This paper investigates the classical Brezis–Nirenberg problem. Motivated by the well-known result of Brezis and Nirenberg [17] on the existence of a least energy solution for ε >0 and q in a certain range, we focus on its asymptotic properties. For the limit ε→ 0 , we determine the blow-up rate and location, and establish a refined energy estimate. Leveraging these estimates, we prove the solution’s uniqueness and nondegeneracy under mild assumptions on Ω . For the opposite limit ε→ +∞ , we similarly derive the vanishing rate, an energy estimate, and the corresponding uniqueness and nondegeneracy. A key finding is that the asymptotic behavior is jointly influenced by the dimension N, the domain geometry, the subcritical exponent q, and the parameter ε .

In this paper, we study the existence, non-existence and asymptotic behavior of positive ground states for the nonlinear Choquard equation: \begin{equation}\label{0.1} -Δu+\varepsilon u=\big(I_α\ast F(u)\big)F'(u),\quad u\in H^1(\mathbb R^N), \end{equation} where $F(u)=|u|^{\frac{N+α}{N}}+G(u)$ with $G(u)=\int_0^ug(s)ds$, $N\geq3$ is an integer, $I_α$ is the Riesz potential of order $α\in(0,N)$ and $\varepsilon>0$ is a frequency parameter. Under some mild subcritical growth assumptions on $g\in C([0,\infty), [0,\infty))$, we establish a sharp threshold result for the existence of ground states, and an asymptotic characterization of the ground state solutions as $\varepsilon\to 0$. In particular, if $g(s)\sim s^{q-1}$ as $s\to 0$ for some $q\in (\frac{N+α}{N}, \frac{N+α}{N-2})$, then if $q<\frac{N+α+4}{N}$, \eqref{0.1} admits a ground state for all $\varepsilon>0$, and if $q\ge \frac{N+α+4}{N}$, then a threshold phenomena occur: there exists $\varepsilon_q>0$ such that \eqref{0.1} has no ground state for $\varepsilon\in (0,\varepsilon_q)$ and admits a ground state for $\varepsilon>\varepsilon_q$. If $g(s)\simeq as^{q-1}$ as $s\to 0$ for some $a>0$ and $q\in (\frac{N+α}{N}, \min\{\frac{N+α}{N-2}, \frac{N+α+4}{N}\})$, we show that as $\varepsilon \to 0$, the ground state solutions of \eqref{0.1}, after a suitable rescaling, converges in $H^1(\mathbb R^N)$ to a particular solution of the Hardy-Littlewood-Sobolev critical equation $u=\frac{N+α}{N}(I_α*|u|^{\frac{N+α}{N}})|u|^{\frac{N+α}{N}-2}u$. It turns out that the limit profiles are determined solely by the locations of $(a,q)$ in $(0,+\infty)\times (\frac{N+α}{N}, \min\{\frac{N+α}{N-2}, \frac{N+α+4}{N}\})$. We also establish a novel sharp asymptotic characterization of such a rescaling.

We study the existence and asymptotic behavior of normalized solutions to the following Choquard equation -Δ u + λ u =μ g(u) + γ (I_α * |u|^N+α/N)|u|^N+α/N-2u in ℝ^N under the L^2 -norm constraint ∫ _ℝ^Nu^2 dx =c^2 . Here γ >0 , N≥ 1 , I_α is the Riesz potential of order α∈ (0,N) , μ >0 is a parameter and the unknown λ appears as a Lagrange multiplier. In a mass supercritical setting on g, by establishing a novel compactness lemma and some prior energy estimate, we find regions in the (c,μ ) –parameter space such that the corresponding equation admits a positive radial ground state solution, and then study the asymptotic profiles of the ground states as (c,μ ) varies. In particular, we show that as μ or c tends to 0 (resp. μ or c tends to +∞ ), after a suitable rescaling the ground state solutions converge in H^1(ℝ^N) to a particular solution of some limit equations. Our main results are new even for the power type nonlinearity g(u)= |u|^q-2u with 2+4/N0 . Particularly, if 2+4/N< q<2^* , we show that there exist two constants 0<η _1≤η _2<∞ such that the corresponding equation has a positive radial least action solution if and only if η≥η _1 and admits two positive solutions if η >η _2 . To the best of our knowledge, this seems to be the first result concerning the non-existence and multiplicity of positive solutions to Choquard type equations involving the lower critical exponent.

