1. CMB Online first
 Steinerberger, Stefan

An Endpoint Alexandrov Bakelman Pucci estimate in the Plane
The classical AlexandrovBakelmanPucci estimate
for the Laplacian states
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c_{s,n} \operatorname{diam}(\Omega)^{2\frac{n}{s}}
\left\ \Delta u
\right\_{L^s(\Omega)}$$
where $\Omega \subset \mathbb{R}^n$, $u \in C^2(\Omega) \cap
C(\overline{\Omega})$ and $s \gt n/2$. The inequality fails for
$s = n/2$. A Sobolev embedding result of Milman and Pustylnik,
originally phrased in a slightly different context, implies an
endpoint inequality: if $n \geq 3$ and $\Omega \subset \mathbb{R}^n$
is bounded, then
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c_n
\left\ \Delta u
\right\_{L^{\frac{n}{2},1}(\Omega)},$$
where $L^{p,q}$ is the Lorentz space refinement of $L^p$. This
inequality fails for $n=2$ and we prove a sharp substitute result:
there exists $c\gt 0$ such that for all $\Omega \subset \mathbb{R}^2$
with finite measure
$$ \max_{x \in \Omega}{ u(x)} \leq \max_{x \in \partial \Omega}{u(x)}
+ c \max_{x \in \Omega} \int_{y \in \Omega}{ \max
\left\{ 1, \log{
\left(\frac{\Omega}{\xy\^2}
\right)}
\right\}
\left \Delta u(y)
\right dy}.$$
This is somewhat dual to the classical TrudingerMoser inequality;
we also note that it is sharper than the usual estimates given
in Orlicz spaces, the proof is rearrangementfree.
The Laplacian can be replaced by any uniformly elliptic operator
in divergence form.
Keywords:AlexandrovBakelmanPucci estimate, second order Sobolev inequality, TrudingerMoser inequality Categories:35A23, 35B50, 28A75, 49Q20 

2. CMB Online first
3. CMB 2018 (vol 61 pp. 738)
 CruzUribe, David; Rodney, Scott; Rosta, Emily

PoincarÃ© Inequalities and Neumann Problems for the $p$Laplacian
We prove an equivalence between weighted PoincarÃ© inequalities
and
the existence of weak solutions to a Neumann problem related
to a
degenerate $p$Laplacian. The PoincarÃ© inequalities are
formulated in the context of degenerate Sobolev spaces defined
in
terms of a quadratic form, and the associated matrix is the
source of
the degeneracy in the $p$Laplacian.
Keywords:degenerate Sobolev space, $p$Laplacian, PoincarÃ© inequalities Categories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35 

4. CMB 2018 (vol 61 pp. 787)
5. CMB Online first
 Rosales, Leobardo

Generalizing Hopf's boundary point lemma
We give a Hopf boundary point lemma for weak solutions of linear
divergence form uniformly elliptic equations, with HÃ¶lder
continuous toporder coefficients and lowerorder coefficients
in a Morrey space.
Keywords:partial differential equation, divergence form, Hopf boundary point lemma Categories:35B50, 35A07 

6. CMB 2017 (vol 61 pp. 473)
 Awonusika, Richard; Taheri, Ali

A spectral identity on Jacobi polynomials and its analytic implications
The Jacobi coefficients $c^{\ell}_{j}(\alpha,\beta)$ ($1\leq
j\leq \ell$, $\alpha,\beta\gt 1$) are linked to the Maclaurin
spectral expansion of the Schwartz kernel of functions of the
Laplacian on a compact rank one symmetric space. It
is proved that these coefficients can be computed by transforming
the even derivatives of the the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$ ($k\geq 0, \alpha,\beta\gt 1$) into a spectral sum associated with
the Jacobi operator. The first few coefficients are explicitly
computed and a direct trace
interpretation of the Maclaurin coefficients is presented.
Keywords:Jacobi coefficient, LaplaceBeltrami operator, symmetric space, Maclaurin expansion, Jacobi polynomial Categories:33C05, 33C45, 35A08, 35C05, 35C10, 35C15 

