In this paper, we investigate a sequence of multi-bubble solutions {u(k)} to the following mean field equation: (-Delta)(n)u(k) = rho(k) h(x)e(uk)/integral(Omega)he(ukdx), in Omega, B(j)u(k) = 0, j=0, ..., n - 1, on partial derivative Omega, where Omega is a bounded and smooth domain in R-2n with n >= 2 as a positive integer, h is a C-2,C-beta positive function, rho(k) are constants such that 0 < rho(k) <= C, for some constant C and B-j, j = 0, ..., n - 1 stand for either Navier or Dirichlet boundary conditions. We show that (after passing to a subsequence if necessary) lim(k ->+infinity) rho(k) = 2(2n+1) n!pi(n)m for some positive integer m. Furthermore, we obtain the following sharp estimates of rho(k): rho(k) - 2(2n+1)n!pi(n)m = c(0) Sigma(m)(j=1)(h(p(k,j)))(-1/n) (epsilon k,j2)[1/2(2n+1)n!pi(n) Delta log h(p(k,j)) +Delta R-2n(p(k,j), p(k,j)) + Sigma(i not equal j)Delta G(2n)(p(k,j), p(k,i))] + o(Sigma(m)(j=1) is an element of(2)(k,j)), where c(0) is a positive generic constant, G(2n) is the Green function of (-Delta)(n) with either Navier or Dirichlet boundary conditions, R-2n is the regular part of G(2n), p(k,j) is the local maximum point of uk in a neighborhood of p(j) with p(j) as the blow-up point of {u(k)} for each j = 1, ..., m, and log (2n)!4(n) /is an element of(2n)(k,j) = u(k)(p(k,j)) - log(integral(Omega) he(uk)dx). Our approach extends the works of Chen-Lin [Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math.55 (2002) 728-771] and Lin-Wei [Sharp estimates for bubbling solutions of a fourth order mean field equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)6(4) (2007) 599-630], which studied the second-order and fourth-order equations, to the general even-order equation. Moreover, this result also hold for other boundary condition, if boundary blow-up can be excluded.

In this paper, the authors study the elliptic system arising from the study of Maxwell-Chern-Simons model. They show that there exists a family of non-topological solutions with magnetic field concentrated at some of the vortex points as the two physical parameters satisfying almost optimal conditions where the limiting profile is the singular Liouville equation.

We consider a reaction-diffusion system for color pattern formation with two activators and one inhibitor. Each of the activators models one of the colors being switched on, for example the first activator could represent the color blue and the second activator the color yellow. If both colors are present the pattern will have green color since the color green is achieved by a mixture of the colors blue and yellow. We prove rigorous results on the existence and stability of spikes for which one of the colors or both of them are switched on. To the best of our knowledge, this paper is the first study of spike solutions for a reaction-diffusion system with two activator and one inhibitor systems and arbitrary strength of the self-activation and cross-activation terms. We classify the different types of solutions which can exist depending on the choice of interaction parameters between the components and we show which of them are stable or unstable. In particular, solutions with spikes for both activators in the same position can be stable when cross-activation dominates over self-activation. On the other hand, solutions with a spike for only one activator and zero concentration for the other activator can be stable when self-activation dominates over cross-activation. The rigorous approach is based on analytical methods such as Green's function, Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical results are confirmed by numerical simulations.

We study the limiting behavior of solutions to nonlinear Schr & ouml;dinger equations -epsilon(2)Delta u(epsilon) + u(epsilon) = u(epsilon)(p), u(epsilon) > 0 in R-n, as epsilon -> 0, where p is Sobolev subcritical. These solutions are assumed to have infinitely many peaks. We derive the interaction form between the limiting peak points. This is achieved by first describing the main order term of u pound and providing a very precise estimate on the error by the reverse Lyapunov-Schmidt reduction method, and then extracting information from the reduction equation in a limiting way. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We present Liouville-type results for stable solutions and finite Morse index solutions of the polyharmonic Hénon elliptic equation (-Δ )^mu=|x|^a|u|^p-1u, in ℝ^n, where m≥ 3 , p>1 , a≥ 0 and n is a large dimension. To construct the classification theorem of homogeneous stable solutions, we exhibit a concise and explicit form for a critical exponent p_a(n,m) which is known as the Joseph-Lundgren exponent when taking a=0 . Based on this, we obtain the desired results by establishing the monotonicity formula, applying some energy estimates and finally using a blow-down analysis.

