Principal spectral curves for Lane–Emden fully nonlinear type systems and applications

Radial solvability for Pucci-Lane-Emden systems in annuli

“…Then we choose small enough such that . Since on , then by the maximum principle for the Lane-Emden system for fully nonlinear operators with weights in [13] we get in . Since then in , but this contradicts the fact that . Case 1: …”

“…If has a fixed point then and solve with in , where Now, the definition of first eigenvalue for the fully nonlinear weighted Lane-Emden system in [13] yields Thus we choose large enough such that in order to derive a contradiction.…”

“…We consider the annulus . Then and solve while and satisfy where Hence, by the definition of first eigenvalue for the fully nonlinear Lane-Emden systems in [13], we have for all , where and , since is fixed. Recall the energy is increasing in by proposition 2.3.…”

Classification of radial solutions for fully nonlinear equations with Hardy potential

“…Let us fix the operator M + , since for M − it is analogous. The solvability of the sublinear PDE in the annulus a consequence of Theorem 1.16 in [15] with p = q, since in the annulus the weights are bounded, so integrable in any Lebesgue space. We give a proof of this fact for reader's convenience.…”

“…Observe that the moving planes method cannot be applied in the annulus to obtain symmetry in the lack of convexity of the domain. We mention that existence of a positive annular solution (not necessarily radial) of (1), (3) follows by [18], since in this case we have a bounded weight; while uniqueness comes for instance from [15]. However, knowing that the solution obtained is radial turns out to be a consequence of the dynamical system approach.…”