Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.     

Multiple fixed-sign solutions for a system of difference equations with Sturm–Liouville conditions

We consider the following system of difference equations:${\mathrm{\Delta }}^m{u}_i(k)+P_i(k,u_1(k),u_2(k),\dots ,u_n(k))=0,k\in \{0,1,\dots ,N\},i=1,2,\dots ,n$together with Sturm–Liouville boundary conditions${\mathrm{\Delta }}^j{u}_i(0)=0,0⩽j⩽m-3\text{,}$$\zeta {\mathrm{\Delta }}^{m-2}{u}_i(0)-\eta {\mathrm{\Delta }}^{m-1}{u}_i(0)=0,\omega {\mathrm{\Delta }}^{m-2}{u}_i(N+1)+\delta {\mathrm{\Delta }}^{m-1}{u}_i(N+1)=0\text{,}$where $m⩾2,N⩾m-1,\zeta >0,\omega >0,\eta ⩾0,\delta ⩾\omega ,\zeta \omega (N+1)+\zeta \delta +\eta \omega >0$. By using two different fixed point theorems, we develop criteria for the existence of three solutions of the system which are of fixed signs on $\{0,1,\dots ,N+m\}$. Examples are also included to illustrate the results obtained.     

Large solutions to the Monge–Ampère equations with nonlinear gradient terms: Existence and boundary behavior

In this paper, we obtain conditions about the existence and boundary behavior of (strictly) convex solutions to the Monge–Ampère equations with boundary blow-up

$\det ?D^{2}u(x)=b(x)f(u(x))\pm |\mathrm{?}u{|}^q,x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

and

$\det ?D^{2}u(x)=b(x)f(u(x))(1+|\mathrm{?}u{|}^q),x\in \mathrm{\Omega },u{|}_{?\mathrm{\Omega }}=+\infty ,$

where Ω is a strictly convex, bounded smooth domain in ${\mathbb{R}}^N$ with $N\ge 2$, $q\in [0,N]$ (or $q\in [0,N)$), $b\in C^{\infty }(\mathrm{\Omega })$ which is positive in Ω, but may vanish or blow up on the boundary, $f\in C[0,\infty )$, $f(0)=0$, and f is strictly increasing on $[0,\infty )$ (or $f\in C(\mathbb{R})$, $f(s)>0,?s\in \mathbb{R}$, and f is strictly increasing on $\mathbb{R}$).     

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system

$\{\begin{array}{cc}{u}_t=\mathrm{?}?(D(u)\mathrm{?}u)?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)\hfill & \hfill \\ \phantom{u_t=}+\mu u(1?u?w),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Here $\chi >0$, $\xi >0$ and $\mu >0$, $D(u)\ge c_D{u}^{m?1}$ for all $u>0$ with some $c_D>0$ and $D(u)>0$ for all $u\ge 0$. It is shown that if the ratio $\frac{\chi }{\mu }$ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.     

Elastic beam problem with higher order derivatives

We apply a fixed point theorem to obtain sufficient conditions for existence of triple positive solutions of the symmetric nonlinear fourth-order boundary value problem$u^{(4)}(t)=a(t)f(t,u(t),|u^{\prime }(t)|,u^{″}(t),|u^{‴}(t)|),t\in (-1,1)\text{,}$$u(-1)=u^{\prime }(-1)=0\text{,}$$u(-t)=u(t),t\in [-1,1]\text{,}$where f satisfies certain growth conditions. In the process, we obtain nontrivial a priori bounds on the derivatives of the solution.     

Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

Global convergence method for singularly perturbed boundary value problems

We give uniformly convergent splines difference scheme for singularly perturbed boundary value problems(1)$-\epsilon u^{″}+p(x)u^{\prime }+q(x)u=f(x),u(a)={\alpha }_0,u(b)={\alpha }_1,$by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration. We prove the $\mathrm{O}(\min (h^2,{\epsilon }^2))$ order of uniform convergence with respect to small parameter $\epsilon$ at nodes on uniform mesh and $\mathrm{O}(\min (h,\epsilon ))$ order of uniform global convergence with respect to the approximate solution given by $S(x)={\sum }_{i=1}^N{S}_{{\Delta }_i}(x)H(x_i-x)$ where H is the Heaviside function, which is the approximation for the closed form of the exact solution.     

On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition

For a competition-diffusion system involving the fractional Laplacian of the form

$?{(?\mathrm{\Delta })}^{s}u=u{v}^2,?{(?\mathrm{\Delta })}^{s}v=v{u}^2,u,v>0\text{in}{\mathbb{R}}^N,$

with $s\in (0,1)$, we prove that the maximal asymptotic growth rate for its entire solutions is 2s. Moreover, since we are able to construct symmetric solutions to the problem, when $N=2$ with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when $N\ge 3$. Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.