Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.     

The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation

The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δu and the magnetic diffusion Δb by $?{(?\mathrm{\Delta })}^{\alpha }u$ and $?{(?\mathrm{\Delta })}^{\beta }b$, respectively, the resulting equations with $\alpha \ge \frac{5}{4}$ and $\alpha +\beta \ge \frac{5}{2}$ then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion $?{(?{\mathrm{\Delta }}_h)}^{\frac{5}{4}}b$, where ${\mathrm{\Delta }}_h={?}_1^2+{?}_2^2$.     

Moser–Trudinger inequality involving the anisotropic Dirichlet norm (∫ΩFN(∇u)dx)1N on W01,N(Ω)

We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm ${({\int }_{\mathrm{\Omega }}{F}^N(\mathrm{?}u)dx)}^{\frac{1}{N}}$ on $W_0^{1,N}(\mathrm{\Omega })$ for $N\ge 2$. Here F is convex and homogeneous of degree 1, and its polar $F^o$ represents a Finsler metric on ${\mathbb{R}}^N$. Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.     

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system

$\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_1{v}_t=\mathrm{\Delta }v?v+w\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_2{w}_t=\mathrm{\Delta }w?w+u\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty )\hfill \end{array}$

under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^4$ with positive parameters ${\tau }_1,{\tau }_2,\chi >0$ and nonnegative smooth initial data $(u_0,v_0,w_0)$.Global existence and boundedness of solutions were shown if ${6u_{0}6}_{L^1(\mathrm{\Omega })}<{(8\pi )}^2/\chi$ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ${6u_{0}6}_{L^1(\mathrm{\Omega })}>{(8\pi )}^2/\chi$. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has $8\pi /\chi$-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in ${\mathbb{R}}^4$.     

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and concentration behavior of minimizers for $i_V(c)=\underset{u\in S_c}{\inf }?I_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ and

$I_V(u)=\frac{1}{2}\underset{{\mathbb{R}}^N}{\int }(a|\mathrm{?}u{|}^2+V(x)|u{|}^2)+\frac{b}{4}{(\underset{{\mathbb{R}}^N}{\int }|\mathrm{?}u{|}^2)}^2?\frac{1}{p}\underset{{\mathbb{R}}^N}{\int }|u{|}^p,$

where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of $i_V(c)$ for $2<p<2^{?}$ when $V(x)\ge 0$, $V(x)\in {L^{\infty }}_{loc}({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }?V(x)=+\infty$. For the case $p\in (2,\frac{2N+8}{N})\backslash \{4\}$, we prove that the global constraint minimizers $u_c$ of $i_V(c)$ behave like

$u_c(x)\approx \frac{c}{|Q_p{|}_2}{\left(\frac{m_c}{c}\right)}^{\frac{N}{2}}{Q}_p(\frac{m_c}{c}x?z_c),$

for some $z_c\in {\mathbb{R}}^N$ when c is large, where $Q_p$ is, up to translations, the unique positive solution of $?\frac{N(p?2)}{4}\mathrm{\Delta }{Q}_p+\frac{2N?p(N?2)}{4}{Q}_p=|Q_p{|}^{p?2}{Q}_p$ in ${\mathbb{R}}^N$ and $m_c={(\frac{\sqrt{a^2{D}_1^2?4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2})}^{\frac{1}{2}}$, $D_1=\frac{Np?2N?4}{2N(p?2)}$ and $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization

