Positive solutions of Kirchhoff type elliptic equations involving the critical Sobolev exponent (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

We report our recent studies on Kirchhoff type elliptic equations involving the critical Sobolev exponent. The interaction between the Kirchhoff type nonlocality and the Sobolev criticality leads us to several new phenomena, techniques and results depending on the dimension of the domain. More precisely, if the dimension is equal to 3, we observe the multiplicity of solutions induced by the nonlocal coefficient. If it is 4, we encounter an additional difficulty in proving the existence of solutions because of the lack of the Ambrosetti-Rabinowitz type condition. With the aid of the well known nonexistence result by the Pohozaev identity, we overcome this difficulty and give a positive answer for the solvability. For higher dimension, the Kirchhoff type nonlocality may break the. umiqueness of solutions of an associated limiting problem. This crucially affects the behavior of Palais-Smale sequences. Because of this, we need nontrivial modification for the concentration compactness analysis. Introducing a new technique based on the method of the Nehari manifold and the fibering map, we succeed in showing the existence of two solutions. This report is based on our talk entitled Two positive solutions of the Kirchhoff type elliptic problem with critical nonlinearity in high dimension on RIMS workshop "Analysis on Shapes of Solutions to Partial Differential Equations" on November 9-11, 2016. This report includes ajoint work with Prof. Shibata at Tokyo Institute of Technology.