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 $|u{|}^{q?2}u$.     

Concentration on curves for a nonlinear Schrödinger problem with electromagnetic potential

We prove the existence of solutions to the nonlinear Schrödinger equation ${\epsilon }^2{(i\mathrm{?}+\mathit{A})}^{2}u+V(y)u?|u{|}^{p?1}u=0$ in ${\mathbb{R}}^2$ with a magnetic potential $\mathit{A}=(A_1,A_2)$. Here V represents the electric potential, the index p is greater than 1. Along some sequence $\{{\epsilon }_n\}$ tending to zero we exhibit complex-value solutions that concentrate along some closed curves.     

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 $\alpha <0$.     

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.     

Classification of cyclic codes over a non-Galois chain ring Zp[u]/〈u3〉

We explicitly determine generators of cyclic codes over a non-Galois finite chain ring ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$, where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$ and four types of non-principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$, which are associated with cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$. We then obtain a mass formula for cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$.     

Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields

We characterize all finite metabelian 2-groups G whose abelianizations $G^{\mathrm{ab}}$ are of type $(2,2^n)$, with $n\ge 2$, and for which their commutator subgroups $G^{\prime }$ have $\text{rank}=2$. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group ${\mathrm{Cl}}_2(k)?(2,2^n)$, $n\ge 2$, such that the rank of ${\mathrm{Cl}}_2(k^1)=2$ where $k^1$ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.     

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.     

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.     

Existence and nonexistence of positive solutions for a static Schrödinger–Poisson–Slater equation

In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth

$?△u+(u^2?\frac{1}{|4\pi x|})u=\mu |u{|}^{p?1}u+|u{|}^{4}u,\text{in}{\mathbb{R}}^3,$

where $\mu >0$ and $p\in (11/7,5)$. For the case of $p\in (2,5)$. We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of $p=2$, we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of $p\in (11/7,2)$, we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for $\mu \in (0,{\mu }^{?})$ with some ${\mu }^{?}>0$. The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case.     

Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion

This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space ${\mathbb{R}}^3$. We establish that, in the inviscid resistive case, the energy ${6b(t)6}_2^2$ vanishes and ${6u(t)6}_2^2$ converges to a constant as time tends to infinity provided the velocity is bounded in $W^{1?\alpha ,\frac{3}{\alpha }}({\mathbb{R}}^3)$; in the viscous non-resistive case, the energy ${6u(t)6}_2^2$ vanishes and ${6b(t)6}_2^2$ converges to a constant provided the magnetic field is bounded in $W^{1?\beta ,\infty }({\mathbb{R}}^3)$. In summary, one single diffusion, being as weak as ${(?\mathrm{\Delta })}^{\alpha }b$ or ${(?\mathrm{\Delta })}^{\beta }u$ with small enough $\alpha ,\beta$, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.     

The regularity of a semilinear elliptic system with quadratic growth of gradient

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form

(0.1)$\mathrm{\Delta }u=Q(x,u,\mathrm{?}u),$

for $u:{\mathbb{R}}^n\to {\mathbb{R}}^K$, Q has quadratic growth in ?u. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When $n=2$, such a system does not have smooth regularity in general for $W^{1,2}$ weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for $n=2$) and F. Béthuel (for $n\ge 3$), assert that a $W^{1,n}$ weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1), that a $W^{1,n}$ weak solution of the system is smooth for $n\ge 3$. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a $W^{2,n/2}$ weak solution of such system is always smooth, for $n\ge 5$. We also construct various examples, and these examples show that our regularity results are optimal in various sense.     

On discrete Brunn-Minkowski and isoperimetric type inequalities

We show that the lattice point enumerator ${\mathrm{G}}_n(?)$ satisfies${\mathrm{G}}_n{(tK+sL+{(?1,?t+s?)}^n)}^{1/n}\ge t{\mathrm{G}}_n{(K)}^{1/n}+s{\mathrm{G}}_n{(L)}^{1/n}$ for any $K,L?{\mathbb{R}}^n$ bounded sets with integer points and all $t,s\ge 0$.We also prove that a certain family of compact sets, extending that of cubes ${[?m,m]}^n$, with $m\in \mathbb{N}$, minimizes the functional ${\mathrm{G}}_n(K+t{[?1,1]}^n)$, for any $t\ge 0$, among those bounded sets $K?{\mathbb{R}}^n$ with given positive lattice point enumerator.Finally, we show that these new discrete inequalities imply the corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.