Chapter 3 Positive solutions to semi-linear and quasi-linear elliptic equations on unbounded domains

Abstract

Abstract In this survey we describe recent results on the existence and nonexistence of positive solution to semi-linear and quasi-linear second-order elliptic equations. A typical example is the equation –Δ u = | x | –σ u q in an exterior of the ball in ℝ N or in a cone-like domain in ℝ N . The equations of this type exhibit a phenomenon of presence of critical exponents in the range of the parameter q ∈ ℝ, which separate the zones of the existence from the zones of the nonexistence. The values of the critical exponents depends on the geometry of the domain, the type of the operator in the main part (divergent or nondivergent), the behaviour of the coefficients in lower order terms at infinity. We investigate these dependencies mostly in the cases of the exterior and cone-like domains. The proofs are often based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the corresponding second-order elliptic operator, comparison principles and Hardy's inequality in exterior domains. To construct the barriers in the cases of equations with non-smooth coefficients we obtain detailed estimates at infinity of small and large solutions to the corresponding linear equations. Some of the results for the equations with first order term are new and have not been published before. In discussions we list some open problems in this area.