Positive solutions to second-order semilinear and quasilinear elliptic equations in unbounded domains
Seminar Room 1, Newton Institute
We survey some recent results on the existence and nonexistence of positive super-solutions to semilinear and quasi-linear elliptic equations equation (*): Lu=up in G, where G is an unbounded domain in RN (N > 2), L is a second order uniformly elliptic operator in divergence form with measurable coefficients and p in R. We prove the existence of critical exponents p- < 1 and p+ >1 such that equation (*) has no positive (super) solution if and only if p in [p-, p+] and show that the values and the properties of p-=p-(L, G) and p+=p+(L, G) essentially depend both on the geometry of the domain G and the coefficients of the elliptic operator L. The focus will be mainly on exterior domains and cone-like domains. . We also examine the same problem for the non-divergence type operators L and for some classes of quasi-linear operators with critical potentials. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behaviour of super-harmonic functions associated to the operator, Phragmën--Lindelöf type comparison arguments and an improved version of Hardy's inequality.