In this paper, we consider the following nonlinear Schrödinger equations with the critical Sobolev exponent and mixed nonlinearities:

\[\left\{\begin{aligned} & -\Delta u+\lambda u=t|u|^{q-2}u+|u|^{2^{*}-2}u\quad\text{in }\mathbb{R}^{N},\\ & u\in H^{1}(\mathbb{R}^{N}), \end{aligned}\right.\]

$N\geq 3$

$t>0$

$\lambda >0$

$2< q<2^{*}=\frac {2N}{N-2}$

1. (1) the above equation has two positive radial solutions for $N=3$, $2< q<4$ and $t>0$ sufficiently large, which gives a rigorous proof of the numerical conjecture in [J. Dávila, M. del Pino and I. Guerra. Non-uniqueness of positive ground states of non-linear Schrödinger equations. Proc. Lond. Math. Soc. 106 (2013), 318–344.];

2. (2) there exists $t_q^{*}>0$ for $2< q\leq 4$ such that the above equation has ground-states for $t\geq t_q^{*}$ in the case of $2< q<4$ and for $t>t_4^{*}$ in the case of $q=4$, while the above equation has no ground-states for $0< t< t_q^{*}$ for all $2< q\leq 4$, which, together with the well-known results on ground-states of the above equation, almost completely solve the existence of ground-states, except for $N=3$, $q=4$ and $t=t_4^{*}$.

$0<\overline {t}_{a,q}<+\infty$

$2< q<2+\frac {4}{N}$

$t>\overline {t}_{a,q}$

$\int _{\mathbb {R}^{N}}|u|^{2}{\rm d}x=a^{2}$

\[\left\{ \begin{aligned} & -\Delta u+\lambda u+(x_1^{2}+x_2^{2})u=|u|^{p-2}u\quad\text{in }\mathbb{R}^{3},\\ & u\in H^{1}(\mathbb{R}^{3}),\quad \int_{\mathbb{R}^{3}}|u|^{2}{\rm d}x=r^{2}, \end{aligned}\right.\]

$x=(x_1,x_2,x_3)\in \mathbb {R}^{3}$

$\frac {10}{3}< p<6$

$r>0$

$(u, \lambda )$

$\lambda$

$r>0$

In this paper, we study the boundary blow-up problem related to the infinity Laplacian

$$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$, we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.

In this paper we introduce new birth-and-death processes with partial catastrophe and study some of their properties. In particular, we obtain some estimates for the mean catastrophe time, and the first and second moments of the distribution of the process at a fixed time t. This is completed by some asymptotic results.

Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns. 21 (1996), 363–394); Grenier (Commun. Pure Appl. Math. 50 (1997), 821–865); Ukai (J. Math. Kyoto Univ. 26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity 24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

In this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.

In this paper, we focus on the formation behaviour of the kinetic Cucker–Smale model for initial datum without compact support for the position variable. Comparing with the case of compact support, the attractive force between particles is weak. First, we obtain the existence and uniqueness of the classical solution to the kinetic Cucker–Smale model by standard approximation method. Second, by using the characteristic flow, we overcome the difficulty brought by the weak attractive force between particles through some estimates and establish the formation behaviour, i.e., consensus of velocity, of the classical solution to the kinetic Cucker–Smale model. Finally, for the measure-valued solution to the kinetic Cucker–Smale model, the formation behaviour is also established.

In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations

\[ -\Delta_{\gamma} u + a(z)u = f(u) \quad \mbox{in}\ \mathbb{R}^{N}, \]

$\Delta _{\gamma }$

$z:=(x,y)\in \mathbb {R}^{m}\times \mathbb {R}^{k}$

$m+k=N\geqslant 3$

$f$

$a$

$a=1$

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

We investigate a reaction–diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components, we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk domains are given by a reaction–diffusion equation for the traces of the bulk solutions, including nonlinear reaction kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenisation results for surface diffusion. To cope with the nonlinear terms, we derive strong two-scale convergence results.

In this paper, we consider an initial-boundary value problem of Hsieh's equation with conservative nonlinearity. The global unique solvability in the framework of Sobolev is established. In particular, one of our main motivations is to investigate the boundary layer (BL) effect and the convergence rates as the diffusion parameter $\beta$ goes zero. It is shown that the BL-thickness is of the order $O(\beta ^{\gamma })$ with $0<\gamma <\frac {1}{2}$. We need to point out that, different from the previous work on nonconservative form of Hsieh's equations, the conservative nonlinearity $(\psi ^{\beta }\theta ^{\beta })_x$ implies that new nonlinear term $\psi _x^{\beta }\theta ^{\beta }$ needs to be handled. It is important that more regularities on the solution to the limit problem are required to obtain the convergence rates and BL-thickness. It is more difficult for initial-boundary problem due to the lack of boundary conditions (especially, higher-order derivatives) prevents us from applying the integration by part to derive the energy estimates directly. Thus it is more complicated than the case of nonconservative form. Consequently more subtle mathematical analysis needs to be introduced to overcome the difficulties.

In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system

\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]

$u$

$v$

$D\in C^{3}([0,\infty ))$

$0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$

$\xi >0$

${K_D}>0$

$\alpha >0$

$\Omega$

We prove the existence of nontrivial ground state solutions of the critical quasilinear Hénon equation $\displaystyle -\Delta _p u=|x|^{\alpha _1}|u|^{p^{*}(\alpha _1)-2}u-|x|^{\alpha _2}|u|^{p^{*}(\alpha _2)-2}u\ \ {\rm in}\ \mathbb {R}^{N}.$ It is a new problem in the sense that the signs of the coefficients of critical terms are opposite.

Savin [‘$\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$ and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$. We also prove an existence result of standing waves and prove their orbital stability.

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]