In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -Δu=\displaystyle\left(\int\limits_Ω\frac{u^{p+1}(y)}{|x-y|^α}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ Ω, \quad \ \ u=0, \ \ &\mbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $α\in (0,2)$ and $p>1$ is a positive parameter. Unlike the higher-dimensional case, we prove that the least energy solutions $u_{p}$ neither blow up nor vanish, and develop only one peak as $p\to+\infty$ under suitable assumptions on $Ω$. In contrast, the modified solutions $pu_p$ exhibit blow-up behavior analogous to that observed in higher dimensions. Furthermore, as $α\to 0$, the main results of this paper become consistent with the known conclusions for the corresponding Lane-Emden equation.

By using variational methods, under some mild one side local conditions, we obtain infinitely many standing waves and homoclinic solutions for time-dependent and time-independent discrete nonlinear Schrodinger equations and systems with unbounded potentials, respectively. The main novelties of this paper are: (1) the primitive functions of nonlinearities may be sign-changing; (2) the conditions required on the primitive functions of nonlinearities are local in nature, therefore, the primitive functions of nonlinearities may be super-quadratic, asymptotically-quadratic and sub-quadratic at zero or infinity; (3) our results can be applied to many mathematical models.

In this paper, we consider the following well-known Brezis-Nirenberg problem -Δ u= u^2^*-1+ε u^q-1, u>0, in Ω, u=0, on ∂Ω, where N≥ 3, Ω is a smooth and bounded domain in ^N, ε>0 is a small parameter, q∈ (2,2^*) and 2^*:=2N/N-2 denotes the critical Sobolev exponent. The existence of solutions to the above problem has been obtained by many authors in the literature. However, as far as the authors know, the asymptotic behavior of solutions to the above problem is still open. Here we first describe the asymptotic profile of solutions to the above problem as ε→ 0. Then, we derive the exact blow-up rate and characterize the concentration speed and the location of concentration points in the general case of multi-peak solutions. Finally, we prove the uniqueness, nondegeneracy and count the exact number of blow-up solutions. The main results in this paper give a complete picture of multi-peak blow-up phenomena in the framework of Brezis-Peletier conjecture in the case of sub-critical perturbation. On the other hand, compared with the special case q=2 previously studied in the literature, we observe that the exponent q has a significant impact on the asymptotic behavior, uniqueness and nondegeneracy of solutions in addition to the geometry of domain Ω and space dimension N which is already known in the literature.

We study asymptotic behavior of positive ground state solutions of the nonlinear Choquard equation with a Sobolev critical attractive local perturbation where N≥ 3 is an integer, p∈ (N+α/N, N+α/N-2 ) , 2^*=2N/N-2 is the Sobolev critical exponent, I_α is the Riesz potential of order α∈ (0,N) and ε >0 is a parameter. We show that as ε→∞ , after suitable rescalings the ground state solutions u_ε of (P_ε ) converge to a particular solution of the critical local Emden–Fowler equation. The rescalings are implicit and depend in a non-trivial way on the exponent p and the space dimension N=3,4 or N≥ 5 . We establish a sharp asymptotic characterisation of such rescalings, as well as the blow-up rates or asymptotics of the L^2 and other relevant norm of u_ε . As a follow up of our main results, we also obtain the existence, multiplicity and asymptotic behaviour of positive normalized solutions of a mass constrained problem associated to (P_ε ) with mass normalization constraint ∫ _ℝ^N|u|^2=c^2 , as c→ 0 and c→∞ .

