Let $F: \mathbb R^{n}\rightarrow [0,+\infty) $ be a convex function of class $C^{2}(\mathbb R^{n}→\{0\})$ which is even and positively homogeneous of degree 1, and its polar $F^{0}$ represents a Finsler metric on $\mathbb R^{n}$. The anisotropic Sobolev norm in $W^{1,n} (\mathbb{R}^{n} )$ is defined by \begin{equation*} \|u\|_{F}=\bigg(\int_{\mathbb R^{n}}(F^{n}(\nabla u)+|u|^{n})dx\bigg)^{1/n}. \end{equation*} In this paper, the following sharp anisotropic Moser–Trudinger inequality involving $L^{n}$ norm \[ \underset{u\in W^{1,n}( \mathbb{R}^{n}), \Vert u \Vert _{F}\leq 1}{\sup}\int_{ \mathbb{R} ^{n}}\Phi ( \lambda_{n} \vert u \vert ^{\frac{n}{n-1}} ( 1+\alpha \Vert u \Vert _{n}^{n} ) ^{\frac{1}{n-1}} ) dx<+\infty \] in the entire space $\mathbb{R}^n$ for any $0<\alpha<1$ is established, where $\Phi ( t ) ={\rm e}^{t}-\sum_{j=0}^{n-2} \frac{t^{j}}{j!}$, $\lambda_{n}=n^{\frac{n}{n-1}}\kappa_{n}^{\frac{1}{n-1}}$ and $\kappa_{n}$ is the volume of the unit Wulff ball in $\mathbb{R}^n$. It is also shown that the above supremum is infinity for all $\alpha\geq1$. Moreover, we prove the supremum is attained, that is, there exists a maximizer for the above supremum when $\alpha>0$ is sufficiently small.

In this paper we study the existence of nontrivial solutions to the well-known Brezis-Nirenberg problem involving the fractional $p$-Laplace operator in unbounded cylinder type domains. By means of the fractional Poincaré inequality in unbounded cylindrical domains, we first study the asymptotic property of the first eigenvalue $\lambda_{p,s}(\widehat{\omega_{\delta}})$ with respect to the domain $\widehat{\omega_{\delta}}$. Then, by applying the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove the existence results. The present work complements the results of Mosconi-Perera-Squassina-Yang [The Brezis-Nirenberg problem for the fractional $p$-Laplacian. Calc. Var. Partial Differential Equations, 55(4), 25 pp. 2016] to unbounded domains and extends the classical Brezis-Nirenberg type results of Ramos-Wang-Willem [Positive solutions for elliptic equations with critical growth in unbounded domains. In: Chapman Hall/CRC Press, Boca Raton, 2000, 192-199] to the fractional $p$-Laplacian setting.

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^\alpha=f.$$ It is shown that when $5/3 \leq\alpha < 7/3$, the $\frac{3\alpha-5}{\alpha-1}$-dimensional parabolic Hausdorff measure of $\mathcal{S}$ is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [SIAM J. Math. Anal., 51, 228-255 (2019)] for $\alpha=2$ and $f=0$. The same result is valid for a 3D modified Navier-Stokes system.

This paper deals with the following prescribed boundary mean curvature problem in $\mathbb{B}^N$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=0,u>0,~~& y\in\mathbb{B}^N,\\ \dfrac{\partial u}{\partial \nu}+\dfrac{N-2}{2}u=\dfrac{N-2}{2}\tilde{K}(y)u^{2^\sharp-1}, ~~&y\in\mathbb{S}^{N-1}, \end{array} \right. \end{equation*} where $\tilde{K}(y)=\tilde{K}(|y'|,\tilde{y})$ is a bounded nonnegative function with $y=(y',\tilde{y})\in \mathbb{R}^2\times\mathbb{R}^{N-3}$, $2^\sharp=\frac{2(N-1)}{N-2}$. Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if $N\geq 5$ and $\tilde{K}(r,\tilde{y})$ has a stable critical point $(r_0,\tilde{y}_0)$ with $r_0>0$ and $\tilde{K}(r_0,\tilde{y}_0)>0$, then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of $\tilde{K}(r,\tilde{y})$.

We prove Liouville type theorems for stable and finite Morse index $H^1_{\rm loc}\cap L^\infty_{\rm loc}$ solutions of the nonlinear Schrödinger equation \[-\Delta u + \lambda|x|^a u=|x|^b|u|^{p-1}u \quad\text{ in } \mathbb{R}^N, \] where $N\ge2$, $\lambda>0$, $a,b>-2$ and $p>1$. Our analysis reveals that all stable solutions of the equation must be zero for all $p>1$. Furthermore, finite Morse index solutions must be zero if $N\ge3$ and $p\ge\frac{N+2+2b}{N-2}$. The main tools we use are integral estimates, a Poho\v{z}aev type identity and a monotonicity formula.

In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies $C^{1,\text{Dini}}$ condition at a boundary point, and the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg $C^{1,\text{Dini}}$ domains.

