On a general \(\mathrm{SU}(3)\) Toda system
Let \(\mu>-2\). The authors consider the following generalized Toda system \[ \begin{cases} -\Delta u=2 e^u+\mu e^v \, \text{ in}\,\, \mathbb{R}^2,\\ -\Delta v=2 e^v+\mu e^u \, \text{ in}\,\, \mathbb{R}^2\\ \int_{\mathbb{R}^2}e^udx<\infty, \, \, \int_{\mathbb{R}^2}e^vdx<\infty. \end{cases} \] It is known that the couple \((U_\mu,U_\mu)\), where \(U_\mu(x)=\log\frac{64}{(2+\mu)(8+|x|^2)^2}\), is a (trivial) solution to the above system. The authors prove that for \(\mu_n=2\frac{2-n-n^2}{2+n+n^2}\), \(n\in \mathbb{N}\), the points \((\mu_n,U_{\mu_n},U_{\mu_n})\) are radial bifurcation points for the curve of solutions \((\mu,U_\mu,U_\mu)\). In particular, for each \(\varepsilon>0\) small enough, the authors prove the existence of radial solutions \((u_\varepsilon,v_\varepsilon)\), with \((u_\varepsilon,v_\varepsilon)\rightarrow (U_{\mu_n},U_{\mu_n})\) as \(\varepsilon\rightarrow 0\), of the type \[ u_\varepsilon(|x|)=U_{\mu_\varepsilon}(x)+\varepsilon P_n(\frac{8-|x|^2}{8+|x|^2})+\varepsilon (z_{1,\varepsilon}+z_{2,\varepsilon}), \] \[ v_\varepsilon(|x|)=U_{\mu_\varepsilon}(x)+\varepsilon P_n(\frac{8-|x|^2}{8+|x|^2})+\varepsilon (z_{1,\varepsilon}-z_{2,\varepsilon}), \] where \(P_n\) are the Legendre polynomials and \(z_{1,\varepsilon},z_{2,\varepsilon}\) are uniformly bounded functions of \(L^\infty(\mathbb{R}^2)\) such that \(z_{1,0}=z_{2,0}=0\). The proof relies on a bifurcation result by \textit{M. G. Crandall} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 8, 321--340 (1971; Zbl 0219.46015)].
scientific article; zbMATH DE number 6544037
scientific article