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Search: MSC category 35J ( Elliptic equations and systems [See also 58J10, 58J20] )

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1. CMB Online first

He, Yubo; Qin, Dongdong; 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|^{q-1}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 stateCategories:35J20, 35J60, 35J25

2. CMB Online first

Zhang, Tao; Zhou, Chunqin
 Classification of solutions for harmonic functions with Neumann boundary value In this paper, we classify all solutions of $\left\{ \begin{array}{rcll} -\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+}, \\ \dfrac{\partial u}{\partial t}&=&-c|x|^{\beta}e^{u} \quad &\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\}, \\ \end{array} \right.$ 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 solutionsCategories:35A05, 35J65

3. CMB 2017 (vol 60 pp. 536)

 The Gradient of a Solution of the Poisson Equation in the Unit Ball and Related Operators In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions. Keywords:MÃ¶bius transformation, Poisson equation, Newtonian potential, Cauchy transform, Bessel functionCategories:35J05, 47G10

4. CMB 2017 (vol 60 pp. 422)

Tang, Xianhua
 New Super-quadratic 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 1-periodic 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 super-quadratic condition is obtained, which is weaker than some well known results. Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov typeCategories:35J20, 35J60

5. CMB 2016 (vol 59 pp. 606)

Mihăilescu, Mihai; Moroşanu, Gheorghe
 Eigenvalues of $-\Delta_p -\Delta_q$ Under Neumann Boundary Condition The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ 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 well-known 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 methodsCategories:35J60, 35J92, 46E30, 49R05

6. 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 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 nonlinearitiesCategory:35J66

7. 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 principleCategories:35J25, 35J92

8. 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 functionalsCategories:35J50, 35J55

9. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ 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 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 SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

10. 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

11. 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 iterationCategories:35B33, 35B40, 35J60

12. 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 Littlewood-type theorem to show that the approach region introduced by Kor\'anyi and Taylor (1983) is best possible. Keywords:boundary behavior, Helmholtz equationCategories:31B25, 35J05

13. 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 embeddingCategories:35J65, 35B33

14. 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 TheoremCategories:35J20, 35J60, 35J85

15. 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 $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ 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, stabilityCategories:35P30, 35P60, 35J70

16. CMB 2006 (vol 49 pp. 144)

Taylor, Michael
 Scattering Length and the Spectrum of $-\Delta+V$ Given a non-negative, locally integrable function $V$ on $\RR^n$, we give a necessary and sufficient condition that $-\Delta+V$ have purely discrete spectrum, in terms of the scattering length of $V$ restricted to boxes. Category:35J10

17. CMB 2004 (vol 47 pp. 504)

Cardoso, Fernando; Vodev, Georgi
 High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation We prove an uniform H\"older continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation. Categories:35B37, 35J15, 47F05

18. 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 quasi-lin\'eaire elliptique sont pr\'esent\'ees. Categories:58E05, 35J20

19. CMB 2001 (vol 44 pp. 346)

Wang, Wei
 Positive Solution of a Subelliptic Nonlinear Equation on the Heisenberg Group In this paper, we establish the existence of positive solution of a nonlinear subelliptic equation involving the critical Sobolev exponent on the Heisenberg group, which generalizes a result of Brezis and Nirenberg in the Euclidean case. Keywords:Heisenberg group, subLapacian, critical Sobolev exponent, extremalsCategories:35J20, 35J60

20. CMB 2001 (vol 44 pp. 210)

Leung, Man Chun
 Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n-2}}=0$ 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 $g=u^{4/(n-2)}g_o$ 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 $L^{2n/(n-2)}$-norm of the solution and show that it has slow decay. Keywords:positive solution, conformal scalar curvature equation, growth estimateCategories:35J60, 58G03

21. CMB 1997 (vol 40 pp. 464)

Kuo, Chung-Cheng
 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 non-linearity $g$ 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 Landesman-Lazer 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:Landesman-Lazer condition, Leray Schauder degreeCategories:35J65, 47H11, 47H15

22. 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 $$\div (|Du|^{m-2}Du) \geq p(x)u^{\sigma}, \quad x \in \Rs^N,$$ 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
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