The purpose of this paper is to investigate the existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity. In the first scenario, under an explicit smallness assumption on the potential, we establish the existence of a mountain pass solution at a positive energy level. In the second scenario, if the mass is smaller and depends on the potential, we identify a local minimizer at a negative energy level and a mountain pass solution at a positive energy level.
In this paper, we establish weighted norm inequalities for the harmonic function in the upper half-space, and corresponding reverse inequalities are also obtained. These weighted Morrey-Lorentz estimates generalize the L^p -isometry of the Poisson semigroup in Stein-Weiss’s book [Introduction to Fourier analysis on Euclidean spaces, 1971]. The novelty of our results lies in the weight constructed from a product of two weights μ and ω . The former weight μ arises from the structure of the elliptic operator, while the latter ω is in the Muckenhoupt class with respect to μ . This is totally different from the two-weight theory. As applications, weighted norm inequalities and their reversed for the harmonic/caloric function related to the Neumann/Dirichlet problem are considered.
We apply Bochner’s technique combined with a mild Dirichlet integral condition to show that a complete constant mean curvature (CMC) spacelike hypersurface immersed with positive semi-definite modified second fundamental form in a Lorentzian product space ℝ_1× M^n , whose Riemannian base M^n has nonnegative sectional curvature, must be a slice {t}× M^n . In particular, we conclude that the spacelike hyperplanes of the Lorentz–Minkowski space 𝕃_1^n+1 are the only complete CMC spacelike hypersurfaces immersed in 𝕃_1^n+1 , whose modified second fundamental form is positive semi-definite and the height function with respect to a spacelike hyperplane satisfies such a Dirichlet integral condition. A version of the classical Calabi–Bernstein Theorem to the context of Lorentzian product spaces is also given.
In this paper we investigate the multiplicity of normalized solutions to the Choquard equation -ε ^2Δ u+V( x)u=λ u+ε ^-α(I_α * F(u))f(u) in ℝ^N, subject to the prescribed mass constraint ∫ _ℝ^N|u|^2 dx=a^2ε ^N, where ε , a> 0 , α∈ (0,N) , N≥ 3 , I_α denotes the Riesz potential, and the parameter λ∈ℝ arises as a Lagrange multiplier associated with the mass constraint. The nonlinearity f:ℝ→ℝ has L^2 -supercritical growth, while the potential V:ℝ^N →ℝ is assumed to be a C^1 -function. Under suitable assumptions on V and f, we relate the number of normalized solutions to the topology of the set where the potential V attains its minimum. The proof relies on Ljusternik–Schnirelmann theory. Our results improve and complement earlier works by Li–Ye (J. Math. Phys. 55(12):121501, 2014) and Moroz–Van Schaftingen (Calc. Var. Partial Differ. Equ. 52(1–2):199–235, 2015) by establishing multiplicity of normalized solutions concentrating at the potential well V.
In this paper, we establish some new results on the well-posedness and longtime dynamics for the second-order evolution equation with degenerate energy damping: u_tt+Au+φ (‖ A^1/2u‖ ^2+‖ u_t‖ ^2)u_t+f(u)=0 , which is a infinite dimensional version of the Krasovskii model, where A is a positive self-adjoint operator defined densely in a Hilbert space H, φ is a nonnegative real function, and ‖·‖ represents the norm of H. The main contributions are that: (i) We prove the existence and the energy decay of the global solutions of the model as well as the dissipativity of the associated dynamical system under the relaxed condition: φ∈ C (ℝ^+)∩ C^1(0,+∞ ) , with φ (0)=0, φ (s)>0, ∀ s>0 , which removes the longstanding restrictions on this issue in literature: φ (s) is either uniformly Lipschitz continuous or φ (s)∼ s^q with q≥ 1/2 . (ii) Especially when φ (s)=γ s^q , with γ >0, q∈ (0,2] , we establish the existence of the compact global attractor of the associated dynamical system, which extends the range of the exponent q from either q∈ [1,1+δ ), 0<δ≪ 1 (for related beam model) or q∈ [1/2,1] (for related wave model) in literature before to q∈ (0,2] . The methods proposed here allow our overcoming the degeneration of the nonlocal energy damping, extending the range of the exponent q and improving greatly the results in literature before.