7. CMB 2017 (vol 61 pp. 353)
 Qin, Dongdong; He, Yubo; Tang, Xianhua

Ground State and Multiple Solutions for Kirchhoff Type Equations with Critical Exponent
In this paper, we consider the following
critical Kirchhoff type equation:
\begin{align*}
\left\{
\begin{array}{lll}

\left(a+b\int_{\Omega}\nabla u^2
\right)\Delta u=Q(x)u^4u + \lambda u^{q1}u,~~\mbox{in}~~\Omega,
\\
u=0,\quad \text{on}\quad \partial \Omega,
\end{array}
\right.
\end{align*}
By using variational methods that are constrained to the Nehari
manifold,
we prove that the above equation has a ground state solution
for the case when $3\lt q\lt 5$.
The relation between the number of maxima of $Q$
and the number of positive solutions for the problem is also
investigated.
Keywords:Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state Categories:35J20, 35J60, 35J25 

8. CMB 2017 (vol 61 pp. 438)
 Zhang, Tao; Zhou, Chunqin

Classification of Solutions for Harmonic Functions with Neumann Boundary Value
In this paper, we classify all solutions of
\[
\left\{
\begin{array}{rcll}
\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+},
\\
\dfrac{\partial u}{\partial t}&=&cx^{\beta}e^{u} \quad
&\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\},
\\
\end{array}
\right.
\]
with the finite conditions
\[
\int_{\partial \mathbb{R}^{2}_{+}}x^{\beta}e^{u}ds \lt C,
\qquad
\sup\limits_{\overline{\mathbb{R}^{2}_{+}}}{u(x)}\lt C.
\]
Here, $c$ is a positive number and $\beta \gt 1$.
Keywords:Neumann problem, singular coefficient, classification of solutions Categories:35A05, 35J65 

9. CMB 2017 (vol 61 pp. 376)
 Sebbar, Abdellah; AlShbeil, Isra

Elliptic Zeta Functions and Equivariant Functions
In this paper we establish a close connection between three
notions attached to a modular subgroup. Namely the set of weight
two meromorphic modular forms, the set of equivariant functions
on the upper halfplane commuting with the action of the modular
subgroup and the set of elliptic zeta functions generalizing
the Weierstrass zeta functions. In particular, we show that the
equivariant functions can be parameterized by modular objects
as well as by elliptic objects.
Keywords:modular form, equivariant function, elliptic zeta function Categories:11F12, 35Q15, 32L10 

10. CMB 2017 (vol 60 pp. 536)
11. CMB 2017 (vol 61 pp. 142)
 Li, Bao Qin

An Equivalent Form of Picard's Theorem and Beyond
This paper gives an equivalent form of Picard's
theorem via entire solutions of the functional equation $f^2+g^2=1$,
and then its improvements and applications to certain nonlinear
(ordinary and partial) differential equations.
Keywords:entire function, Picard's Theorem, functional equation, partial differential equation Categories:30D20, 32A15, 35F20 

12. CMB 2017 (vol 60 pp. 422)
 Tang, Xianhua

New Superquadratic Conditions for Asymptotically Periodic SchrÃ¶dinger Equations
This paper is dedicated to studying the
semilinear SchrÃ¶dinger equation
$$
\left\{
\begin{array}{ll}
\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N},
\\
u\in H^{1}({\mathbf{R}}^{N}),
\end{array}
\right.
$$
where $f$ is a superlinear, subcritical nonlinearity. It focuses
on the case where
$V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1periodic
in each of $x_1, x_2, \ldots, x_N$ and
$\sup[\sigma(\triangle +V_0)\cap (\infty, 0)]\lt 0\lt \inf[\sigma(\triangle
+V_0)\cap (0, \infty)]$,
$V_1\in C(\mathbf{R}^N)$ and $\lim_{x\to\infty}V_1(x)=0$. A new superquadratic
condition is obtained,
which is weaker than some well known results.
Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of NehariPankov type Categories:35J20, 35J60 

