Existence of sign changing solutions for an equation with a weighted p-Laplace operator

Carmen Cortázar (1) , Jean Dolbeault (2) , Marta Garcia-Huidobro (1) , Raul Manásevich (3, 4)

1 Departamento de Matemáticas [Santiago de Chile]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
4 DIM - Departamento de Ingeniería Matemática [Santiago]

We consider radial solutions of a general elliptic equation involving a weighted $p$-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.

Mots clés: p-Laplace operator; Nodal solutions; Nodes; Shooting method; Compact support principle; Double zero; Hamiltonian systems; Energy methods

Domaines: Equations aux dérivées partielles [math.AP]

Carmen Cortázar, Jean Dolbeault, Marta Garcia-Huidobro, Raul Manásevich. Existence of sign changing solutions for an equation with a weighted p-Laplace operator. Nonlinear Analysis: Theory, Methods and Applications, 2014, 110, pp.1-22. ⟨hal-00985507v2⟩