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The KP‐I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution, which is a traveling wave, and the KP‐I equation in this case reduces to the Boussinesq equation. In this paper we classify all the 'lump‐type' solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman–Miller for the Schrödinger equation, we show that the lump‐type solutions are rational and their τ$\tau $ functions have to be polynomials of degree k(k+1)$k(k+1)$ for some integer k$k$. In particular, this implies that the lump solution is the unique ground state of the KP‐I equation (as conjectured by Klein–Saut). The problem studied in this paper was mentioned in Airault–McKean–Moser, our result can be regarded as a two‐dimensional analogy of their theorem on the classification of rational solutions for the KdV equation. [ABSTRACT FROM AUTHOR]

We investigate the existence of blowing-up solutions to the following singular Liouville problem$ -\Delta u = \lambda V_ \lambda(|x|) e^u-4\pi N {\mathit{\boldsymbol{\delta}}}_0\;\hbox{ in } B_1,\quadu = 0 \;\hbox{ on }\partial B_1, $where $ B_1 $ is the unit ball in $ \mathbb{R}^2 $ centered at the origin, $ V_ \lambda(|x|) $ is a positive smooth potential, $ N $ is a positive integer ($ N\geq 1 $), $ {\mathit{\boldsymbol{\delta}}}_0 $ defines the Dirac measure with pole at $ 0 $, and $ \lambda>0 $ is a small parameter. If the potential $ V_ \lambda(|x|) $ satisfies some suitable assumptions in terms of the first $ 2(N+1) $ derivatives at $ 0 $, then we find a solution which exhibits a non-simple blow-up profile as $ \lambda \to 0^+ $. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of blowing-up solutions to the following boundary value problem - Δ u = λ V (x) e u - 4 π N δ 0 in B 1 , u = 0 on ∂ B 1 , where B 1 is the unit ball in R 2 centered at the origin, V(x) is a positive smooth potential, N is a positive integer ( N ≥ 1 ). Here δ 0 defines the Dirac measure with pole at 0, and λ > 0 is a small parameter. We assume that N = 1 and, under some suitable assumptions on the derivatives of the potential V at 0, we find a solution which exhibits a non-simple blow-up profile as λ → 0 + . [ABSTRACT FROM AUTHOR]

We show that the bounded solutions to the fractional Helmholtz equation, (− Δ) s u = u for 0 < s < 1 in ℝ n , are given by the bounded solutions to the classical Helmholtz equation (− Δ) u = u in ℝ n for n ≥ 2 when u is additionally assumed to be vanishing at ∞. When n = 1 , we show that the bounded fractional Helmholtz solutions are again given by the classical solutions A cos x + B sin x. We show that this classification of fractional Helmholtz solutions extends for 1 < s ≤ 2 and s ℕ when u C ∞ (ℝ n). Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation ψ (− Δ) u = ψ (1) u in ℝ n , when ψ is complete Bernstein and certain regularity conditions are imposed on the associated weight a (t). [ABSTRACT FROM AUTHOR]

In this paper, we study the local behavior of positive singular solutions to the equation (− Δ) σ u = u n n − 2 σ i n B 1 ∖ { 0 } (− Δ) σ u = u n n − 2 σ i n B 1 ∖ { 0 } where (− Δ) σ (− Δ) σ is the fractional Laplacian operator, 0 < σ < 1 0 < σ < 1 and n n − 2 σ n n − 2 σ is the critical Serrin exponent. We show that either u u can be extended as a continuous function near the origin or there exist two positive constants c 1 c 1 and c 2 c 2 such that c 1 | x | 2 σ − n (− ln | x |) − n − 2 σ 2 σ ≤ u (x) ≤ c 2 | x | 2 σ − n (− ln | x |) − n − 2 σ 2 σ i n B 1 ∖ { 0 }. c 1 | x | 2 σ − n (− ln ⁡ | x |) − n − 2 σ 2 σ ≤ u (x) ≤ c 2 | x | 2 σ − n (− ln ⁡ | x |) − n − 2 σ 2 σ i n B 1 ∖ { 0 }. [ABSTRACT FROM AUTHOR]