The Keller–Segel–Navier–Stokes system

(?)$\{\begin{array}{ccc}\hfill n_t+u?\mathrm{?}n& \hfill =\hfill & \mathrm{\Delta }n?\chi \mathrm{?}?(n\mathrm{?}c)+\rho n?\mu n^2,\hfill \\ \hfill c_t+u?\mathrm{?}c& \hfill =\hfill & \mathrm{\Delta }c?c+n,\hfill \\ \hfill u_t+(u?\mathrm{?})u& \hfill =\hfill & \mathrm{\Delta }u+\mathrm{?}P+n\mathrm{?}?+f(x,t),\hfill \\ \hfill \mathrm{?}?u& \hfill =\hfill & 0,\hfill \end{array}$

is considered in a bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with smooth boundary, where $?\in W^{1,\infty }(\mathrm{\Omega })$ and $f\in C^1(\overline{\mathrm{\Omega }}\times [0,\infty ))$, and where $\chi >0,\rho \in \mathbb{R}$ and $\mu >0$ are given parameters.It is proved that under the assumption that ${\sup }_{t>0}?{\int }_t^{t+1}{6f(?,s)6}_{L^{\frac{6}{5}}(\mathrm{\Omega })}ds$ be finite, for any sufficiently regular initial data $(n_0,c_0,u_0)$ satisfying $n_0\ge 0$ and $c_0\ge 0$, the initial-value problem for (?) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in $L^1(\mathrm{\Omega })\times L^6(\mathrm{\Omega })\times L^2(\mathrm{\Omega };{\mathbb{R}}^3)$.Moreover, under the explicit hypothesis that $\mu >\frac{\chi \sqrt{{\rho }_{+}}}{4}$, these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying

$\begin{array}{c}(n(?,t),c(?,t))\to (\frac{{\rho }_{+}}{\mu },\frac{{\rho }_{+}}{\mu })\text{in }{L}^1(\mathrm{\Omega })\times L^p(\mathrm{\Omega })\hfill \\ \text{for all }p\in [1,6)\text{as }t\to \infty .\hfill \end{array}$

Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that $u(?,t)\to 0$ in $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ as $t\to \infty$.     

Random attractor for the 3D viscous primitive equations driven by fractional noises

We develop a new and general method to prove the existence of the random attractor (strong attractor) for the primitive equations (PEs) of large-scale ocean and atmosphere dynamics under non-periodic boundary conditions and driven by infinite-dimensional additive fractional Wiener processes. In contrast to our new method, the common method, compact Sobolev embedding theorem, is to obtain the time-uniform a priori estimates in some Sobolev space whose regularity is higher than the solution space. But this method can not be applied to the 3D stochastic PEs with the non-periodic boundary conditions. Therefore, the existence of universal attractor (weak attractor) was established in previous works (see [15], [16]). The main idea of our method is that we first derive that $\mathbb{P}$-almost surely the solution operator of stochastic PEs is compact. Then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set. We should point out that our method has some advantages over the common method of using compact Sobolev embedding theorem, i.e., using our method we only need to obtain time-uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system, while the common method need to establish time-uniform a priori estimates in a more regular functional space than the solution space. Take the stochastic PEs for example, as the unique strong solution to the stochastic PEs belongs to $C([0,T];{(H^1(?))}^3)$, in view of our method, we only need to obtain the time-uniform a priori estimates in the solution space ${(H^1(?))}^3$ to prove the existence of random attractor for this stochastic system, while the common method need to establish time-uniform a priori estimates for the solution in the functional space ${(H^2(?))}^3$. However, time-uniform a priori estimates in ${(H^2(?))}^3$ for the solution to stochastic PEs are too difficult to be established. The present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by Wiener noises, fractional noises and even jump noises. In a forth coming paper, using this new method we [46] prove the existence of random attractor for the stochastic nematic liquid crystals equations. This is the first result about the long-time behavior of stochastic nematic liquid crystals equations.     

