Symmetry results for cooperative elliptic systems via linearization

In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in \RN, $N \geq 2$. More precisely we prove that solutions having Morse index $j \leq N$ are foliated Schwarz symmetric if the nonlinearity is convex and a full coupling condition is satisfied along the solution

Symmetry of n-mode positive solutions for two-dimensional H\'enon type systems

We provide a symmetry result for n-mode positive solutions of a general class of semi-linear elliptic systems under cooperative conditions on the nonlinearities. Moreover, we apply the result to a class of H\'enon systems and provide the existence of multiple n-mode positive solutions.Comment: 8 page

A new dynamical approach of Emden-Fowler equations and systems

We give a new approach on general systems of the form (G){[c]{c}% -\Delta_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=\epsilon_{1}|x| ^{a}u^{s}v^{\delta}, -\Delta_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=\epsilon_{2}|x|^{b}u^{\mu}v^{m}, where $Q,p,q,\delta,\mu,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $\epsilon_{1}=\pm1,$ $\epsilon_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $\epsilon_{1}=\epsilon_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is variational. We give an important result on existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0.$ In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$ and $\delta=m+1$ and $\mu=s+1.$Comment: 43 page

Sectional symmetry of solutions of elliptic systems in cylindrical domains

In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in \cite{DaPaSys, DaGlPa1, Pa, PaWe}.Comment: arXiv admin note: text overlap with arXiv:1209.5581, arXiv:1206.392

A symmetry result for cooperative elliptic systems with singularities

We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess "holes" or "small vacancies" (measured in terms of capacity) along which the solution may diverge. The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system