p ≠ q , where 0.3 either p ≠ - q and ℘ ′ (p) + ℘ ′ (q) = 0 ∀ τ , or q = - p , ℘ ″ (p) = 0 and g 2 (τ) ≠ 0. Here E τ = C / Λ τ , Λ τ is the lattice generated by ω 1 = 1 and ω 2 = τ , τ ∈ H , the upper half plane. We prove, among other things that (i) If (0.1) has a solution u then there is a solution u p of (0.2) with p = (p , q) satisfying (0.3). Moreover, u p (z) is continuous with respect to p and uniformly converges to u(z) in any compact subset of E τ \ { 0 } as p → 0, however, u p blows up at z = 0 . This provides an example for describing blowing-up phenomena without concentration;(ii) If (0.2) is invariant under the change z → - z , i.e., (p, q) is either (ω i 2 , ω j 2) i ≠ j for any τ ∈ H , or q = - p and ℘ ″ (p) = 0 if g 2 (τ) ≠ 0 , then (0.1) has the same number of even solutions as (0.2). The converse of (i) remains open. In this paper, we establish a connection between curvature equations and the elliptic KdV theory. The results (i) and (ii) are proved by using this connection. [ABSTRACT FROM AUTHOR]

This article is the second part of our study of the spectrum σ (L n ; τ) of the Lamé operator L n = d 2 d x 2 - n (n + 1) ℘ (x + z 0 ; τ) in L 2 (R , C) , where n ∈ N , ℘ (z ; τ) is the Weierstrass elliptic function with periods 1 and τ , and z 0 ∈ C is chosen such that L n has no singularities on R . An endpoint of σ (L n ; τ) is called a cusp if it is an intersection point of at least three semi-arcs of σ (L n ; τ) . We obtain a necessary and sufficient condition for the existence of cusps in terms of monodromy datas and prove that σ (L n ; τ) has at most one cusp for fixed τ . We also consider the case n = 2 and study the distribution of τ 's such that σ (L 2 ; τ) has a cusp. For any γ ∈ Γ 0 (2) and the fundamental domain γ (F 0) , where F 0 : = { τ ∈ H | 0 ⩽ Re τ ⩽ 1 , | z - 1 2 | ⩾ 1 2 } is the basic fundamental domain of Γ 0 (2) , we prove that there are either 0 or 3 τ 's in γ (F 0) such that σ (L 2 ; τ) has a cusp and also completely characterize those γ 's. To prove such results, we will give a complete description of the critical points of the classical modular forms e 1 (τ) , e 2 (τ) , e 3 (τ) , which is of independent interest. [ABSTRACT FROM AUTHOR]

This paper, the third in a series, completes our description of all (radial) solutions on C ∗ of the tt*-Toda equations 2 (w i) t t ¯ = - e 2 (w i + 1 - w i) + e 2 (w i - w i - 1) , using a combination of methods from p.d.e., isomonodromic deformations (Riemann–Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data, holomorphic data, and asymptotic data of the global solutions. [ABSTRACT FROM AUTHOR]

In this paper, we show that as τ → - 1 ∞ , any zero of the Lamé function converges to either ∞ or a finite point p satisfying Re p = 1 2 and e 2 π i p being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line Re z = 1 2 . We also compute explicitly the eigenvalue set of the corresponding L 2 [ 0 , 1 ] eigenvalue problem for such KdV potentials, thus extends Takemura's works (Commun Math Phys 235:467–494, 2003) and (Electron J Differ Equ 2004(15):1–30, 2004). Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy et al. (Trans Am Math Soc 358:603–656, 2006) and the Darboux transformation. [ABSTRACT FROM AUTHOR]

We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter τ, we introduce a conceptual proof by applying Painlevé VI equation. As an application of this simple zero property, we obtain the smoothness of the degeneracy curves of trivial critical points for some multiple Green function. [ABSTRACT FROM AUTHOR]

We are concerned with Chern–Simons gauge theories of rank 2 in R 2 such as SU(3), SO(5), and G 2 Chern–Simons theory. One can reduce the self-dual equations of these models to a 2 × 2 system of elliptic equations. There exist three types of solutions of these elliptic system classified according to their asymptotic behaviors, namely, topological, nontopological, and mixed type solutions. In this paper, we consider the existence of radial mixed type solutions of these models. By counting the degree of the corresponding operator, we show the existence under a reasonable condition. Our result on the SU(3) Chern–Simons theory is an improvement of the existence result in Choe et al. (2017). Also, our results for the SO(5) and G 2 Chern–Simons theories are entirely new, to which the variational approach developed in Choe et al. (2017) cannot be applied. [ABSTRACT FROM AUTHOR]

