Radial solutions of a semilinear elliptic equation at a critical exponent
I consider the problem of finding radial solutions of a semilinear elliptic equation which are not necessarily positive. The case of interest occurs when the nonlinear term is a power with a critical exponent plus some lower order term. It is known from the work of Brezis and Nirenberg that the lower order terms can force either the existence or non-existence of positive solutions depending on its power. In this paper it is shown that a similar effect occurs for non-positive solutions but the power cut-off is different. This leads to the interesting case in which a positive solution exists but there are no non-positive ones. My techniques are to construct a compactified phase portrait using appropriate scalings. The existence is then determined by tracking the contortions of an appropriate invariant manifold.
scientific article; zbMATH DE number 4113130
scientific article