We consider the existence of three non-trivial smooth solutions for nonlinear elliptic problems driven by the p-Laplacian. Using variational arguments, coupled with the method of upper and lower solutions, critical groups and suitable truncation techniques, we produce three non-trivial smooth solutions, two of which have constant sign. The hypotheses incorporate both coercive and non-coercive problems in our framework of analysis.

We have derived radial sky-subtracted surface brightness profiles of a sample of 42 face-on nearby early-type spiral galaxies, classified as unbarred, from SDSS data as well as our own observations (INT/WFC) at La Palma. A key objective was to derive the fraction of truncated profiles. We find that only a small minority 7% of the galaxies are obviously truncated within the limits of the measured surface brightness range, while some 20% show exponential disc profiles but with a shallower inner portion near the centre: classical “Freeman Type II” profiles (Freeman, 197). Half of the discs show a single unbroken exponential (“type I”) brightness profile with no evidence of truncation out to the edge of detectability, while the remainder show “antitruncations”: external profiles less steep than those of the internal disc (Type III). A general profile classification scheme based on these and other recent related measurements is presented, and will be of use in probing disc formation scenarios.

Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).