On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

On the positive radial solutions of a class of singular semilinear elliptic equations

In this paper, we consider the following elliptic equation(0.1)$\mathit{div}\left(A\right(|x|\left)\mathrm{?}u\right)+B(|x|)u^p=0\text{in}{\mathbb{R}}^n,$ where $p>1$, $n?3$, $A(|x|)>0$ is differentiable in ${\mathbb{R}}^n?\{0\}$ and $B(|x|)$ is a given nonnegative Hölder continuous function in ${\mathbb{R}}^n?\{0\}$. The asymptotic behavior at infinity and structure of separation property of positive radial solutions with different initial data for (0.1) are discussed. Moreover, the existence and separation property of infinitely many positive solutions for Hardy equation and an equation related to Caffarelli–Kohn–Nirenberg inequality are obtained respectively, as special cases.     

On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

Harnack inequality for quasilinear elliptic equations on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations

$\mathrm{div}\left(\frac{F^{\prime }(|\mathrm{?}u|)}{|\mathrm{?}u|}\mathrm{?}u\right)=h$

of p-Laplacian type on Riemannian manifolds, where an even function $F\in C^1\left(\mathbb{R}\right)\cap C^2(0,\infty )$ is supposed to be strictly convex on $(0,\infty )$. Under the assumption that either $F\in C^2(\mathbb{R})$ or its convex conjugate $F^{?}\in C^2(\mathbb{R})$ with some structural condition, we establish a (locally) uniform ABP type estimate and the Krylov–Safonov type Harnack inequality on Riemannian manifolds with the use of an intrinsic geometric quantity to the operator. Here, the $C^2$-regularities of F and $F^{?}$ account for degenerate and singular operators, respectively.     

Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient $f(z)$ of the critical nonlinearity. For sufficiently small $\lambda ,\mu >0$, there are at least k positive solutions of the semilinear elliptic systems$\{\begin{array}{c}?\mathrm{\Delta }u=\lambda g(z){|u|}^{p?2}u+\frac{\alpha }{\alpha +\beta }f(z){|u|}^{\alpha ?2}u{|v|}^{\beta }\text{in}\Omega ;\hfill \\ ?\mathrm{\Delta }v=\mu h(z){|v|}^{p?2}v+\frac{\beta }{\alpha +\beta }f(z){|u|}^{\alpha }{|v|}^{\beta ?2}v\text{in}\Omega ;\hfill \\ u=v=0\text{on}?\Omega ,\hfill \end{array}$ where $0\in \Omega ?{\mathbb{R}}^N$ is a bounded domain, $\alpha >1$, $\beta >1$ and $2<p<\alpha +\beta =2^{?}$ for $N>4$.     

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations on torus

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation${\mathrm{\partial }}_{t}u=i(f(x)\mathrm{\Delta }u+\nabla f(x)·\nabla u+k(x)|u{|}^{2}u)$on ${\mathbb{T}}^2$. We present the $L^2$-concentration property for general initial data and investigate the $L^2$-minimality.     

Elliptic operators with unbounded diffusion,drift and potential terms

We prove that the realization $A_p$ in $L^p({\mathbb{R}}^N),1<p<\infty$, of the elliptic operator $A=(1+|x{|}^{\alpha })\mathrm{\Delta }+b|x{|}^{\alpha ?1}\frac{x}{|x|}?\mathrm{?}?c|x{|}^{\beta }$ with domain $D(A_p)=\{u\in W^{2,p}({\mathbb{R}}^N)|Au\in L^p({\mathbb{R}}^N)\}$ generates a strongly continuous analytic semigroup $T(?)$ provided that $\alpha >2,\beta >\alpha ?2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [4] and in [16]. Moreover we show that $T(?)$ is consistent, immediately compact and ultracontractive.     

Existence of standing waves of nonlinear Schrödinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation ${\epsilon }^2\mathrm{\Delta }u?V(x)u+u^p=0$, $x\in {\mathbb{R}}^N$, we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for $p\in (\frac{N}{N?2},\frac{N+2}{N?2})$ or nonnegative V satisfies ${\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}_{|x|\to \infty }V(x){|x|}^2\log |x|>0$ for $p=\frac{N}{N?2}$.     

Uniqueness of degree-one Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7

For $\epsilon >0$, we consider the Ginzburg–Landau functional for ${\mathbb{R}}^N$-valued maps defined in the unit ball $B^N?{\mathbb{R}}^N$ with the vortex boundary data x on $?B^N$. In dimensions $N\ge 7$, we prove that, for every $\epsilon >0$, there exists a unique global minimizer $u_{\epsilon }$ of this problem; moreover, $u_{\epsilon }$ is symmetric and of the form $u_{\epsilon }(x)=f_{\epsilon }(|x|)\frac{x}{|x|}$ for $x\in B^N$.     

Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms

In this paper we investigate boundary blow-up solutions of the problem

$\{\begin{array}{c}?{\mathrm{\Delta }}_{p(x)}u+f(x,u)=\pm K(x)|\mathrm{?}u{|}^{m(x)}\text{ in }\mathrm{\Omega },\hfill \\ u(x)\to +\infty \text{as }d(x,?\mathrm{\Omega })\to 0,\hfill \end{array}$

where ${\mathrm{\Delta }}_{p(x)}u=\mathrm{div}(|\mathrm{?}u{|}^{p(x)?2}\mathrm{?}u)$ is called the $p(x)$-Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that $K(x)|\mathrm{?}u(x){|}^{m(x)}$ is a small perturbation, to the case in which $\pm K(x)|\mathrm{?}u{|}^{m(x)}$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,?\mathrm{\Omega })$ and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since $f(x,?)$ is not assumed to be monotone in this paper.     

A sharp Balian–Low uncertainty principle for shift-invariant spaces

A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators ${\{f_k\}}_{k=1}^K?L^2({\mathbb{R}}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy ${\int }_{{\mathbb{R}}^d}|x||f_k(x){|}^{2}dx=\infty$, namely, $\stackrel{?}{f_k}?H^{1/2}({\mathbb{R}}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+?}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1/2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1/2$.