# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

June 2014 , Volume 34 , Issue 6

Special issue on qualitative properties of solutions on nonlinear elliptic equations and systems

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2014, 34(6): i-ii doi: 10.3934/dcds.2014.34.6i +[Abstract](1705) +[PDF](96.6KB)
Abstract:
The field of nonlinear elliptic equations/systems has experienced a new burst of activities in recent years. This includes the resolution of De Giorgi's conjecture for Allen-Cahn equation, the classification of stable/finite Morse index solutions for Lane-Emden equation, the regularity of interfaces of elliptic systems with large repelling parameter, Caffarelli-Silvestre extension of fractional laplace equations, the analysis of Toda type systems, etc. This special volume touches several aspects of these new activities.

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2014, 34(6): 2451-2467 doi: 10.3934/dcds.2014.34.2451 +[Abstract](1461) +[PDF](440.1KB)
Abstract:
In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation $\Delta u=F'(u)$, which are monotone in some direction. In this paper we prove the analogous statement for the equation $\Delta u- \langle x,\nabla u\rangle u=F'(u)$, where the Laplacian is replaced by the Ornstein-Uhlenbeck operator. Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain an similar result in infinite dimensions by a limit procedure.
2014, 34(6): 2469-2479 doi: 10.3934/dcds.2014.34.2469 +[Abstract](1518) +[PDF](394.2KB)
Abstract:
We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\mathbb{R}^N$, $-\Delta v=u^q$ in $\mathbb{R}^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
2014, 34(6): 2481-2493 doi: 10.3934/dcds.2014.34.2481 +[Abstract](1464) +[PDF](360.6KB)
Abstract:
We show how certain solutions of the limit equation continue to solutions of the full equations when a parameter is large.
2014, 34(6): 2495-2503 doi: 10.3934/dcds.2014.34.2495 +[Abstract](1333) +[PDF](358.5KB)
Abstract:
Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth open set. We prove that the singular set of any extremal solution of the system \begin{equation*} -\Delta u=\mu e^v , \quad - \Delta v=\lambda e^u\quad\mbox{ in }\Omega, \end{equation*} with $u=v=0$ on $\partial \Omega$, $\mu,\lambda\geq0$, has Hausdorff dimension at most $n-10$.
2014, 34(6): 2505-2511 doi: 10.3934/dcds.2014.34.2505 +[Abstract](1245) +[PDF](303.7KB)
Abstract:
We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at infinity, must be one dimensional in the case of two spatial variables. We also prove the one dimensional symmetry for half-monotone solutions, i.e., for solutions having only one monotone component.
2014, 34(6): 2513-2533 doi: 10.3934/dcds.2014.34.2513 +[Abstract](1592) +[PDF](481.9KB)
Abstract:
We consider Liouville-type theorems for the following Hénon-Lane-Emden system \begin{eqnarray*} \left\{ \begin{array}{lcl} -\Delta u&=& |x|^{a}v^p \ \ in\ \ \mathbb{R}^n,\\ -\Delta v&=& |x|^{b}u^q \ \ in\ \ \mathbb{R}^n, \end{array}\right. \end{eqnarray*} when $p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $\frac{n+a}{p+1}+\frac{n+b}{q+1}>{n-2}$. We show that this is indeed the case in dimension $n=3$ provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case.
Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $n\ge 3$ in the first case (resp., $n\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $1< p < \frac{n+2+2a}{n-2}$ (resp., $1< p < \frac{n+4+2a}{n-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \begin{equation*}\label{sysdim00} n < 2 + 2 (\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}} + \sqrt{ \frac{pq(q+1)}{p+1} - \sqrt{\frac{pq(q+1)}{p+1}}}). \end{equation*}
2014, 34(6): 2535-2560 doi: 10.3934/dcds.2014.34.2535 +[Abstract](1264) +[PDF](516.4KB)
Abstract:
Given a 3-dimensional Riemannian manifold $(M,g)$, we investigate the existence of positive solutions of the Klein-Gordon-Maxwell system $$\left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+au=u^{p-1}+\omega^{2}(qv-1)^{2}u & \text{in }M\\ -\Delta_{g}v+(1+q^{2}u^{2})v=qu^{2} & \text{in }M \end{array}\right.$$ and Schrödinger-Maxwell system $$\left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+u+\omega uv=u^{p-1} & \text{in }M\\ -\Delta_{g}v+v=qu^{2} & \text{in }M \end{array}\right.$$ when $p\in(2,6).$ We prove that if $\varepsilon$ is small enough, any stable critical point $\xi_0$ of the scalar curvature of $g$ generates a positive solution $(u_\varepsilon,v_\varepsilon)$ to both the systems such that $u_\varepsilon$ concentrates at $\xi_0$ as $\varepsilon$ goes to zero.