In this article, we continue with the research initiated in our previous work on singular Liouville equations with quantized singularity. The main goal of this article is to prove that as long as the bubbling solutions violate the spherical Harnack inequality near a singular source, the first derivatives of coefficient functions must tend to zero. [ABSTRACT FROM AUTHOR]

For the generalized surface quasi-geostrophic equation { ∂tθ +u ⋅ ∇ θ = 0, in R2 × (0,T), u = ∇⊥ ψ, ψ = (−Δ)−s θ in R2 × (0,T), 0 < s < 1, we consider for k ≥ 1 the problem of finding a family of k-vortex solutions θε (x,t) such that as ε → 0 { θε (x,t) → ∑j=1k mjδ (x-ξj(t) for suitable trajectories for the vortices x = ξj(t). We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem { (−Δ)sW = (W−1)γ+, in R2, 1 < γ < 1+s}/1−s whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the magnetic Ginzburg–Landau equation with external potentials μV(x) for the type II superconductors. We prove, by reduction arguments, that under suitable conditions on V(x), the magnetic Ginzburg–Landau equation with external potentials in R 2 has infinitely many multi-vortex solutions for μ > 0 being sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in ℝn, whose level set {u = 0} is contained in a half-space. If n ≤ 3, we prove that the solution must be one-dimensional. In dimension n ≥ 4, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained. [ABSTRACT FROM AUTHOR]

In this paper we study the following conformally invariant poly-harmonic equation Δmu = −u3+2/m3−2m in R3, u > 0, with m = 2,3. We prove the existence of positive smooth radial solutions with prescribed volume ∫R3u6/3−2mdx. We show that the set of all possible values of the volume is a bounded interval (0,Λ∗] for m = 2, and it is (0,∞) for m = 3. This is in sharp contrast to m = 1 case in which the volume ∫R3u6/3−2mdx is a fixed value. [ABSTRACT FROM AUTHOR]

Let |$(M, g)$| be a smooth compact |$n$| -dimension Riemannian manifold. We are concerned with the existence of sign-changing blow-up solutions to the following elliptic problem   \begin{eqnarray*} \Delta_g u+h u=|u|^{\frac{4}{n-2}-\varepsilon}u,\ \ \ \ \ in \ \ M, \end{eqnarray*} where |$\Delta_g=-{\rm div}_g(\nabla)$| is the Laplace–Beltrami operator on |$M$|⁠ , |$h $| is a |$\mathcal {C}^1$| function on |$M$|⁠ , |$\varepsilon$| is a small real parameter such that |$\varepsilon$| goes to 0. [ABSTRACT FROM AUTHOR]

For the Neumann sinh-Gordon equation on the unit ball $${B \subset \mathbb {R}^2}$$ we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ ± are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276]. [ABSTRACT FROM AUTHOR]

Abstract: We consider the following anisotropic Emden–Fowler equation where is a smooth bounded domain and a is a positive smooth function. We study here the phenomenon of boundary bubbling solutions which do not exist for the isotropic case a≡constant. We determine the localization and asymptotic behavior of the boundary bubbles, and construct some boundary bubbling solutions. In particular, we prove that if is a strict local minimum point of a, there exists a family of solutions such that tends to in as . This result will enable us to get a new family of solutions for the isotropic problem in rotational torus of dimension . [Copyright &y& Elsevier]

Abstract: We consider the anisotropic Emden–Fowler equation: in Ω, on ∂Ω where is a smooth bounded domain and is a positive, smooth function. We investigate the effect of anisotropic coefficient on the existence of bubbling solutions. We show that at given strict local maximum points of a, there exist solutions with arbitrarily many bubbles. As a consequence, the quantity can approach to +∞ as . These results show a striking difference with the isotropic case (). To cite this article: J. Wei et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: Conventionally, to learn wave collapse and optical turbulence, one must study finite-time blow-up solutions of one-component self-focusing nonlinear Schrödinger equations (NLSE). Here we consider simultaneous blow-up solutions of two-component systems of self-focusing NLSE. By studying the associated self-similar solutions, we prove two components of solutions blow up at the same time. These self-similar solutions may come from solitary wave solutions with multi-bumps forming abundant geometric patterns which cannot be found in one-component self-focusing NLSE. Our results may provide the first step to investigate optical turbulence in two-component systems of NLSE. [Copyright &y& Elsevier]

