This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation and its counterpart with prescribed L^2 -norms which come from physically relevant situations. Here, N ≥ 2 , V:ℝ^N →ℝ is a non-symmetric and non-periodic potential satisfying certain decay conditions, a is a prescribed constant, and λ arises as an unknown Lagrange multiplier. For problem (1), using purely variational methods, we establish the existence of multi-bump positive solutions with either finitely or infinitely many bumps. For the normalized problem (2), we prove the existence of normalized multi-bump positive solutions with a finite number of bumps. The main difficulty comes from the nonsmooth nature of logarithmic nonlinearity, which introduces some challenges to the variational framework. In particular, the corresponding energy functional is not of class C^1 on H^1(ℝ^N) , which prevents the direct application of standard critical point theory for C^1 functionals or any reduction methods for C^1+σ nonlinearities. The main ingredients in this paper are nonsmooth critical point theory, localized variational methods, and a max-min argument. To the best of our knowledge, this paper appears to be the first successful application of the localized variational method to nonsmooth functionals.

We consider the semilinear fully fractional heat equation (∂_t-Δ)^σu = |u|^p-1u in ℝ^n ×ℝ_-, 0 < σ< 1. For n≤ 2σ or 1<p≤n+2σ/n-2σ, we generalize the monotonicity formula and Liouville-type theorem when σ=1 proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for σ=1. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.

Let B_1 be the unit disk in ℝ^2. We consider the harmonic map equation -Δu=|∇ u|^2u, subject to the Dirichlet boundary condition u(e^iθ)=(R,R,√(1-R^2)):=g_R, where 0<R<1 and u: B_1→𝕊^2 is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace g_R, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential X associated with u, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply |u_r|≡ R, u_r· u_θ≡0 on ∂ B_1. Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis–Coron maps.

We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say σ_1, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension N≥ 2, where we prove that in fact σ_1 is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

Abstract For the Liouville equation with singular sources, the most subtle and challenging situation arises when the sources are quantized, namely, when each Dirac mass has a strength equal to a multiple of . In this regime, two types of blow‐up may occur: simple and nonsimple. In this paper, we concentrate on the simple blow‐up case and give complete and optimal estimates. We begin by establishing sharp estimates for a localized problem in a ball and then apply these estimates to the Dirichlet problem with multiple quantized sources. Our results show that the conditions used by del Pino, Esposito, and Musso to construct simple blow‐up solutions are not only sufficient but also necessary. We further obtain a sharp estimate on the total mass.

We prove that for every integer N≥ 3 and α≥1/2, Beckner's inequality 2∫_𝕊^Nu(P_Nu) dw+(N-1)!∫_𝕊^Nu dw-(N-1)!/Nln∫_𝕊^Ne^Nu dw≥ 0 holds for any axially symmetric u∈ H^N/2(𝕊^N) whose center of mass is at the origin. The proof is mainly based on a weighted ℓ ^2 estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer N≥ 3.

For every subsonic speed c∈(0,√(2)), we prove the existence of a finite-energy traveling wave for the planar Gross–Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mariş (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate E(ψ)≤ C_J(I_c(ψ)+(ψ)), c∈ J, where E is the energy, I_c the action at speed c, the real Morse index, J is any compact interval contained in (0,√(2)), and C_J is a positive constant depending only on J. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation u_t=Δu+u^3, n≥ 5. The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As t↗ T, the solution concentrates in a thin tubular region around an (n-4)-dimensional sphere whose radius shrinks to zero at the self-similar scale ξ_r(t)∼√(2(n-4)(T-t)). At the same time, concentration takes place transversely to the sphere at the much smaller scale λ(t)∼ κ_* T-t/|log(T-t)|^n/n-2, for some κ_*>0. More precisely, in cylindrical coordinates r=|x'|, z∈ℝ^3, the leading profile is u(x,t) ∼1/λ(t) U( r-ξ_r(t)/λ(t), z/λ(t)), where U is the Aubin–Talenti bubble in ℝ^4. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale √(T-t), whereas its transverse thickness is governed by the much smaller type II scale λ(t). The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent p=3 is energy-supercritical in dimensions n≥5, but lies below the Joseph–Lundgren exponent for 5≤ n≤ 12, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