We prove the following properties: (1) Let $a\in \Lambda_{1,0,k,k'}^{m_0}({\mathbb R}^{n}\times {\mathbb R}^{n})$ with $m_0=-1|\frac {1} {p}-\frac {1} {2}|(n-1),\ n\geq 2\, (1< p \leq 2,\ k> \frac {n} {p},\ k'> 0;\ 2\le p\le \infty,\ k> \frac {n} {2},\ k'> 0$ respectively). Suppose the phase function $S$ is positively homogeneous in $\xi$-variables, non-degenerate and satisfies certain conditions. Then the Fourier integral operator $T$ is $L^p$-bounded. Applying the method of (1), we can obtain the $L^p$-boundedness of the Fourier integral operator if (2) the symbol $a \in \Lambda_{1,δ,k,k'}^{m_0},\ 0\le δ < 1$, with $m_{0},\, k,\, k'$ and $S$ given as in (1). Also, the method of (1) gives: (3) $a\in \Lambda_{1,δ ,k,k'}^{0},\ 0\leq δ < 1$ and $k,\, k'$ given as in (1), then the $L^{p}$-boundedness of the pseudo-differential operators holds, $1

<\infty$.

We study additional non-isospectral symmetries of multicomponent constrained $N=2$ supersymmetric Kadomtsev—Petviashvili (KP) hierarchies. These symmetries are shown to form an infinite-dimensional non-Abelian superloop superalgebra.

We first prove the $L^2$-boundedness of a Fourier integral operator where it's symbol $a\in S^0_{\frac {1} {2},\frac {1} {2}}({\mathbb R}^n\times {\mathbb R}^n)$ and the phase function $S$ is non-degenerate, satisfies certain conditions and may not be positively homogeneous in $\xi$-variables. Then we use the above property, Paley's inequality, covering lemma of Calderon and Zygmund etc., and obtain the $L^p$-boundedness of Fourier integral operators if (1) the symbol $a\in \Lambda_{k}^{m_0}$ and ${\rm Supp}\ a=E\times {\mathbb R}^n$, with $E$ a compact set of ${\mathbb R}^n (m_0= -|\frac {1} {p}-\frac {1} {2}|n, 1<p\leq 2, k>\frac {n} {2} ; 2<p<\infty, k>\frac {n} {p})$, (2) the symbol $a\in \Lambda_{0,k,k{'}}^{m_0} ( m_{0}=-|\frac {1} {p}-\frac {1} {2}|n, 1<p\leq 2, k>\frac {n} {2}, k'>\frac {n} {p}; 2<p<\infty, k>\frac {n} {p}, k'>\frac {n} {2})$ with the phase function $S(x,\xi)=x\xi+h(x,\xi),x,\xi\in {\mathbb R}^n$ non-degenerate, satisfying certain conditions and $∂_\xi h\in S_{1,0}^{0}({\mathbb R}^n\times {\mathbb R}^n)$, or (3) the symbol $a\in \Lambda_{0,k,k'}^{m_0}$, the requirements for $m_0,k,k'$ are the same as in (2), and $∂_\xi h$ is not in $S_{1,0}^{0} ({\mathbb R}^n\times {\mathbb R}^n)$ but the phase function $S$ is non-degenerate, satisfies certain conditions and is positively homogeneous in $\xi$-variables.

We mainly study the nonexistence of quasi-harmonic spheres and harmonic spheres into spheres of any dimension which omits a neighbourhood of totally geodesic submanifold of co-dimension 2. We will show that such target admits no quasi-harmonic spheres and harmonic spheres.

In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state in Killip-Visan-Zhang [The focusing cubic NLS on exterior domains in three dimensions. Appl. Math. Res. Express. AMRX, 1, 146-180 (2016)] by utilizing the method of Dodson and Murphy [A new proof of scattering below the ground state for the 3d radial focusing cubic NLS. Proc. Amer. Math. Soc., 145, 4859-4867 (2017)] and the dispersive estimate in Ivanovici and Lebeau [Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples. Comp. Rend. Math., 355, 774-779 (2017)], which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimates, we prove that the solution decays at a large time interval, which meets the scattering criterion.

We establish the global well-posedness for the multidimensional chemotaxis model with some classes of large initial data, especially the case when the rate of variation of ln v₀ (v₀ is the chemical concentration) contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D. Besides, we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption, which include specially the situations of d=2, 3 and improve the previous time-decay rates. Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.

This paper is concerned with the following Chern-Simons-Schrödinger equation -Δu + V (|x|)u +(∫_|x|^∞h(s)/su²(s)ds + h²(|x|)/|x|²)u=a(|x|)f(u) in R², where h(s)=∫₀^sl/2u²(l)dl, V, a:R⁺ → R are radially symmetric potentials and the nonlinearity f:R → R is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions. In particular, some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the k-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also solutions of the classical Neumann problem converge to a translating solution.