We consider a nonlinear Robin problem driven by a differential operator with unbalanced growth and a reaction which exhibits the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) term. Using the Nehari method, we show that for all small values of the parameter, the problem has two bounded, ground state (least energy) solutions. In the process of the proof, we establish some auxiliary results which are of independent interest.
This paper investigates the existence of positive solution for a class of semipositone elliptic problems driven by a Φ -Laplacian operator in bounded smooth domains. The nonlinear term is assumed to exhibit critical exponential growth in the framework of Orlicz–Sobolev spaces. By introducing an auxiliary problem and employing variational methods, we overcome the difficulties caused by the negative parameter that characterizes the semipositone nature of the equation. Under suitable conditions on the 𝒩 -function Φ and on the nonlinearity, we establish the existence of at least one weak positive solution for sufficiently small values of the parameter.
In this work we are interested in studying the existence of a solution and the phenomenon of solution concentration for the following class of problems -div (ħ ^2 ϕ (ħ |∇ u |)∇ u) + V(x)b(|u|)u + W'(u) = 0, x ∈ℝ^N, (P_ħ ) where ħ > 0 , u: ℝ^N →ℝ^N+1 , 3 ≤ N < p , Φ is a N-function of the form Φ (t) = ∫ _0^|t| sϕ (s) ds, the function b:(0,+∞ ) → (0,+∞ ) checks the relationship B(t) = ∫ _0^|t| sb(s) ds, where B is a N-function, and V, W satisfy some technical conditions.
We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in the class of Lipschitz domains. The threshold exponent for the Robin case is explicitly recovered and shown to be strictly smaller than in the Dirichlet one.
For a non-empty, bounded, open, and convex set of class C^2 , we consider the Torsional Rigidity associated to the k-Hessian operator. We first prove Pólya type lower bound for the k-Torsional Rigidity in any dimension; then, in order to investigate optimal sets in the Pólya type inequality, we provide two quantitative estimates. We finally prove an upper bound for the eigenvalue of the k-Hessian operator.
We study one-dimensional Schrödinger operators defined as closed operators that are exactly solvable in terms of the Gauss hypergeometric function. We allow the potentials to be complex. These operators fall into three groups. The first group can be reduced to the Gegenbauer equation, up to an affine transformation, a special case of the hypergeometric equation. The two other groups, which we call hypergeometric of the first, resp. second kind, can be reduced to the general Gauss hypergeometric equation. Each of the group is subdivided in three families, acting on the Hilbert space L^2]-1,1[, L^2(ℝ_+) resp. L^2(ℝ) . Motivated by geometric applications of these families, we call them spherical, hyperbolic, resp. deSitterian. All these families are known from applications in Quantum Mechanics: e.g. spherical hypergeometric Schrödinger operators of the first kind are often called trigonometric Pöschl-Teller Hamiltonians. For operators belonging to each family we compute their spectrum and determine their Green function (the integral kernel of their resolvent). We also describe transmutation identities that relate these Green functions. These identities interchange spectral parameters with coupling constants across different operator families. Finally, we describe how these operators arise from separation of variables of (pseudo-)Laplacians on symmetric manifolds. Our paper can be viewed as a sequel to [12], where closed realizations of one-dimensional Schrödinger operators solvable in terms Kummer’s confluent equation were studied.
We characterize the asymptotic behaviour, in the sense of Γ -convergence, of a thin magnetoelastic shallow shell. The compactness is achieved up to rigid motions. For deformations, it relies on an approximation by rigid movements, whereas for magnetizations it is based on a careful consideration of the geometry of the deformed domain. This result is a generalization of [1] by incorporating geometric effects due to vanishing curvature, which constitute the main novelty of the analysis.
In this article we investigate the question of finding a network configuration of minimal length connecting three given points in the Heisenberg group. After proving existence of (possibly degenerate) minimal horizontal triods, we investigate their characterization. We then formulate a horizontal curve shortening flow that deforms any given suitable initial triod into a critical point for the length functional. Numerical experiments based on a stable fully discrete finite element scheme provide useful insights into the rich landscape of this sub-Riemannian geometry.