For the three dimensional axisymmetric Euler flow, we construct a family of solutions with multiple travelling vortex rings, with large speed of order O(|lnɛ|), where ɛ>0 is a small parameter. Our construction is based on the analysis of the following nonlinear elliptic equation: ∂rrψ+3r∂rψ+∂zzψ=−F((ψ−α2|lnɛ|)r2),(r,z)∈R2,ψr(0,z)=0forr=0,for some special functions F, where α is a parameter. The location of the vortex rings is governed by some balancing systems, which can be solved by the polynomial method in several special cases. For the non-swirl case, in the core of each vortex ring, our solutions can be regarded as a rescaled finite mass solution of the Liouville equation. The results can be generalized directly to the case with swirl, for which we also construct different types of solutions with multiple vortex rings.

In this paper, we study a three-component consumer chain model which is based on Schnakenberg type kinetics in a two-dimensional domain. In the model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. Through a rigorous analysis, we show that there exist two different single spike solutions if the feed rates are small. Further, we also establish the stability results: If the time-relaxation constants for both producer and the second consumer vanish, the large amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are positive. We show a new effect that if the time-relaxation constant of the second consumer is bounded, the large-amplitude spike solution is still stable while it is unstable in the one-dimensional case.

Given a smooth bounded domain \begin{document}$ \Omega\subset{{\mathbb R}}^2 $\end{document}, we consider the following Ambrosetti-Prodi problem with Neumann boundary: \begin{document}$ \begin{equation*} \left\{\begin{array}{l} -\Delta u = \left\vert{u}\right\vert^p-\sigma \quad {\mbox {in}} \ \Omega,\\ {\partial u \over \partial \nu} = 0 \quad {\mbox {on}} \ \partial \Omega. \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ p>2 $\end{document}, \begin{document}$ \sigma>0 $\end{document} is a large parameter and \begin{document}$ \nu $\end{document} denotes the outward normal of \begin{document}$ \partial \Omega $\end{document}. We constructed a new class of solutions comprised of a large number of spikes concentrated on a segment of the boundary containing a local minimum point of the mean curvature function and having the same mean curvature at the endpoints. A similar boundary-concentrating phenomenon was obtained for the Lin-Ni-Takagi problem by Ao et al. [3].

In this paper, we will consider the fractional Caffarelli-Kohn-Nirenb erg inequality & nbsp;lambda(integral(Rn)& nbsp;|u(x)|(p)/xI(beta P) dx)(2/p) <= integral(Rn)integral(Rn) (u(x) - u(y))(2)/ |x - y|(n+2 gamma)|x|(alpha)|y|(alpha) dy dx & nbsp;where gamma is an element of (0, 1), n & nbsp;>= 2, and alpha, beta & nbsp;is an element of R satisfy & nbsp;alpha & nbsp;<= beta & nbsp;<= alpha + gamma, -2 gamma < alpha < n - 2 gamma/2,& nbsp;and the exponent p is chosen to be & nbsp;p = 2n/n- 2 gamma + 2(beta - alpha),& nbsp;such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as alpha & nbsp;-> -2 gamma. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenb erg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE.& nbsp;For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non local Wronskian of Ao-Chan-DelaTorre-Fontelos-Gonzalez-Wei (Duke'19). (C) 2022 Elsevier Inc. All rights reserved.

We consider the fractional elliptic problem: where B 1 is the unit ball in ℝ N , N ⩾ 3, s ∈ (0, 1) and p > ( N + 2 s )/( N − 2 s ). We prove that this problem has infinitely many solutions with slow decay O (| x | −2 s /( p −1) ) at infinity. In addition, for each s ∈ (0, 1) there exists P s > ( N + 2 s )/( N − 2 s ), for any ( N + 2 s )/( N − 2 s ) < p < P s , the above problem has a solution with fast decay O (| x | 2 s − N ). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.

We consider the elliptic system problems arising from the Maxwell-Chern-Simons model. In this system, there are two important parameters related to Chern-Simons mass scale and electric charge. Under almost optimal conditions on these two parameters, we show the existence of nontopological condensates with magnetic field concentrated at multi-bubbling points.

We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-\Delta)^\gamma u=fu^p\quad\text{in }\mathbb R^n\setminus\Sigma $$ which are singular at $\Sigma$. Here $\gamma \in (0, \frac{n}{2})$. Among other things, we first prove that if $\Sigma$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}\frac{n-\bf d}{n-{\bf d}-2\gamma},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solution in $\mathbb R^n$. Then we prove that $ (-\Delta)^\sigma u >0$ for all $ \sigma \in (0, \gamma)$, if $\Sigma=\phi$.