13. CMB 2017 (vol 60 pp. 436)
 Weng, Peixuan; Liu, Li

Globally Asymptotic Stability of a Delayed IntegroDifferential Equation with Nonlocal Diffusion
We study a population model with nonlocal diffusion, which
is a delayed integrodifferential equation with double nonlinearity
and two integrable kernels. By comparison method and analytical
technique, we obtain globally asymptotic stability of the zero
solution and the positive equilibrium. The results obtained
reveal that the globally asymptotic stability only depends on
the property of nonlinearity. As application, an example for
a population model with age structure is discussed at the end
of the article.
Keywords:integrodifferential equation, nonlocal diffusion, equilibrium, globally asymptotic stability, population model with age structure Categories:45J05, 35K57, 92D25 

14. CMB 2016 (vol 59 pp. 606)
 Mihăilescu, Mihai; Moroşanu, Gheorghe

Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition
The
eigenvalue problem $\Delta_p u\Delta_q u=\lambdau^{q2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the wellknown case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.
Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods Categories:35J60, 35J92, 46E30, 49R05 

15. CMB 2016 (vol 59 pp. 734)
16. CMB 2016 (vol 59 pp. 417)
 Song, Hongxue; Chen, Caisheng; Yan, Qinglun

Existence of Multiple Solutions for a $p$Laplacian System in $\textbf{R}^{N}$ with Signchanging Weight Functions
In this paper, we consider the quasilinear elliptic
problem
\[
\left\{
\begin{aligned}
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla u^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla u^{p2}\nabla u
\right)
\\
&
\qquad=\frac{\alpha}{\alpha+\beta}H(x)u^{\alpha2}uv^{\beta}+\lambda
h_{1}(x)u^{q2}u,
\\
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla v^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla v^{p2}\nabla v
\right)
\\
&
\qquad=\frac{\beta}{\alpha+\beta}H(x)v^{\beta2}vu^{\alpha}+\mu
h_{2}(x)v^{q2}v,
\\
&u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N}
\end{aligned}
\right.
\]
where $\lambda, \mu\gt 0$, $1\lt p\lt N$,
$1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{Np}$, $0\leq
a\lt \frac{Np}{p}$, $a\leq b\lt a+1$, $d=a+1b\gt 0$, $M(s)=k+l s^{\tau}$,
$k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$
are
continuous functions which change sign in $\mathbb{R}^{N}$. We
will prove that the problem has at least two positive solutions
by
using the Nehari manifold and the fibering maps associated with
the Euler functional for this problem.
Keywords:Nehari manifold, quasilinear elliptic system, $p$Laplacian operator, concave and convex nonlinearities Category:35J66 

17. CMB 2016 (vol 59 pp. 553)
 Kachmar, Ayman

A New Formula for the Energy of Bulk Superconductivity
The energy of a type II superconductor submitted to an external
magnetic field of intensity close to the second critical field
is given by the celebrated Abrikosov energy. If the external
magnetic field is comparable to and below the second critical
field, the energy is given by a reference function obtained as
a special (thermodynamic) limit of a nonlinear energy. In this
note, we give a new formula for this reference energy. In particular,
we obtain it as a special limit of a linear energy defined
over configurations normalized in the $L^4$norm.
Keywords:GinzburgLandau functional Categories:35B40, 35P15, 35Q56 

18. CMB 2016 (vol 59 pp. 542)
 Jiang, Yongxin; Wang, Wei; Feng, Zhaosheng

Spatial Homogenization of Stochastic Wave Equation with Large Interaction
A dynamical approximation of a stochastic wave
equation with large interaction is derived.
A random invariant manifold is discussed. By a key linear transformation,
the random invariant manifold is shown to be close to the random
invariant manifold
of a secondorder stochastic ordinary differential equation.
Keywords:stochastic wave equation, homogeneous system, approximation, random invariant manifold, Neumann boundary condition Categories:60F10, 60H15, 35Q55 