We study the non-existence and multiplicity of positive solutions of the nonlinear Choquard type equation -Delta upsilon + epsilon kappa = (1+u/PuP2u+u/9-2u, in RN, (Pc) where N >= 3 is an integer, pe (Na,], q is an element of (2, Ia is the Riesz potential of order a E (0, N) and 0 is a parameter. We fix one of p,q as a critical exponent (in the sense of Hardy-Littlewood-Sobolev and Sobolev inequalities) and view the others in p,q,e, a as parameters, we find regions in the (p,q, a, c)-parameter space, such that the corresponding equation has no positive ground state or admits multiple positive solutions. This is a counterpart of the Brezis-Nirenberg Conjecture (Brezis and Nirenberg, 1983 [7]) for nonlocal elliptic equation in the whole space. Particularly, some threshold results for the existence of ground states and some conditions which insure two positive solutions. are obtained. These results are quite different in nature from the corresponding local equation with combined powers nonlinearity and reveal the special influence of the nonlocal term. To the best of our knowledge, the only two papers concerning the multiplicity of positive solutions of elliptic equations with critical growth nonlinearity are given by Atkinson and Peletier (1986) [5] for elliptic equation on a ball and Wei and Wu (2023) [40] for elliptic equation with a combined powers nonlinearity in the whole space. The ODE technique is main ingredient in the proofs of the above mentioned papers, however, ODE technique does not work any more in our model equation due to the presence of the nonlocal term. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -Δu=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^α}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $α\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic decomposition. The main result of this paper can be viewed as an extension of the nondegeneracy of solutions for both the planar Liouville equation and the higher-dimensional upper critical Choquard equation.

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation -Δ u+ε u=(I_α∗ |u|^p)|u|^p-2u+ g(u), inℝ^N, (P_ε) where N≥ 3 is an integer, p=N+α/N, or N+α/N-2, I_α is the Riesz potential and ε>0 is a parameter. Under some mild conditions on g(u), we show that as ε→∞, after a suitable rescaling the ground state solutions of (P_ε) converge to a particular solution of some limit equations, and establish a sharp asymptotic characterisation of such a rescaling, which depend in a non-trivial way on the asymptotic behavior of the function g(s) at infinity and the space dimension N. Based on this study, we also present some results on the existence and asymptotic behaviors of positive normalized solutions of (P_ε) with the normalization constraint ∫_ℝ^N|u|^2=a^2. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as a→ 0 and a→∞.

In this paper, by using variational methods we study the existence of positive solutions for the following Kirchhoff type problem: {[ -(a+b∫_Ω|∇ u|^2dx)Δ u+V(x)u=u^5, x∈Ω,; ; u=0, x∈∂Ω, ]. where a>0, b≥0, Ω⊂ℝ^3 is an unbounded exterior domain, ∂Ω≠∅, ℝ^3\Ω is bounded, u∈ D_0^1,2(Ω), and V∈ L^3/2(Ω) is a non-negative continuous function. It turns out that the above Kirchhoff equation has no ground state solution. Nonetheless, by establishing some global compact lemma and constructing a suitable minimax value c at a higher energy level where so called Palais-Smale condition holds, we succeed to obtain a positive solution for such a problem whenever V and the hole ℝ^3∖Ω are suitable small in some senses. To the best of our knowledge, there are few similar results published in the literature concerning the existence of positive solutions for Kirchhoff equation in exterior domains. Our result also holds true in the case Ω=ℝ^3, particularly, if a=1 and b=0, we improve some existing results (such as Benci, Cerami, Existence of positive solutions of the equation -Δ u+a(x)u=u^(N+2)/(N-2) in R^N, J. Funct. Anal., 88 (1990), 90–117) for the corresponding Schrödinger equation in the whole space.

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models.

This paper is concerned with a class of periodic Schrödinger lattice systems with spectrum 0 and saturable nonlinearities. The existence of ground state solitons of the systems under weak assumptions is obtained. The main novelties are as follows. (1) Some new sufficient conditions for the existence of ground state solitons under the “spectral endpoint” assumption are constructed. (2) Our “non-monotonic” conditions make the proofs of the boundedness of the (PS) sequences to be easier. (3) Our result extends and improves the related results in the literature. Besides, some examples are given to illuminate our result.

