We study the quasiconvexity of Burkholder-Šverák function and prove an inequality for radial symmetric functions.

In this paper, we consider the existence of solutions to a class of fourth-order Kirchhoff-type elliptic equations with critical term and linear pertubation

where $\displaystyle2^\#=\frac{2N}{N-4}$ is the critical Sobolev exponent. With the help of the Concentration Compactness Principle, Ekeland's Variational Principle and Mountain Pass Lemma, we show that the (P.S.)$_c$ condition is locally satisfied and then obtain at least two nontrivial weak solutions under some assumptions on $a,\lambda$ and $\sigma$.

The studies on the convergence of Allen-Cahn equation mainly focus on Neumann boundary value problems, but there are few researches on other related boundary value problems. This paper mainly explores the limit varifold induced by the Dirichlet boundary value problem of the parabolic Allen-Cahn equation when the parameters tend to 0, which is Brakke's mean curvature flow.

This paper employs the constrained variational method to study the existence and qualitative properties of the normalized solutions to the Schrödinger equation with a partially confining potential and repulsive perturbation term. Specifically, it addresses the physically significant three dimensional cubic-quintic nonlinear case, which corresponds to the limiting situation of the cigar-shaped model for Bose-Einstein condensates (BEC) with a defocusing quintic nonlinear term. Furthermore, the stability of the corresponding standing wave solutions for the related time-dependent problem is also discussed.

We deduce optimal higher order regularity result for the even order geometrical elliptic system

where all the coefficients $ \{V_l, w_l\}_{l} $ are assumed to have the smallnest regularity and $ f $ lies in the borderline function space $ L\log L(B_1) $. As an application, we also obtain a compactness result.

In this paper, we first defined weighted Pucci operator $ M_{\lambda,\Lambda}^{+}(D(|x|^\alpha D u)) $ and $ M_{\lambda,\Lambda}^{-}(D(|x|^\alpha D u)) $, then we study the nonlinear Liouville theorem for inequalities

and

This paper considers the a priori estimates for a class of degenerate fully nonlinear equations arising in conformal geometry on manifolds with boundary. Based on these a priori estimates, we obtain an existence result using the continuity method.

In this paper, we consider the following Schrödinger equations with general logarithmic nonlinear terms

$\begin{equation*} (-\Delta)^s u = u(\log|u|)^{\alpha}\ \text{in}\ \mathbb{R}^N, \end{equation*}$

where $0<s<1$, $N>2s$, $\alpha\ge 1$ is a constant. By observing the convergent phenomenon of the power-law Schrödinger equation $(-\Delta)^s u = u(|u|^{\sigma}-1)^{\alpha}$ as $\sigma\to 0^+$, we show that the problem has a positive ground state solution if $(-1)^{\alpha}=-1$.

For $\frac{3}{2} < p \leq \infty$, when the norm of the initial data $\|\mathcal{F}_x f_0\|_{L^1 \cap L^p \cap \mathcal{X}^{-p}(\mathbb{R}^3_{\xi}; L^2(\mathbb{R}^3_v))}$ is sufficiently small, we construct global solutions to the Cauchy problem for the non-cutoff Boltzmann equation near equilibrium in the whole space $\mathbb{R}^3$. Here, $\mathcal{F}_x f_0(\xi, v)$ denotes the Fourier transform of $f_0(x, v)$ with respect to the spatial variable $x$, and $\mathcal{X}^{-p}$ is the $L^p$ space incorporating a Hardy potential.Compared to the $L^1_{\xi} \cap L^p_{\xi}$ space used in [15], we consider the low-regularity Sobolev space $L^1_{\xi} \cap L^p_{\xi} \cap \mathcal{X}^{-p}_{\xi}$ in the whole-space framework. Under the energy method framework, we establish a priori estimates to close the argument, thereby obtaining global solutions. In particular, we also derive the decay estimate, for any arbitrarily small $\delta>0,$

In this paper, we study the following biharmonic equation with Navier boundary condition:

$\begin{equation} \left\{ \begin{array}{ll} \Delta^2u=|u|^{p-1}u+f &\mbox{ in } \Omega,\\ \Delta u=u=0 &\mbox{ on } \partial\Omega, \end{array}\tag{$0.1_f$} \right. \end{equation}$

where $1<p<\frac{N+4}{N-4}$ ($1<p<\infty$ for $N=1,\,2,\,3,\,4$), and $\Omega$ is a smooth and bounded domain in $\mathbb{R}^N$ with boundary $\partial\Omega$. We prove that there exists an open dense subset of $L^2(\Omega)$ such that for any $f$ belongs to this set, (0.1f) has infinitely many solutions. This is an application of the topological result for a certain class of functionals developed by [Bahri A. J Funct Anal,1981, 41(3): 397--427].

