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 $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

Local boundedness of weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

In this paper we prove that weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

${?}_{t}u?\mathrm{?}?({(u?z)}^{\alpha }|\mathrm{?}u{|}^{\gamma ?1}\mathrm{?}u)=f$

are locally bounded. Here, u describes the height of the water, z is a given function that represents the land elevation and f is a source term accounting for evaporation, infiltration or rainfall.     

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.     

Stability of attractors for the Kirchhoff wave equation with strong damping and critical nonlinearities

The paper investigates longtime dynamics of the Kirchhoff wave equation with strong damping and critical nonlinearities: $u_{tt}?(1+?{\Vert \mathrm{?}u\Vert }^2)\mathrm{\Delta }u?\mathrm{\Delta }{u}_t+h(u_t)+g(u)=f(x)$, with $?\in [0,1]$. The well-posedness and the existence of global and exponential attractors are established, and the stability of the attractors on the perturbation parameter ? is proved for the IBVP of the equation provided that both nonlinearities $h(s)$ and $g(s)$ are of critical growth.     

Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball

We prove the existence and multiplicity of positive radial solutions to the nonlinear system

$\{\begin{array}{c}\hfill ?\mathrm{\Delta }{u}_i=\lambda K_i(|x|)f_i(u_j)\text{ in }\mathrm{\Omega },\hfill \\ \hfill d_i\frac{?u_i}{?n}+{\stackrel{?}{c}}_i(u_i)u_i=0\text{ on }|x|=r_0,\hfill \\ \hfill u_i(x)\to 0\text{ as }|x|\to \infty ,\hfill \end{array}$

for a certain range of $\lambda >0$, where $i,j\in \{1,2\},i\ne j$, $\mathrm{\Omega }=\{x\in {\mathbb{R}}^N:|x|>r_0>0\}$, $N>2,d_i\ge 0$, $K_i:[r_0,\infty )\to (0,\infty )$, $\stackrel{?}{c}:[0,\infty )\to [0,\infty ),f_i:(0,\infty )\to \mathbb{R}$ are continuous with possible singularity ±∞ at 0 and satisfy a combined superlinear condition at ∞.     

Well posedness of nonlinear parabolic systems beyond duality

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial–boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system${?}_{t}u?\mathrm{div}(\nu (|\mathrm{?}u|)\mathrm{?}u)=?\mathrm{div}f$ with a given strictly positive bounded function ν, such that ${\lim }_{k\to \infty }?\nu (k)={\nu }_{\infty }$ and $f\in L^q$ with $q\in (1,\infty )$. The existence, uniqueness and regularity results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty )$.Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted $L^q$ spaces.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).     

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.     

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Let ${\{{\phi }_i\}}_{i=0}^{\infty }$ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{E}}_n(\mu )$, of random polynomials

$P_n(z):=\sum _{i=0}^n{\eta }_i{\phi }_i(z),$

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that ${\mathbb{E}}_n(|\mathrm{d}\xi |)$ admits an asymptotic expansion of the form

${\mathbb{E}}_n(|\mathrm{d}\xi |)～\frac{2}{\pi }\log ?(n+1)+\sum _{p=0}^{\infty }{A}_p{(n+1)}^{?p}$

(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case ${\mathbb{E}}_n(\mu )$ admits an analogous expansion with the coefficients $A_p$ depending on the measure μ for $p\ge 1$ (the leading order term and $A_0$ remain the same).     

Multiplicity and asymptotic behavior of solutions for quasilinear elliptic equations with small perturbations

This paper considers the following general form of quasilinear elliptic equation with a small perturbation:$\{\begin{array}{c}?{\sum }_{i,j=1}^N{D}_j(a_{ij}(x,u)D_{i}u)+\frac{1}{2}{\sum }_{i,j=1}^N{D}_t{a}_{ij}(x,u)D_{i}u{D}_{j}u\hfill \\ =f(x,u)+\epsilon g(x,u),x\in \mathrm{\Omega },u\in H_0^1(\mathrm{\Omega }),\hfill \end{array}$ where $\mathrm{\Omega }?{\mathbb{R}}^N(N\ge 3)$ is a bounded domain with smooth boundary and $|\epsilon |$ small enough. We assume the main term in the equation to have a mountain pass structure but do not suppose any conditions for the perturbation term $\epsilon g(x,u)$. Then we prove the equation possesses a positive solution, a negative solution and a sign-changing solution. Moreover, we are able to obtain the asymptotic behavior of these solutions as $\epsilon \to 0$.