2014, 34(6): 2561-2580 doi: 10.3934/dcds.2014.34.2561 +[Abstract](1275) +[PDF](470.2KB)
Abstract:
We first obtain Liouville type results for stable entire solutions of the biharmonic equation $-\Delta^2 u=u^{-p}$ in $\mathbb{R}^N$ for $p>1$ and $3 \leq N \leq 12$. Then we consider the Navier boundary value problem for the corresponding equation and improve the known results on the regularity of the extremal solution for $3 \leq N \leq 12$. As a consequence, in the case of $p=2$, we show that the extremal solution $u^{*}$ is regular when $N =7$. This improves earlier results of Guo-Wei [21] ($N \leq 4$), Cowan-Esposito-Ghoussoub [2] ($N=5$), Cowan-Ghoussoub [4] ($N=6$).
2014, 34(6): 2581-2615 doi: 10.3934/dcds.2014.34.2581 +[Abstract](1476) +[PDF](617.7KB)
Abstract:
We study the nonlinear fractional reaction-diffusion equation $∂_t u + (-\Delta)^s u = f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\mathbb{R}^N \setminus \Omega$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $\Omega$ is symmetric with respect to reflection at a hyperplane, say $\{x_1=0\}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak additional technical assumptions, we then show that any nonzero element in the $\omega$-limit set of nonnegative globally bounded solution is even in $x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$.
2014, 34(6): 2617-2637 doi: 10.3934/dcds.2014.34.2617 +[Abstract](1504) +[PDF](502.9KB)
Abstract:
Let $A=(a_{ij})_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: \begin{eqnarray*} \left\{ \begin{array}{lcl} \Delta u_i+\sum_{j=1}^n a_{ij}|x|^{\beta_j}e^{u_j(x)}=0,\quad \mathbb R^2, \quad i=1,...,n\\ \\ \int_{\mathbb R^2}|x|^{\beta_i}e^{u_i(x)}dx<\infty, \quad i=1,...,n \end{array}\right. \end{eqnarray*} where $\beta_1,...,\beta_n$ are constants greater than $-2$. If all $\beta_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate.
2014, 34(6): 2639-2656 doi: 10.3934/dcds.2014.34.2639 +[Abstract](1254) +[PDF](472.7KB)
Abstract:
In the Heisenberg group framework, we study rigidity properties for stable solutions of $(-\Delta_\mathbb{H})^sv=f(v)$ in $\mathbb{H}$, $s\in(0,1)$. We obtain a Poincaré type inequality in connection with a degenerate elliptic equation in $\mathbb{R}^4_+$; through an extension (or lifting") procedure, this inequality will be then used for giving a criterion under which the level sets of the above solutions are minimal surfaces in $\mathbb{H}$, i.e. they have vanishing mean curvature.
2014, 34(6): 2657-2667 doi: 10.3934/dcds.2014.34.2657 +[Abstract](1186) +[PDF](353.0KB)
Abstract:
We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain $\Omega$. We assume that $\Omega$ is symmetric about a hyperplane $H$ and convex in the direction perpendicular to $H$. Each nonnegative solution of such a problem is symmetric about $H$ and, if strictly positive, it is also decreasing in the direction orthogonal to $H$ on each side of $H$. The latter is of course not true if the solution has a nontrivial nodal set. In this paper we prove that for a class of domains, including for example all domains which are convex (in all directions), there can be at most one nonnegative solution with a nontrivial nodal set. For general domains, there are at most finitely many such solutions.
2014, 34(6): 2669-2691 doi: 10.3934/dcds.2014.34.2669 +[Abstract](1284) +[PDF](490.8KB)
Abstract:
For a class of competition-diffusion nonlinear systems involving the $s$-power of the laplacian, $s\in(0,1)$, of the form $(-\Delta)^{s} u_i=f_i(u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k,$ we prove that $L^\infty$ boundedness implies $\mathcal{C}^{0,\alpha}$ boundedness for $\alpha>0$ sufficiently small, uniformly as $\beta\to +\infty$. This extends to the case $s\neq1/2$ part of the results obtained by the authors in the previous paper [arXiv: 1211.6087v1].

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