We are concerned with the Sinh-Gordon equation in bounded domains. We construct blow up solutions with residual mass exhibiting either partial or asymmetric blow up, i.e. where both the positive and negative part of the solution blow up. This is the first result concerning residual mass for the Sinh-Gordon equation showing in particular that the concentration-compactness theory with vanishing residuals of Brezis-Merle can not be extended to this class of problems.

We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

New finite energy traveling wave solutions with small speed are constructed for the three dimensional Gross-Pitaevskii equation. These solutions have the shape of 2n + 1 vortex rings, far away from each other. Among these vortex rings, n + 1 of them have positive orientation and the other n of them have negative orientation. The location of these rings are described by the roots of a sequence of polynomials with rational coefficients. The polynomials found here can be regarded as a generalization of the classical Adler-Moser polynomials and can be expressed as the Wronskian of certain very special functions.

For the generalized surface quasi-geostrophic equation { a m p ; ∂ t θ + u ⋅ ∇ θ = 0 , in R 2 × ( 0 , T ) , a m p ; u = ∇ ⊥ ψ , ψ = ( − Δ ) − s θ in R 2 × ( 0 , T ) , \begin{equation*} \left \{ \begin {aligned} & \partial _t \theta +u\cdot \nabla \theta =0, \quad \text {in } \mathbb {R}^2 \times (0,T), \\ & u=\nabla ^\perp \psi , \quad \psi = (-\Delta )^{-s}\theta \quad \text {in } \mathbb {R}^2 \times (0,T) , \end{aligned} \right . \end{equation*} 0 > s > 1 0>s>1 , we consider for k ≥ 1 k\ge 1 the problem of finding a family of k k -vortex solutions θ ε ( x , t ) \theta _\varepsilon (x,t) such that as ε → 0 \varepsilon \to 0 θ ε ( x , t ) ⇀ ∑ j = 1 k m j δ ( x − ξ j ( t ) ) \begin{equation*} \theta _\varepsilon (x,t) \rightharpoonup \sum _{j=1}^k m_j\delta (x-\xi _j(t)) \end{equation*} for suitable trajectories for the vortices x = ξ j ( t ) x=\xi _j(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 ) s W = ( W − 1 ) + γ , in R 2 , 1 > γ > 1 + s 1 − s \begin{equation*} (-\Delta )^sW = (W-1)^\gamma _+ , \quad \text {in } \mathbb {R}^2, \quad 1>\gamma > \frac {1+s}{1-s} \end{equation*} 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].

In order to study electrically and magnetically charged vortices in fractional quantum Hall effect and anyonic superconductivity, the Maxwell–Chern–Simons (MCS) model was introduced by Lee et al. (Phys Lett B 252:79–83, 1990) as a unified system of the classical Abelian–Higgs model (AH) and the Chern–Simons (CS) model. In this article, the first goal is to obtain the uniform (CS) limit result of (MCS) model with respect to the Chern–Simons parameter, without any restriction on either a particular class of solutions or the number of vortex points, as the Chern–Simons mass scale tends to infinity. The most important step for this purpose is to derive the relation between the Higgs field and the neutral scalar field. Our (CS) limit result also provides the critical clue to answer the open problems raised by Ricciardi and Tarantello (Comm Pure Appl Math 53:811–851, 2000) and Tarantello (Milan J Math 72:29–80, 2004), and we succeed to establish the existence of periodic Maxwell–Chern–Simons vortices satisfying the concentrating property of the density of superconductive electron pairs. Furthermore, we expect that the (CS) limit analysis in this paper would help to study the stability, multiplicity, and bubbling phenomena for solutions of the (MCS) model.

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article "On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program"; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli--Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wro\'nskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Poho\vzaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation.

We prove the existence of positive solutions for the supercritical nonlinear fractional Schrödinger equation (−Δ)su+V(x)u−up=0inRn, with u(x)→0 as |x|→+∞, where p>n+2sn−2s for s∈(0,1),n>2s. We show that if V(x)=o(|x|−2s) as |x|→+∞, then for p>n+2s−1n−2s−1, this problem admits a continuum of solutions. More generally, for p>n+2sn−2s, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of (−Δ)sw=wpinRn, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf’s argument as in the paper by Ao, Chan, DelaTorre, Fontelos, González and Wei on the singular fractional Yamabe problem.