In this paper, we are concerned with the following nonlinear Schrödinger system with quadratic nonlinearities

$\begin{align*} \begin{cases} -\epsilon^2\Delta u_1+V_1(x)u_1=\alpha u_1 u_{2} \ \text{ in } \mathbb{R}^N,\\ -\epsilon^2\Delta u_2+V_2(x)u_2=\frac{\alpha}{2}u_1^2+\beta u_2^{2} \ \text{ in } \mathbb{R}^N, \end{cases} \end{align*}$

where$2\leq N<6$, $\epsilon>0$is a small parameter,$\alpha>0$and$\alpha >\beta,$ the functions $V_i$ are positive, $V_i$, $|\nabla V_i| \in L^\infty(\mathbb{R}^N).$ As $\epsilon$ goes to zero, applying the finite dimensional reduction method, we construct synchronized solution which concentrates at the non-degenerate critical point of a new function constructed by the potential functions $V_{i}(x)(i=1,2).$ Moreover, by the contradiction argument combining some local Pohozeav identities and the blow-up technique we prove the uniqueness of this single-peak solution. Our results extend the results in [Gross M. Ann Inst H Poincar'e C Anal Non Lin'eaire, 2002] about a single Schrödinger equation to our system.

This paper studies the existence of normalized solutions for quasilinear Schrödinger equation with $L^2$-subcritical general nonlinearity. By using the compactness concentration principle, Schwartz symmetric technique and scaling method, this paper proves the existence and nonexistence of global least energy normalized solutions and the existence of local minimal normalized solutions. The main results can be viewed an extension of the results concerning about the existence of normalized solutions to the quasilinear equation with a pure power nonlinearity.

In this paper, we consider the existence and asymptotic properties of normalized solutions to a fourth-order Schrödinger equation with a positive second-order dispersion coefficient. In the mass supercritical regime, we study two types of local minimization problems and prove their equivalence in order to avoid the dependence of mass with respect to the locally constraint radius. Then, we prove the compactness of the corresponding minimizing sequences and the existence of ground states. Furthermore, by utilizing subtle energy estimates and analysis, we derive the asymptotic behavior of the ground state and the Lagrange multiplier as the parameter vanishes. This paper removes the radial symmetry condition in (Sci China Math, 2023, 66: 1237--1262), and provides an alternative but more transparent proof than that of (J Differential Equations, 2022, 330: 1--65).

JeanJean-Lu obtained the existence of ground states for the Schrödinger equation with mixed-type nonlinearities in the reference [Calc Var Partial Differential Equations, 2022]. Based on this fact, by analyzing the quantitative relationship between energy, frequency and mass, we prove that under appropriate rescalings, the ground state obtained in [Calc Var Partial Differential Equations, 2022] converges to the ground state of the classical Schrödinger equation with a single nonlinear term (when the mass declines) or converges to the ground state of the corresponding Thomas Fermi equation (when the mass tends to infinity). In particular, our conclusion holds true for the cubic-quintic nonlinear Schrödinger equation.

This paper investigates the asymptotic symmetry and monotonicity of solutions to a class of nonlocal first-order Hamilton-Jacobi equations. By extending the classical results on the nonlinear term $H(t,u)$ from the literature [Adv Math, 2021, 377: Art 107463] to the more general case of $H(t,x,u,\nabla u)$, we overcome the limitations of the original theoretical framework. The study employs the asymptotic moving plane method proposed in [Adv Math, 2021, 377: Art 107463] as the core tool. However, to address the new challenges posed by the gradient term $\nabla$u in the Hamiltonian, we make critical improvements to the construction of lower solution methods. This expands the applicability of the approach, enabling it to handle a broader range of nonlinear term types.

This paper derives the sub-Laplacian estimates of the Riemannian distance function on foliated Riemannian manifolds, and applies this result to establish the gradient estimates and Liouville-type theorems for foliated subelliptic harmonic maps from complete non-compact foliated Riemannian manifolds to Cartan-Hadamard manifolds. }