1. CMB Online first
 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 

2. 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 

3. 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 

4. 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 

5. 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 

6. CMB 2011 (vol 55 pp. 663)
 Zhou, Chunqin

An Onofritype Inequality on the Sphere with Two Conical Singularities
In this paper, we give a new proof of the Onofritype inequality
\begin{equation*}
\int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{
\frac{1}{4\pi(\beta+1)} \int_S \nabla u^2 \,ds^2 +
\frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\}
\end{equation*}
on the sphere $S$ with Gaussian curvature $1$ and with conical
singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for
$\beta\in (1,0)$; here $p_1$ and $p_2$ are antipodal.
Categories:53C21, 35J61, 53A30 

7. CMB 2011 (vol 55 pp. 537)
8. CMB 2009 (vol 52 pp. 555)
 Hirata, Kentaro

Boundary Behavior of Solutions of the Helmholtz Equation
This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewoodtype theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 

9. CMB 2008 (vol 51 pp. 140)
 Rossi, Julio D.

First Variations of the Best Sobolev Trace Constant with Respect to the Domain
In this paper we study the best constant of the Sobolev trace
embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$
is a bounded smooth domain in $\RR^N$. We find a formula for the
first variation of the best constant with respect to the domain.
As a consequence, we prove that the ball is a critical domain when
we consider deformations that preserve volume.
Keywords:nonlinear boundary conditions, Sobolev trace embedding Categories:35J65, 35B33 

10. CMB 2007 (vol 50 pp. 356)
 Filippakis, Michael E.; Papageorgiou, Nikolaos S.

Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities
In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Keywords:$p$Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem Categories:35J20, 35J60, 35J85 

11. CMB 2006 (vol 49 pp. 358)
 Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed

On the Principal Eigencurve of the $p$Laplacian: Stability Phenomena
We show that each point of the principal eigencurve of the
nonlinear problem
$$
\Delta_{p}u\lambda m(x)u^{p2}u=\muu^{p2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.
Keywords:$p$Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stability Categories:35P30, 35P60, 35J70 

12. CMB 2006 (vol 49 pp. 144)
13. CMB 2004 (vol 47 pp. 515)
 Frigon, M.

Remarques sur l'enlacement en thÃ©orie des points critiques pour des fonctionnelles continues
Dans cet article, \`a partir de la notion d'enlacement introduite
dans ~\cite{F} entre des paires d'ensembles $(B,A)$ et $(Q,P)$,
nous \'etablissons l'existence d'un point critique d'une
fonctionnelle continue sur un espace m\'etrique lorsqu'une de ces
paires enlace l'autre. Des renseignements sur la localisation du
point critique sont aussi obtenus. Ces r\'esultats conduisent \`a
une g\'en\'eralisation du th\'eor\`eme des trois points critiques.
Finalement, des applications \`a des probl\`emes aux limites pour
une \'equation quasilin\'eaire elliptique sont pr\'esent\'ees.
Categories:58E05, 35J20 

14. CMB 2004 (vol 47 pp. 504)
15. CMB 2001 (vol 44 pp. 346)
16. CMB 2001 (vol 44 pp. 210)
 Leung, Man Chun

Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n2}}=0$ in $\R^n$
We construct unbounded positive $C^2$solutions of the equation
$\Delta u + K u^{(n + 2)/(n  2)} = 0$ in $\R^n$ (equipped
with Euclidean metric $g_o$) such that $K$ is bounded between two
positive numbers in $\R^n$, the conformal metric $g=u^{4/(n2)}g_o$
is complete, and the volume growth of $g$ can be arbitrarily fast
or reasonably slow according to the constructions. By imposing natural
conditions on $u$, we obtain growth estimate on the $L^{2n/(n2)}$norm
of the solution and show that it has slow decay.
Keywords:positive solution, conformal scalar curvature equation, growth estimate Categories:35J60, 58G03 

17. CMB 1997 (vol 40 pp. 464)
 Kuo, ChungCheng

On the solvability of a Neumann boundary value problem at resonance
We study the existence of solutions of the semilinear equations (1)
$\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on
$\partial \Omega$ in which the nonlinearity $g$ may grow
superlinearly in $u$ in one of directions $u \to \infty$ and $u \to
\infty$, and (2) $\triangle u + g(x,u)=h$, ${\partial u \over
\partial n} = 0$ on $\partial \Omega$ in which the nonlinear term $g$
may grow superlinearly in $u$ as $u \to \infty$. The purpose of this
paper is to obtain solvability theorems for (1) and (2) when the
LandesmanLazer condition does not hold. More precisely, we require
that $h$ may satisfy $\int g^\delta_ (x) \, dx < \int h(x) \, dx = 0<
\int g^\gamma_+ (x)\,dx$, where $\gamma, \delta$ are arbitrarily
nonnegative constants, $g^\gamma_+ (x) = \lim_{u \to \infty} \inf
g(x,u) u^\gamma$ and $g^\delta_ (x)=\lim_{u \to \infty} \sup
g(x,u)u^\delta$. The proofs are based upon degree theoretic arguments.
Keywords:LandesmanLazer condition, Leray Schauder degree Categories:35J65, 47H11, 47H15 

18. CMB 1997 (vol 40 pp. 244)
 Naito, Yūki; Usami, Hiroyuki

Nonexistence results of positive entire solutions for quasilinear elliptic inequalities
This paper treats the quasilinear elliptic inequality
$$
\div (Du^{m2}Du) \geq p(x)u^{\sigma},
\quad x \in \Rs^N,
$$
where $N \geq 2$, $m > 1$, $ \sigma > m  1$, and $p \colon \Rs^N
\rightarrow (0, \infty)$ is continuous. Sufficient conditions are
given for this inequality to have no positive entire solutions. When
$p$ has radial symmetry, the existence of positive entire solutions can
be characterized by our results and some known results.
Categories:35J70, 35B05 
