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Search: MSC category 35 ( Partial differential equations )

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1. CMB Online first

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

2. CMB Online first

Castro, Alfonso; Fischer, Emily
Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres
We show that a class of semilinear Laplace-Beltrami 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 Pohozaev-type 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 $|s|^{p-1}s$ for $|s|$ large with $1 \lt p \lt (n+5)/(n-3)$.

Keywords:Laplace-Beltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, sub-super critical nonlinearity
Categories:58J05, 35A24

3. CMB 2015 (vol 58 pp. 651)

Tang, Xianhua
Ground State Solutions of Nehari-Pankov 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 Nehari-Pankov 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 Nehari-Pankov type, strongly indefinite functionals
Categories:35J50, 35J55

4. CMB 2015 (vol 58 pp. 471)

Demirbas, Seckin
Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on Torus
In a previous paper, we proved that $1$-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s\gt \frac{1-\alpha}{2}$ and globally well-posed for $s\gt \frac{10\alpha-1}{12}$. 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 well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness 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

5. 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 Opial-type 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, Opial-type inequality, Hölder's inequality, partial differential operator, partial differential equation
Categories:26D10, 35A23

6. CMB Online first

Song, Hongxue; Chen, Caisheng; Yan, Qinglun
Existence of multiple solutions for a $p$-Laplacian system in $\textbf{R}^{N}$ with sign-changing weight functions
In this paper, we consider the quasi-linear elliptic problem \[ \left\{ \begin{aligned} & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla u|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla u|^{p-2}\nabla u \right) \\ & \qquad=\frac{\alpha}{\alpha+\beta}H(x)|u|^{\alpha-2}u|v|^{\beta}+\lambda h_{1}(x)|u|^{q-2}u, \\ & -M \left(\int_{\mathbb{R}^{N}}|x|^{-ap}|\nabla v|^{p}dx \right){\rm div} \left(|x|^{-ap}|\nabla v|^{p-2}\nabla v \right) \\ & \qquad=\frac{\beta}{\alpha+\beta}H(x)|v|^{\beta-2}v|u|^{\alpha}+\mu h_{2}(x)|v|^{q-2}v, \\ &u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N} \end{aligned} \right. \] where $\lambda, \mu\gt 0$, $1\lt p\lt N$, $1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{N-p}$, $0\leq a\lt \frac{N-p}{p}$, $a\leq b\lt a+1$, $d=a+1-b\gt 0$, $M(s)=k+l s^{\tau}$, $k\gt 0$, $l, \tau\geq0$ 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

7. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger 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 Hölder 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 second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz 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:Musielak-Orlicz-Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

8. 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 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2.

Keywords:quantum limits, Eisenstein series, scattering poles
Categories:11F72, 8G25, 35P25

9. CMB 2012 (vol 56 pp. 814)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
Quantum Limits of Eisenstein Series and Scattering States
We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

Keywords:quantum limits, Eisenstein series, scattering poles
Categories:11F72, 58G25, 35P25

10. CMB 2011 (vol 56 pp. 378)

Ma, Li; Wang, Jing
Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases
In this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.

Keywords:Gross-Pitaevskii equation, sharp threshold, global existence, blow up
Categories:35Q55, 35A05, 81Q99

11. CMB 2011 (vol 56 pp. 659)

Yu, Zhi-Xian; Mei, Ming
Asymptotics and Uniqueness of Travelling Waves for Non-Monotone Delayed Systems on 2D Lattices
We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed 2D lattice equations with non-monotone birth functions. First, with the help of Ikehara's Theorem, the a priori asymptotic behavior of travelling wave is exactly derived. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. These results complement earlier results in the literature.

Keywords:2D lattice systems, traveling waves, asymptotic behavior, uniqueness, nonmonotone nonlinearity
Category:35K57

12. CMB 2011 (vol 56 pp. 3)

Aïssiou, Tayeb
Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori
We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof.

Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits
Categories:58G25, 81Q50, 35P20, 42B05

13. CMB 2011 (vol 55 pp. 555)

Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang
Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$.

Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates
Categories:42B20, 42B25, 35S05, 47G30

14. CMB 2011 (vol 55 pp. 663)

Zhou, Chunqin
An Onofri-type Inequality on the Sphere with Two Conical Singularities
In this paper, we give a new proof of the Onofri-type 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

15. CMB 2011 (vol 55 pp. 736)

Hernández, Eduardo; O'Regan, Donal
Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations
In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

Keywords:neutral equations, mild solutions, classical solutions
Categories:35R10, 34K40, 34K30

16. CMB 2011 (vol 55 pp. 537)

Kang, Dongsheng
Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents
In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy-Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iteration
Categories:35B33, 35B40, 35J60

17. CMB 2011 (vol 55 pp. 623)

Pan, Jiaqing
The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations
In this paper, we consider the Cauchy problem $$ \begin{cases} u_{t}=\Delta(u^{m}), &x\in{}\mathbb{R}^{N}, t>0, N\geq3, \\ % ^^----- here u(x,0)=u_{0}(x), &x\in{}\mathbb{R}^{N}. \end{cases} $$ We will prove that: (i) for $m_{c}
Keywords:fast diffusion equations, Cauchy problem, continuous dependence on nonlinearity
Categories:35K05, 35K10, 35K15

18. CMB 2011 (vol 55 pp. 249)

Chang, Der-Chen; Li, Bao Qin
Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases, where $X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$, $X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$ are linearly independent operators with $p_j$ being arbitrary polynomials in $\mathbf{C}^2$.

Keywords:entire solution, eiconal equation, polynomial, transcendental function
Categories:32A15, 35F20

19. CMB 2011 (vol 55 pp. 3)

Agarwal, Ravi P.; Mustafa, Octavian G.
On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations
We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations.

Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equation
Categories:34E10, 34C10, 35Q35

20. CMB 2011 (vol 55 pp. 88)

Ghanbari, K.; Shekarbeigi, B.
Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a connected bounded domain in $\mathbb{R}^n$. Let $0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues of the following Dirichlet problem: $$ \begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in D u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial D}=0, \end{cases} $$ where $V(x)$ is a nonnegative potential, and $\rho(x)\in C(\bar{D})$ is positive. We prove the following inequalities: $$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}} {\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times \frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2}, $$ $$\frac{n^2k^2}{8(n+2)}\leq \Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr] \times\sum_{i=1}^k\mu_i^{1/2}. $$

Keywords:biharmonic operator, eigenvalue, eigenvector, inequality
Category:35P15

21. CMB 2011 (vol 54 pp. 249)

Dattori da Silva, Paulo L.
A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies
This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic.

Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$)
Categories:35A01, 58Jxx

22. CMB 2010 (vol 54 pp. 28)

Chang, Yu-Hsien; Hong, Cheng-Hong
Generalized Solution of the Photon Transport Problem
The purpose of this paper is to show the existence of a generalized solution of the photon transport problem. By means of the theory of equicontinuous $C_{0}$-semigroup on a sequentially complete locally convex topological vector space we show that the perturbed abstract Cauchy problem has a unique solution when the perturbation operator and the forcing term function satisfy certain conditions. A consequence of the abstract result is that it can be directly applied to obtain a generalized solution of the photon transport problem.

Keywords:photon transport, $C_{0}$-semigroup
Categories:35K30, 47D03

23. CMB 2010 (vol 54 pp. 126)

Jin, Yongyang; Zhang, Genkai
Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups
We prove that the fundamental solutions of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in $\alpha$ with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups.

Categories:22E30, 35R03, 43A80

24. CMB 2010 (vol 53 pp. 674)

Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba
Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions
Categories:58J05, 35P30

25. CMB 2010 (vol 53 pp. 737)

Vougalter, Vitali
On the Negative Index Theorem for the Linearized Non-Linear Schrödinger Problem
A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky, and Vougalter based on the variational principle for the quadratic form of a self-adjoint operator. It is the negative index theorem for a linearized NLS operator in three dimensions.

Categories:35Q55, 81Q10
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