Abstract: We consider the following Brezis–Nirenberg problem on where D is a geodesic ball on with geodesic radius , and is the Laplace–Beltrami operator on . We prove that for any and for every with sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394]. To cite this article: W. Chen, J. Wei, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

We establish some general theorems for the existence and nonexistence of ground state solutions of steady-stateNcoupled nonlinear Schrödinger equations. The sign of coupling constantsß ij’s is crucial for the existence of ground state solutions. When allß ij’s are positive and the matrix S is positively definite, there exists a ground state solution which is radially symmetric. However, if allß ij’s are negative, or one ofß ij’s is negative and the matrix S is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric whenN=3. [ABSTRACT FROM AUTHOR]

We study stationary solutions of the Cahn--Hilliard equation in a bounded smooth domain that have an interior spherical interface (bubbles). We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exists such a solution whose bubble center lies close to a given nondegenerate peak point. Our construction uses, among others, the Liapunov--Schmidt reduction method and exponential asymptotics. [ABSTRACT FROM AUTHOR]

For Liouville equations with singular sources, the interpretation of the equation and its impact are most significant if the singular sources are quantized: the strength of each Dirac mass is a multiple of 4 π. However the study of bubbling solutions around a quantized singular source is particularly challenging: near the singular source, the spherical Harnack inequality may not hold and there are multiple local maximums of bubbling solutions all swarming to the singular source. In this article we seek to provide a complete understanding of the blowup picture in this core difficulty and we establish two major types of results: first we prove that not only the first derivatives of coefficient functions tend to zero at the singular source, the second derivatives also have a vanishing estimate. Second we derive pointwise estimates for bubbling solutions to be approximated by global solutions, which have crucial applications in a number of important projects. Since bubbling solutions near a singular source are very commonly observed in geometry and physics, it seems that the estimates in this article can also be applied to many related equations and systems with various backgrounds. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of blowing-up solutions to the following boundary value problem − Δ u = λ e u − 4 π N λ δ 0 in Ω , u = 0 on ∂ Ω , where Ω is a smooth and bounded domain in R 2 such that 0 ∈ Ω , N λ is a positive number close to an integer N (N ≥ 1) from the right side, δ 0 defines the Dirac measure with pole at 0, and λ > 0 is a small parameter. We assume that Ω is (N + 1) -symmetric and the regular part of the Green's function satisfies a nondegeneracy condition (both assumptions are verified if Ω is the unit ball) and we find a solution which exhibits a non-simple blow-up profile as λ → 0 +. [ABSTRACT FROM AUTHOR]

We prove the existence of smooth non-radial entire solution to Δ 2 u + u - q = 0 in R 3 , u > 0 , for q > 1 . This answers an open question raised by McKenna and Reichel (Electron J Differ Equ 37:1–3, 2003). [ABSTRACT FROM AUTHOR]

Abstract We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R 2 , where the intraspecies interaction (− a 1 , − a 2) and the interspecies interaction − β are both attractive, i. e , a 1 , a 2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L 2 -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β ↗ β ⁎ = a ⁎ + (a ⁎ − a 1) (a ⁎ − a 2) , where 0 < a i < a ⁎ : = ‖ w ‖ 2 2 (i = 1 , 2) is fixed and w is the unique positive solution of Δ w − w + w 3 = 0 in R 2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. [ABSTRACT FROM AUTHOR]

We prove the existence of nontrivial global minimizers of the Allen–Cahn equation in dimension 8 and above. More precisely, given any strict area-minimizing Lawson's cone, there is a family of global minimizers whose nodal sets are asymptotic to this cone. As a consequence of Jerison–Monneau's program we then establish the existence of many new counter-examples to the De Giorgi conjecture whose nodal sets are different from the Bombieri–De Giorgi–Giusti minimal graph. [ABSTRACT FROM AUTHOR]