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and $\lambda\to +\infty$, where $N=3$ or $N= 4$, $2"0$, $b\ge 0$ are constants and $\lambda>0$ is a parameter. In particular, we prove that in the case $2"

In this paper, we study asymptotic behavior of positive ground state solutions for the nonlinear Choquard equation: -Δ u+ε u=(I_α∗ F(u))F'(u), u∈ H^1(ℝ^N), where F(u)=|u|^N+α/N-2+G(u), N≥3 is an integer, I_α is the Riesz potential of order α∈(0,N), and ε>0 is a parameter. Under some mild subcritical growth assumptions on G(u), we show that as ε→∞, the ground state solutions of (), after a suitable rescaling, converge to a particular solution of the critical Choquard equation -Δ u=N+α/N-2(I_α*|u|^N+α/N-2)|u|^N+α/N-2-2u. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the asymptotic behavior of G(u) at infinity and the space dimension N=3, N=4 or N≥5.

In this paper, we consider the following Choquard equations with double nonlocal nonlinearities: −Δu=λu+μ(Iα∗|u|p)|u|p−2u+(Iβ∗|u|N+βN−2)|u|N+βN−2−2uinRN,u∈H1(RN),∫RN|u|2=c2,where N≥5, N+αN0, Iα and Iβ are the Riesz potentials with α, β ∈(0,N) and the frequency λ∈R is unknown and appears as Lagrange multiplier. In Ye et al. (2022), Yang and his co-authors have proved the existence of ground states and mountain-pass type solutions under different assumptions. In this paper, we will prove the asymptotic behaviors of ground states and mountain-pass solutions as μ→0 (or c→0).

We extend the results of Akahori et al. (2019) on ground states. We discuss the existence of ground states to the double-type nonlinear p-Laplace problem involving the Sobolev critical exponent in R-N:-Delta(p)u + |u|(p-2)u = |u|(p & lowast;-2)u + lambda|u|(q-2)u , u is an element of W-1(,p )(R-N),where N >= 2, lambda > 0, 1 < p < N , p < q < p(& lowast;) , p(& lowast;) := N-p/(N - p) is the Sobolev critical index, Delta(p)u = div(|del u|(p-2) del u) is the p-Laplace operator. We show that: (i) if p < q < p & lowast; , then there exists lambda(0) > 0 so that ground states exist for all lambda > lambda(0); (ii) if max(p , p(& lowast;) - p/p-1 ) < q < p(& lowast;) , then there exist ground states for all lambda > 0.

This paper is concerned with the existence of solutions to the problem

This paper is concerned with the existence of solutions to the problem -(a+ b∫_ℝ^N|∇ u|^2 dx )Δ u +V(x)u+λ u = |u|^p-2u, x ∈ℝ^N, λ∈ℝ^+ where a, b>0 are constants, V ≥ 0 is a potential, N ≥ 1, and p ∈ (2+ 4/N,2^*). We use a more subtle analysis to revisit the limited problem(V ≡ 0), and obtain a new energy inequality and bifurcation results. Based on these observations, we establish the existence of bound state normalized solutions under different assumptions on V. These conclusions extend some known results in previous papers.

该文考察了一维格点上均为单位质量粒子的FPU型格点问题.这个系统的动力学方程描述如下(q)n=U'(qn+1-qn)-U'(qn-qn-1),n ∈ Z,其中U是相邻两个粒子相互作用产生的位势,qn(t)是第n个粒子在时刻t的状态.通过直接使用通常的变分方法,比起Pankov[10],Zhang和Ma[20]之前的工作,该文在更加宽泛的条件下研究了这类系统的基态行波解(即具有最小能量的非平凡行波解)的存在性.并且文中还讨论了孤立基态行波的单调性.