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann-Hilbert problem. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the SU(3) Chern-Simons theory in $$\mathbb {R}^2$$ . It is believed that the self-dual equations of the theory possess three types of solutions, classified according to their (spatial) limiting behaviors. Solutions corresponding to the topological and the nontopological limiting behaviors have been shown to exist in the literature under various conditions. However, solutions satisfying the third limiting behavior, which we call mixed type solutions, have not yet been shown to exist explicitly. The purpose of this article is to show the existence of these mixed type solutions in a radially symmetric situation using the variational analysis incorporated with the bubbling argument. [ABSTRACT FROM AUTHOR]

It is well known that the presence of multiple constraints of non-Abelian relativisitic Chern-Simons-Higgs vortex equations makes it difficult to develop an existence theory when the underlying Cartan matrix K of the equations is that of a general simple Lie algebra and the strongest result in the literature so far is when the Cartan subalgebra is of dimension 2. In this paper we overcome this difficulty by implicitly resolving the multiple constraints using a degree-theorem argument, utilizing a key positivity property of the inverse of the Cartan matrix deduced in an earlier work of Lusztig and Tits, which enables a process that converts the equality constraints to inequality constraints in the variational formalism. Thus this work establishes a general existence theorem that settles a long-standing open problem in the field regarding the general solvability of the equations. [ABSTRACT FROM AUTHOR]

In this paper, we study mixed-type solutions of $${SU(3)}$$ Chern-Simons system (see (1.4) below) on a two dimensional flat torus. Nolasco and Tarantello (Commun Math Phys 213:599-639, ), among other things, Nolasco and Tarantello obtained solutions of (1.4) as minimizers of several functionals closely related to (1.4), and showed that if $${N_1+N_2=1}$$ , then one of those minimizers turns out to be a mixed-type solution, that is, one component tends to $${\ln\frac{1}{2}}$$ pointwise a.e. and the other component converges to a solution of a mean field equation. We call these kinds of solutions mixed-type (I) solutions. In this paper, we prove two main results: (i) the asymptotic analysis of mixed-type (I) solutions with arbitrary configuration of vortex points, and (ii) the existence of mixed-type (I) solutions under a non-degenerate condition. This non-degenerate condition also ensures some uniqueness result. In particular, our results imply that when $${N_1+N_2=1}$$ , there are only two mixed-type (I) solutions of (1.4). [ABSTRACT FROM AUTHOR]

We study qualitative properties of positive singular solutions to a two-coupled elliptic system with critical exponents. This system is related to coupled nonlinear Schrödinger equations with critical exponents for nonlinear optics and Bose-Einstein condensates. We prove a sharp result on the removability of the same isolated singularity for both two components of the solutions. We also prove the nonexistence of positive solutions with one component bounded near the singularity and the other component unbounded near the singularity. These results will be applied in a subsequent work where the same system in a punctured ball will be studied. [ABSTRACT FROM AUTHOR]

We consider a nonlinear elliptic equation derived from the Chern-Simons gauged $$O(3)$$ sigma model on a flat torus. Recently, it has been conjectured that there might exist a stable nontopological entire solution in $$\mathbb {R}^2$$ under certain conditions. In this paper, we construct bubbling stable solutions on a flat torus which blow up at a vortex point. Our result shows that some global condition on the coefficients in Bartolucci et al. (Ann Inst Henri Poincaré Anal Non Linéaire. doi:, ) is necessary for the equivalence result between stable solutions and topological solutions. Moreover, we investigate other types of bubbling solutions including mountain pass solutions and analyze the asymptotic behavior of general solutions on a flat torus. [ABSTRACT FROM AUTHOR]

In Guest et al. () (part I) we computed the Stokes data for the smooth solutions of the tt*-Toda equations whose existence we had previously established by p.d.e. methods. Here we formulate the existence problem as a Riemann-Hilbert problem, based on this Stokes data, and solve it under certain conditions (Theorem 5.4). In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data (Theorem 5.5). We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions (Theorem 4.1), clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, we resolve some conjectures of Cecotti and Vafa concerning the positivity of S + S (where S is the Stokes matrix) and the unimodularity of the eigenvalues of the monodromy matrix (Theorem 5.6). In particular, we show that 'unitarity implies regularity' for the tt*-Toda equations. [ABSTRACT FROM AUTHOR]

