Regular solutions to elliptic equations

Abstract

A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by  linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given  elliptic problem can be analyzed by the relationship between the  spectrum of the operator and the behavior of the nonlinearity near infinity and at zero.  Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping  principle best suited for using continuous dependence to ordinary  differential equations on initial conditions is presented. Another appendix on the phase plane analysis as needed to take advantage  of initial conditions is also included. For studies on singular solutions  the reader is referred to Ardila et al., Milan J. Math (2014)  and references therein. See also https://ejde.math.txstate.edu/special/02/c2/abstr.html