In this paper, we consider the anisotropic Allen-Cahn equation -div a(Du)+W'(u)=0 in ℝ^N, where a(p):=DH(p) with H(p)=1/2F(p)^2 and F a uniformly elliptic integrand, and W(u)=1/4(1-u^2)^2. Based on the Mooney-Yang anisotropic minimal graph, we prove that the anisotropic Allen-Cahn equation admits a stable solution for N≥4 in the weak sense, whose level sets are not hyperplanes. As a byproduct, we also construct a smooth solution of the above anisotropic Allen-Cahn equation for N≥5 that is monotone in one direction but is not one-dimensional.

In this paper, we consider the Brezis-Nirenb erg problem { - triangle u = lambda u + |u|(2 & lowast; -2) u, in ohm, u = 0, on partial derivative ohm, where ohm is a smoothly bounded domain of R-N with N >= 3, lambda > 0 is a parameter, and 2(& lowast;) = 2N/N - 2 is the critical Sobolev exponent. First, we recall the history of the Brezis-Nirenb erg problem and then provide new results of it in dimension six. Finally, we also list some open questions on the Brezis-Nirenb erg problem.

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin–Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

We study the blow-up behavior of solutions to the singular Liouville equation Δũ+λe^ũ=4παδ_0 in B, ũ=0 on ∂ B, where α>0, λ>0 and B⊂ℝ^2 is the unit disk. Our main results give a complete classification of all blow-up solutions and determine the exact number of solutions to the above equation. More precisely, for fixed α>0 and λ∈(0,λ_α), the singular Liouville equation has exactly ⌈ α⌉+2 solutions (up to rotation): a unique minimal energy solution; a unique singular sequence blowing up at the origin; and for each 1≤ m≤⌈ α⌉, a unique m-peak sequence whose blow-up points are the vertices of a regular m-gon centered at the origin. This result answers the questions raised in Bartolucci-Montefusco and Bartolucci . We also prove the non-degeneracy of these solutions. Thus we provide a full description of the blow-up structure for the singular Liouville equation on the disk.

In this article we are concerned with the existence of blow-up solutions to the following boundary value problem -Δv= λV(x) |x|^2e^v B_1, v=0 ∂ B_1, where B_1 is the unit ball in ℝ^2 centered at the origin, V(x) is a positive smooth potential, and λ>0 is a small parameter. We find necessary and sufficient conditions on the potential V for the existence of a blow-up sequence of solutions tending to infinity near the origin as λ→ 0^+. In particular, we obtain a second-order classification of the coefficient function V for which (simple) blow-up occurs at the origin.

In 2023, H. Brezis [Some of my favorite open problems, Rend. Lincei 34(2) (2023) 307-335] published a list of his "favorite open problems", which he described as challenges he had "raised throughout his career and has resisted so far". In this paper, we shall provide a partial answer to this question by presenting the existence of sign-changing solutions to the equation whenever the parameter is small enough. Our construction is based on the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to Yamabe problem.

For every integer n≥27, we construct a smooth metric on 𝕊^n that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized σ_2-Yamabe equation admits a noncompact family of positive Γ_2^+-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the σ_2 problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension n=27.

