Positive solutions to second order semi-linear elliptic equations

Here G ