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For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,$$-\\Delta u+V_{\\mu, \\nu} u=(I_\\alpha\\ast \\vert u \\vert ^{({N+\\alpha})/{N}}){ \\vert u \\vert }^{{\\alpha}/{N}-1}u,\\quad {\\rm in} \\ {\\open R}^N, $$where$N\\ges 3$, V μ,ν :ℝ N  → ℝ is an external potential defined for μ, ν > 0 and x  ∈ ℝ N by V μ,ν ( x ) = 1 − μ/(ν 2  + | x | 2 ) and$I_\\alpha : {\\open R}^N \\to 0$is the Riesz potential for α ∈ (0, N ), we exhibit two thresholds μ ν , μ ν  > 0 such that the equation admits a positive ground state solution if and only if μ ν  < μ < μ ν and no ground state solution exists for μ < μ ν . Moreover, if μ > maxμ ν , N 2 ( N  − 2)/4( N  + 1), then equation still admits a sign changing ground state solution provided$N \\ges 4$or in dimension N = 3 if in addition 3/2 < α < 3 and$\\ker (-\\Delta + V_{\\mu ,\\nu }) = \\{ 0\\} $, namely in the non-resonant case.

In this paper, we study the following quasilinear Schrödinger equation − Δ u + V ( x ) u − [ Δ ( 1 + u 2 ) 1 / 2 ] u 2 ( 1 + u 2 ) 1 / 2 = h ( u ) ,   x ∈ ℝ N , where N ≥ 3 , 2 * = 2 N N − 2 , V(x) is a potential function. Unlike V ∈ C²(ℝ N ), we only need to assume that V ∈ C¹(ℝ N ). By using a change of variable, we prove the non-existence of ground state solutions with Berestycki-Lions conditions, which contain the superliner case: lim s → + ∞ h ( s ) s = + ∞ and asymptotically linear case: lim s → + ∞ h ( s ) s = η . Our results extend and complement the results in related literature.

Based on their finding that the circularly polarized Beltrami fields are solutions to their eigenvalue equation (7), the authors of preceding Comment claim that the linearly polarized basis is inapplicable to light waves in an isotropic chiral medium. We show that the analysis is flawed. The reason is twofold. Firstly, we prove that the Beltrami fields are, by definition, not a polarization basis for both the electric and magnetic fields of light waves in the chiral medium. Secondly, we demonstrate that linearly polarized and circularly polarized waves in the chiral medium are on the same footing if the electric and magnetic fields, rather than the Beltrami fields, are considered.

In this paper, we give some examples of non-star-shaped bounded domains of R2 where, for a class of nonlinear elliptic Dirichlet problems involving supercritical Sobolev exponents, there exists only the trivial identically zero solution (notice that a well-known result of Pohozaev concerns only the star-shaped domains).Unlike the case of previous papers (Molle and Passaseo (2020, 2021, 2023, 2025)) where we proved nonexistence results for nontrivial solutions in thin domains sufficiently close to prescribed curves, in the present paper, the domains do not need to be thin.

The aim of this paper is to investigate limit cycles of the generalized Rayleigh–Liénard oscillator x ˙ = φ ( y ) , y ˙ = - g ( x ) - f ( x , y ) y . A criterion on the uniqueness and stability of limit cycles is given. We apply the criterion to two generalized Rayleigh–Liénard systems and get the uniqueness of limit cycles. Moreover, the stability and locations of limit cycles are obtained if they exist.

We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the β-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases.

In this article, under some weaker assumptions on and , the authors aim to study the existence of nontrivial radial solutions and nonexistence of nontrivial solutions for the following Schrödinger-Poisson system with zero mass potential where . In particular, as a corollary for the following system: a sufficient and necessary condition is obtained on the existence of nontrivial radial solutions.

This paper studies a first‐price common‐value auction in which bidders do not know the number of their competitors. In contrast to the case of common‐value auctions with a known number of rival bidders, the inference from winning is not monotone, and a “winner's blessing” emerges at low bids. As a result, bidding strategies may not be strictly increasing, but instead may contain atoms. Moreover, an equilibrium fails to exist when the expected number of competitors is large and the bid space is continuous. Therefore, we consider auctions on a grid. On a fine grid, high‐signal bidders follow an essentially strictly increasing strategy, whereas low‐signal bidders pool on two adjacent bids on the grid. The solutions of a “communication extension” based on Jackson, Simon, Swinkels, and Zame (2002) capture the equilibrium bidding behavior in the limit, as the grid becomes arbitrarily fine.

We consider the boundary Hardy–Hénon equation \\[ -\\Delta u=(1-|x|)^{\\alpha} u^{p},\\ \\ x\\in B_1(0), \\] where $B_1(0)\\subset \\mathbb {R}^{N}$ $(N\\geq 3)$ is a ball of radial $1$ centred at $0$, $p>0$ and $\\alpha \\in \\mathbb {R}$. We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$, we establish the estimate of positive solutions. When $\\alpha \\leq -2$ and $p>1$, we give some conclusions with respect to nonexistence. When $\\alpha >-2$ and $1< p<({N+2})/({N-2})$, we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\\leq 1$ and $\\alpha \\leq -2$, we show the nonexistence of positive solutions. When $0< p<1$, $\\alpha >-2$, we give some results with respect to existence and uniqueness of positive solutions.

We consider a wave inequality in an exterior domain of R N , involving the product of two nonlinear convolution terms. The problem is considered under an inhomogeneous Dirichlet-type boundary condition. We establish sufficient conditions depending on the parameters of the problem, under which we have existence/nonexistence of weak solutions.