In this paper, we study non-topological solutions ( u, u) to the SU(3) Chern-Simons system, satisfying the asymptotic behavior $${u_k(x)=-2\alpha_k\ln |x|+O(1)}$$ for some $${\alpha_k > 1, k = 1, 2}$$ . We show that solutions remain uniformly bounded as long as ( α, α) lies in S (see Sect. 1), and then we prove the existence of non-topological solutions for any $${(\alpha_1,\alpha_2)\in S_N}$$ . For this purpose, we have to study the phenomena of partial blowup at infinity, and show that S is the optimal range of ( α, α) such that the partial blowup could occur if ( α, α) is on some part of $${\partial S_N}$$ . [ABSTRACT FROM AUTHOR]

We consider the mean field equation: where $$\varOmega \subset \mathbb {R}^2$$ is an open and bounded domain of class $$C^1$$ . In his 1992 paper, Suzuki proved that if $$\varOmega $$ is a simply-connected domain, then Eq. (1) admits a unique solution for $$\rho \in [0,8\pi )$$ . This result for $$\varOmega $$ a simply-connected domain has been extended to the case $$\rho =8\pi $$ by Chang, Chen and the second author. However, the uniqueness result for $$\varOmega $$ a multiply-connected domain has remained a long standing open problem which we solve positively here for $$\rho \in [0,8\pi ]$$ . To obtain this result we need a new version of the classical Bol's inequality suitable to be applied on multiply-connected domains. Our second main concern is the existence of solutions for (1) when $$\rho =8\pi $$ . We obtain a necessary and sufficient condition for the solvability of the mean field equation at $$\rho =8\pi $$ which is expressed in terms of the Robin's function $$\gamma $$ for $$\varOmega $$ . For example, if Eq. (1) has no solution at $$\rho =8\pi $$ , then $$\gamma $$ has a unique nondegenerate maximum point. As a by product of our results we solve the long-standing open problem of the equivalence of canonical and microcanonical ensembles in the Onsager's statistical description of two-dimensional turbulence on multiply-connected domains. [ABSTRACT FROM AUTHOR]

In this paper, we study the entire radial solutions of the self-dual equations arising from the relativistic SU(3) Chern-Simons model proposed by Kao and Lee (Phys Rev D 50:6626-6632, ) and Dunne (Phys Lett B 345:452-457, ; Nuclear Phys B 433:333-348, ). Understanding the structure of entire radial solutions is one of the fundamental issues for the system of nonlinear equations. In this paper, we prove that any entire radial solutions must be one of topological, non-topological and mixed type solutions, and completely classify the asymptotic behaviors at infinity of these solutions. Even for radial solutions, this classification has remained an open problem for many years. As an application of this classification, we prove that the two components u and v have intersection at most finite times. [ABSTRACT FROM AUTHOR]

We study the existence of bubbling solutions for the the following Chern-Simons-Higgs equation: where Ω is a torus. If k = 1, for any critical point q of the associated sum of the Green functions, we introduce a quantity D( q) (see (1.11) below). We show that for any non-degenerate critical point q with D( q) < 0, the above problem has a solution u satisfying that ε → 0, u blows up at q. The calculations in this paper also show that, if a sequence of solutions u blows up at q as ε → 0, then q must be a critical point of the associated sum of the Green functions, and $${D(q) \leqq 0}$$ . So, the condition D( q) < 0 is almost necessary to obtain our result. We also construct solutions with k bubbles for $${k \geqq 2}$$ . [ABSTRACT FROM AUTHOR]

In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU( n + 1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to $${\mathbb{CP}^n}$$. In order to compute the Leray-Schcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our main goals in this paper are (i) to obtain a sharp convergence rate, (ii) to completely determine the locations, and (iii) to derive the $${\partial _z^2}$$ condition, a unexpected and important geometric constraint. [ABSTRACT FROM AUTHOR]

We consider the following Toda system [Equation not available: see fulltext.] where γ>−1, δ is Dirac measure at 0, and the coefficients a form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: This generalizes the classification result by Jost and Wang for γ=0, $\forall\;1\leq i\leq n$. (ii) We prove that if γ+ γ+⋯+ γ∉ℤ for all 1≤ i≤ j≤ n, then any solution u is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, where 0 ≦ s < s ≦ 2, $${0 \ne \lambda \in \mathbb {R}}$$ and $${0 \in\partial \Omega}$$ . The existence (or nonexistence) for least-energy solutions has been extensively studied when s = 0 or s = 0. In this paper, we prove that if 0 < s < s < 2 and the mean curvature of $${\partial \Omega}$$ at 0 H(0) < 0, then (0.1) has a least-energy solution. Therefore, this paper has completed the study of (0.1) for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of (0.1) with $${\Omega =\mathbb {R}_{+}^{N}}$$ under different situations of s, s and λ. [ABSTRACT FROM AUTHOR]

