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 $|u{|}^{q?2}u$.     

New results on permutation binomials of finite fields

After a brief review of the existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of ${\mathbb{F}}_{q^2}$ of the form $X^n(X^{d(q-1)}+a)$, where n and d are positive integers and $a\in {\mathbb{F}}_{q^2}^{⁎}$. Our contributions include two nonexistence results: (1) If q is even and sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{3(q-1)}+a)$ is not a PB of ${\mathbb{F}}_{q^2}$. (2) If $2\le d|q+1$, q is sufficiently large and $a^{q+1}\ne 1$, then $X^n(X^{d(q-1)}+a)$ is not a PB of ${\mathbb{F}}_{q^2}$ under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with $n=1$.     

Long-time behavior for 3D MHD equations with nonlinear damping

The existence of global attractors is proved for the MHD equations with damping terms $|u{|}^{\alpha ?1}u$ and $|B{|}^{\beta ?1}B$ $(\alpha ,\beta ?1)$ on a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^3$. First we establish the well-posedness of strong solutions. Then, the continuity of the corresponding semigroup is verified under the assumption $\alpha ,\beta <5$, which is guided by Gagliardo-Nirenberg inequality. Finally, the system is shown to possess an $(\mathbb{V},\mathbb{V})$-global attractor and an $(\mathbb{V},{\mathbf{\text{H}}}^2)$-global attractor.     

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 $\alpha <0$.