Classification of Solutions to Mixed Order Elliptic System with General Nonlinearity
Shaolong Peng
Abstract
.In this paper, we consider the mixed order elliptic system \(\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)=f(u,v), \\ (-\Delta ) v(x)=g(u,v), \end{cases} \) where \(u\geq 0\) , \(\alpha \in (0,2)\) , \(v\) may change signs. We aim to study the classification results of solutions to the above semilinear elliptic system in \(\mathbb R^{2}\) . We first derive the equivalence between the above PDE system and the corresponding IE (integral equation) system. Then, applying the method of moving spheres in integral form combined with integral inequalities, under certain assumptions, we give a complete classification of the classical solutions to the above system in \(\mathbb R^{2}\) .Keywordsclassification of solutionssemilinear elliptic systemmethod of moving spheresmixed ordergeneral nonlinearityMSC codes35M3035B0635J4735J61