We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of non-Abelian multiple vortex equations governing coupled SU( N) and U(1) gauge and Higgs fields which may be embedded in a supersymmetric field theory framework. When the underlying domain is doubly periodic, we prove the existence and uniqueness of an n-vortex solution under a necessary and sufficient condition explicitly relating the domain size to the vortex number n and the Higgs boson masses. When the underlying domain is the full plane, we use a constructive approach to establish the existence and uniqueness of an n-vortex solution. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem where B is a ball in R n. For a small d>0, we show the uniqueness (up to rotation) of the one-bubbling solution which concentrates at a point of the boundary. [ABSTRACT FROM AUTHOR]

In this article, we construct self-dual N-vortex solutions with a large magnetic flux Φ of (2+1)-dimensional relativistic Chern-Simons model, provided that the coupling constant κ is small and the cites of vorticity satisfies . Our solutions exhibit the bubbling phenomenon at each p j. Near each vortex p j, solutions are locally asymptotically symmetric with respect to p j, and the curvature F12 tends to a sum of Dirac measures as κ tends to zero. By a heuristic argument, it is shown that (0.1) is also a necessary condition for the existence of multi-vortex solutions which has a locally asymptotically symmetric vortex at . [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the existence of non-negative weak solution u of having a given closed set S as its singular set. We prove that when $$\frac{n}{{n - 2}}< p< \frac{{n + 2\sqrt {n - 1} }}{{n - 4 + 2\sqrt {n - 1} }}$$ and S is a closed subset of Ω, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal scalar curvature equation for n ≥ 9, we construct a weak solution $$u \in L^{\frac{{n + 2}}{{n - 2}}} \left( {S^n } \right)$$ of $$L_0 u + L^{\frac{{n + 2}}{{n - 2}}} = 0$$ such that Sn is the singular set of u where L0 is the conformal Laplacian with respect to the standard metric of Sn. When n = 4 or 6, this kind of solution has been constructed by Pacard. [ABSTRACT FROM AUTHOR]

A conjecture about the nodal line of a second eigenfunction states that the nodal line of a second eigenfunction divides the domain Ω by intersecting with the boundary of Ω transversely, where Ω is a bounded convex domain of R. We prove this conjecture provided Ω has a symmetry. Also, we prove the multiplicity of the second eigenvalue is two at most provided Ω is a bounded convex domain of R. [ABSTRACT FROM AUTHOR]

In this article, we prove there are no positive smooth solutions of $$ \Delta u +K(x) u^{\frac{n+2}{n-2}} =0 \quad \text{ in } \mathbb{R}^n , \leqno(0.1) $$ where $K(x)\in C^1(\mathbb{R}^n)$ satisfies one of the following conditions: (i) K is a subharmonic function in ${\mathbb{R}}^n$ with $K(\infty) =\lim\limits_{|x|\to +\infty}K(x) >0$ , and the derivative $|\nabla K(x)|$ satisfies $$ c_1|x|^{-(l+1)}\leq |\nabla K(x)|\leq c_2|x|^{-(l+1)} , $$ where $l>\frac{1}{2}$ for $n=3$ , $l>1$ for $4\leq n\leq 6$ and $l\geq \frac{n-2}{2}$ for $n\geq 7$ . (ii) $K(x) \neq \text{constant}$ is nondecreasing along each ray $\{ t\xi | t\geq 0\}$ for any unit vector $\xi$ in $\mathbb{R}^n$ and $\lim\limits_{|x|\to +\infty}K(x)=K(\infty) >0$ . (iii) $K(x_1, \cdots , x_n) \equiv K(x_1)$ is nondecreasing in $x_1$ , $K\equiv K_1$ for $x_1\leq b$ and $K\equiv K_2>0$ for $x_1\geq a$ , where $K_1,K_2,a$ and b are constants. Various generalizations to a more general class of nonlinearities are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we prove that solutions minimizing the nonlinear functional among the Sobolev space H (Ω) are unique when Ω is bounded convex domain in ℝ. This uniqueness's results is equivalent to saying that solutions obtained from the Mountain Pass Lemma for the equation Δ u+ u =0 are unique. We also prove that the level set of the unique solution is strictly convex. [ABSTRACT FROM AUTHOR]