We construct global growing, bounded, and decaying solutions to the 1-equivariant harmonic map flow from R-2 into S-2 {v(t) = v(rr) + vr/ r - sin(2v)/ 2r(2) , (r, t) is an element of R+ x (t(0), infinity) v(r, t(0)) = v(0)(r), r is an element of R+ for t(0) sufficiently large and the initial data v(0)(r) satisfying v(0)(0) = pi and |v(0)(r)| less than or similar to t(0)(max{0, gamma/ 2-1})r(1-gamma) as r ->infinity, gamma > 1. These global solutions exhibit the following trichotomy longtime asymptotic behavior ||v(r)(center dot,t)|| L-infinity([0,infinity)) similar to {t(gamma/2-1) ln t if 1 < gamma < 2 1 if gamma = 2 ln t if gamma>2. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We classify the singular limits relative to a free boundary problem arising in plasma physics in dimension d=2, under suitable natural integral bounds. It turns out that one of the asymptotic behaviors allowed corresponds to the Dancer-Yan spikes (J. London Math. Soc. (78) 2008, 639–662). Interestingly enough, roughly speaking and unlike the higher dimensional case, it is not true that any solution in the limit is a Dancer-Yan spike. Indeed, the spiking structure is more rich and we succeed in a detailed description of the singular behavior by a careful analysis, from local to global, of the tiny difference between the maximum value of the spikes and their “vanishing level” defining the free boundary.

Motivated by the average partition function of c free bosons (Afhkami-Jeddi et al. ) and the average of the genus 1 partition function over the Narain moduli space (Maloney-Witten ), we investigate ratios of theta functions. In this paper, we completely classify the minimizers (or maximizers) for ratios of theta and Epstein zeta functions. We find that the hexagonal lattice plays a pivotal role there. These results have direct applications in conformal and Liouville field theory via partition functions. Additionally, they yield the minima of differences of theta and Epstein zeta functions, which have implications for the mathematics of crystallization and interacting particle theory ().

Let q be a nondegenerate lump type solution to the KP-I (Kadomtsev-Petviashvili-1) equation 89-2 root 2829-3 root 28((829)2)-2829 = 0. We show the existence of travelling wave solutions with the form ue (zct, y), for the GP (Gross-Pitaevskii) equation i partial t Psi + Delta*Psi + (1 - |Psi| <^> 2) * Psi = 0 in R2, with travelling speed c = sqrt(2) - epsilon <^> 2 and u_{c} = 1 + i*epsilon*q + \mathcal{O}(epsilon <^> 2) This proves the existence of finite energy solutions in the so-called Jones-Roberts program within the transonic regime c \in (sqrt(2) - epsilon <^> 2, sqrt(2)) The main ingredients in our proof are detailed point-wise estimates for the Green functions associated to a family of fourth order hypoelliptic operators. In view of the classification of lump type solutions of the KP-I equation, our proof also indicates that for fixed small s, there should exist a sequence of travelling wave solutions to GP equation, with energy tends to infinity. (c) 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, Al training, and similar technologies.

We consider the nonlinear heat equations with Neumann boundary conditions {u(t) = Delta u in R-+(4) x (0, T), -du/dx(4)((x) over tilde, 0, t) = u(2)((x) over tilde, 0, t) in R-3 x (0, T). We establish the existence of a finite-time blow-up solution. Specifically, for any sufficiently small T > 0 and any k distinct points q(1), ... , q(k) is an element of R-3, there exists an initial datum u(0) such that the corresponding solution u(x, t) blows up exactly at q(1), ... , q(k) as t NE arrow T. Furthermore, when t NE arrow T, the solution admits the asymptotic profile u(x, t) = Sigma(k)(j=1) U-mu j(t),U-xi j(t)(x) + Z(0)*(x) + o(1) as t NE arrow T, where U-mu j(t),U-xi j(t)(x) := mu(-1)(j)(t)U(x-xi(j)(t)/mu(j)(t)), x is an element of R-+(4) and Z(0)* is an element of C-0(infinity)(R-+(4)) satisfying Z(0)*(q(j), 0) < 0 for all j = 1, ... , k. Here, U(y) denotes the harmonic extension to R-+(4) of the positive radially symmetric solution (U) over tilde to the fractional Yamabe problem (-Delta)(1/2) (U) over tilde = (U) over tilde (2) in R-3. For some constants beta(j) > 0, the scaling parameters mu(j)(t) and the translation parameters xi(j)(t) satisfy mu(j)(t) = beta(j)vertical bar log 2T vertical bar(T-t)/vertical bar log(T-t)vertical bar(2) (1 + o(1)) -> 0, xi(j)(t) -> (q(j), 0) as t NE arrow T.