Global regularity for a class of 2D generalized tropical climate models

This paper establishes the global existence and regularity of solutions to a two-dimensional (2D) tropical climate model (TCM) with fractional dissipation. The inviscid counterpart of this model was derived by Frierson, Majda and Pauluis [8] as a model for tropical geophysical flows. This model reflects the interaction and coupling among the barotropic mode u, the first baroclinic mode v of the velocity and the temperature θ. The systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances. Mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization. The aim here is the global regularity with the least dissipation. We prove two main results: first, the global regularity of the system with ${(?\mathrm{\Delta })}^{\beta }v$ and ${(?\mathrm{\Delta })}^{\gamma }\theta$ for $\beta >1$ and $\beta +\gamma >\frac{3}{2}$; and second, the global regularity of the system with ${(?\mathrm{\Delta })}^{\beta }v$ for $\beta >\frac{3}{2}$. The proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal. The key tools employed here include the maximal regularity for general fractional heat operators, the Littlewood–Paley decomposition and Besov space techniques, lower bounds involving fractional Laplacian and simultaneous estimates of several coupled quantities.     

Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:

$\{\begin{array}{cc}\hfill & ?{\epsilon }^2\mathrm{\Delta }u+V(x)u+{\lambda }_1|u{|}^{2}u+{\lambda }_2(K?|u{|}^2)u=0\text{ in }{\mathbb{R}}^3,\hfill \\ \hfill & u>0,u\in H^1({\mathbb{R}}^3),\hfill \end{array}$

where ε is a small positive parameter, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, ? denotes the convolution, $K(x)=\frac{1?3{\cos }^2?\theta }{|x{|}^3}$ and $\theta =\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector x. Under certain assumptions on $({\lambda }_1,{\lambda }_2)\in {\mathbb{R}}^2$, we construct a family of positive solutions $u_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around the local minima of V as $\epsilon \to 0$. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.     

Global existence and decay estimate of solutions to magneto-micropolar fluid equations

We are concerned with magneto-micropolar fluid equations (1.3)–(1.4). The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the magneto-micropolar-Navier–Stokes (MMNS) system, we obtain global existence and large time behavior of solutions near a constant states in ${\mathbb{R}}^3$. Appealing to a refined pure energy method, we first obtain a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms ${\dot{H}}^{?s}$ $(0\le s<\frac{3}{2})$ or homogeneous Besov norms ${\dot{B}}_{2,\infty }^{?s}$ $(0<s\le \frac{3}{2})$, we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual $L^p?L^2$ $(1\le p\le 2)$ type of the decay rates without requiring that the $L^p$ norm of initial data is small. At last, we derive a weak solution to (1.3)–(1.4) in ${\mathbb{R}}^2$ with large initial data.     

Regularity theory for Ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where T is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $?C{6M6}^{?(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.     

On the global Cauchy problem for the Hartree equation with rapidly decaying initial data

This paper is concerned with the Cauchy problem for the Hartree equation on ${\mathbb{R}}^n,n\in \mathbb{N}$ with the nonlinearity of type $(|?{|}^{?\gamma }?|u{|}^2)u,0<\gamma <n$. It is shown that a global solution with some twisted persistence property exists for data in the space $L^p\cap L^2,1\le p\le 2$ under some suitable conditions on γ and spatial dimension $n\in \mathbb{N}$. It is also shown that the global solution u has a smoothing effect in terms of spatial integrability in the sense that the map $t?u(t)$ is well defined and continuous from $\mathbb{R}?\{0\}$ to $L^{p^{\prime }}$, which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat $L^p$-spaces are also presented. The local and global results are proved by combining arguments by Carles–Mouzaoui with a new functional framework introduced by Zhou. Furthermore, it is also shown that the global results can be improved via generalized dispersive estimates in the case of one space dimension.     

Uniqueness of critical points and maximum principles of the singular minimal surface equation

We consider a smooth solution $u>0$ of the singular minimal surface equation $\sqrt{1+|Du{|}^2}\text{ div}(Du/\sqrt{1+|Du{|}^2})=\alpha /u$ defined in a bounded strictly convex domain of ${\mathbb{R}}^2$ with constant boundary condition. If $\alpha <0$, we prove the existence a unique critical point of u. We also derive some $C^0$ and $C^1$ estimates of u by using the theory of maximum principles of Payne and Philippin for a certain family of Φ-functions. Finally we deduce an existence theorem of the Dirichlet problem when $\alpha <0$.