This paper is the latter part of our series concerning infinite concentration and oscillation phenomena on supercritical semilinear elliptic equations in discs. Our supercritical setting admits two types of nonlinearities, the Trudinger-Moser type e^u^p with p>2 and the multiple exponential one exp(⋯ (exp(u^m))) with m>0 . In the first part, we accomplished the analysis of infinite concentration phenomena on any blow-up solutions. In this second part, we proceed to the study of infinite oscillation phenomena based on the concentration estimates obtained in the first part. As a result, we provide a precise description of the asymptotic shapes of the graphs of blow-up solutions near the origin. Two types of oscillation and intersection properties are observed here depending on the choice of the growth. Moreover, it allows us to show several oscillation behaviors around singular solutions. This leads to a natural sufficient condition for infinite oscillations of bifurcation diagrams which ensure the existence of infinitely many solutions. We successfully apply this condition to certain classes of nonlinearities including the two types mentioned above.
We establish optimal results on limits at infinity for functions in fractional Sobolev spaces.
The aim of this paper is to prove that the A_∞ -nerve of two quasi-equivalent A_∞ -categories (linear over a commutative ring) are weak-equivalent in the Joyal model structure. As a consequence we prove that the A_∞ -nerve of a pretriangulated A_∞ -category is a stable ∞ -category.
In the context of a Banach space X endowed with a right quaternionic structure, we examine a bounded right linear operator T defined on X. We introduce and investigate the spherical local spectrum of T, which serves as the quaternionic counterpart to the local spectrum of bounded operators on complex Banach spaces. To generalize the concept of the single-valued extension property originally formulated in the complex setting to the quaternionic framework, we adopt the definition provided by Colombo et al. (Spectral theory on the S-spectrum for quaternionic operators. Birkhäuser, Basel, 2018). This definition posits the existence of a unique right slice hyperholomorphic extension of the pseudo-resolvent R_s(T)=𝒬_s(T)(xs̅-Tx) for any axially symmetric open neighborhood 𝒰 such that ρ _s(T)⊆𝒰 . Furthermore, leveraging the properties of the quaternionic right Banach space, we establish and prove several fundamental results analogous to those in the complex setting, extending them to the quaternionic context. This systematic approach enables the development of local spectral theory within quaternionic right Banach spaces. Additionally, we introduce and analyze key subsets of X that are pivotal to quaternionic local spectral theory. These subsets encompass the spherical local spectral subspace of T, the spherical global spectral subspace of T, and the spherical quasi-nilpotent part of T.
The multi-marginal Monge Kantorovich problem (MKP) has become a pivotal framework in probability theory, mathematical physics, and economics. Despite its significance, the complexity of its solution structure and uniqueness continues to pose difficulties. In this work, we examine the properties of optimal transport plans under the m -twist condition, which ensures that any optimal plan is confined to the union of graphs of at most m transport maps, offering a precise characterization of solutions. Beyond this structural result, we explore alternative formulations of (MKP) problem, including its decomposition into two-marginal transport problems, which offer a decompositional approach. Specifically, we show that (MKP) can be rewritten as the center-marginal Monge Kantorovich problem (C-MKP) or adjacent-marginal Monge Kantorovich problem (A-MKP), each of which retains structural properties from the classical two-marginal setting. Our results refine the understanding of optimality and uniqueness in (MKP), bridging between two-marginal classical optimal transport theory and multi-marginal formulations. They have implications for applications in density functional theory, economics, and quantum transport by providing rigorous conditions for when solutions are concentrated on finitely many maps. By introducing new structural conditions techniques, this work advances our understanding of multi-marginal optimal transport.
In this paper, we establish sharp up-to-the-boundary gradient estimates for weak solutions of the Dirichlet problem involving nonlinear partial differential equations (PDEs) with non-standard growth conditions and unbounded source terms. Solutions are analyzed within a suitable Orlicz–Sobolev framework. Leveraging these estimates, we derive optimal boundary gradient bounds for a class of free boundary problems, and subsequently prove Lipschitz regularity for a flame propagation model. Our results extend those available in the literature and allow for the treatment of applications in mathematical physics under even more challenging scenarios.