19. CMB 2015 (vol 59 pp. 73)
 Gasiński, Leszek; Papageorgiou, Nikolaos S.

Positive Solutions for the Generalized Nonlinear Logistic Equations
We consider a nonlinear parametric elliptic equation driven
by a nonhomogeneous differential
operator with a logistic reaction of the superdiffusive type.
Using variational methods coupled with suitable truncation
and comparison techniques,
we prove a bifurcation type result describing the set of positive
solutions
as the parameter varies.
Keywords:positive solution, bifurcation type result, strong comparison principle, nonlinear regularity, nonlinear maximum principle Categories:35J25, 35J92 

20. CMB 2015 (vol 58 pp. 723)
 Castro, Alfonso; Fischer, Emily

Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear LaplaceBeltrami Equations on Spheres
We show that a class of semilinear LaplaceBeltrami equations
on the unit sphere
in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions.
The solutions to
these equations are the solutions to a two point boundary value
problem for a
singular ordinary differential equation. We prove the existence
of such solutions
using energy and phase plane analysis. We derive a
Pohozaevtype
identity
in
order to prove that the energy to an associated initial value
problem tends
to infinity as the energy at the singularity tends to infinity.
The nonlinearity is allowed to grow as fast as $s^{p1}s$ for
$s$ large
with $1 \lt p \lt (n+5)/(n3)$.
Keywords:LaplaceBeltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, subsuper critical nonlinearity Categories:58J05, 35A24 

21. CMB 2015 (vol 58 pp. 651)
 Tang, Xianhua

Ground State Solutions of NehariPankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$
This paper is concerned with the following
elliptic system of Hamiltonian type
\[
\left\{
\begin{array}{ll}
\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
u, v\in H^{1}({\mathbb{R}}^{N}),
\end{array}
\right.
\]
where the potential $V$ is periodic and $0$ lies in a gap of
the spectrum of $\Delta+V$, $W(x, s, t)$ is
periodic in $x$ and superlinear in $s$ and $t$ at infinity.
We develop a direct approach to find ground
state solutions of NehariPankov type for the above system.
Especially, our method is applicable for the
case when
\[
W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{\alpha_iu+\beta_iv}g_i(x,
t)t\mathrm{d}t
+\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x,
t)t\mathrm{d}t,
\]
where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne
0$ and $a_j\gt b_j^2$, $g_i(x, t)$
and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every
$x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.
Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of NehariPankov type, strongly indefinite functionals Categories:35J50, 35J55 

22. CMB 2015 (vol 58 pp. 471)
 Demirbas, Seckin

Almost Sure Global Wellposedness for the Fractional Cubic SchrÃ¶dinger Equation on Torus
In a previous paper, we proved that $1$d periodic fractional
SchrÃ¶dinger equation with cubic nonlinearity is locally wellposed
in $H^s$ for $s\gt \frac{1\alpha}{2}$ and globally wellposed for
$s\gt \frac{10\alpha1}{12}$. In this paper we define an invariant
probability measure $\mu$ on $H^s$ for $s\lt \alpha\frac{1}{2}$,
so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$
such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally
wellposed for initial data in $\Omega$. We see that this fills
the gap between the local wellposedness and the global wellposedness
range in almost sure sense for $\frac{1\alpha}{2}\lt \alpha\frac{1}{2}$,
i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense.
Keywords:NLS, fractional Schrodinger equation, almost sure global wellposedness Category:35Q55 

23. CMB 2015 (vol 58 pp. 486)
 Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong

Inequalities for Partial Derivatives and their Applications
We present various weighted integral inequalities for partial
derivatives acting on products and compositions of functions
which are applied to establish some new Opialtype inequalities
involving functions of several independent variables. We also
demonstrate the usefulness of our results in the field of partial
differential equations.
Keywords:inequality for integral, Opialtype inequality, HÃ¶lder's inequality, partial differential operator, partial differential equation Categories:26D10, 35A23 

24. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

25. CMB 2013 (vol 56 pp. 827)
 Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.

Erratum to ``Quantum Limits of Eisenstein Series and Scattering States''
This paper provides an erratum to Y. N. Petridis,
N. Raulf, and M. S. Risager, ``Quantum Limits
of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published
online 20120203, http://dx.doi.org/10.4153/CMB20112002.
Keywords:quantum limits, Eisenstein series, scattering poles Categories:11F72, 8G25, 35P25 
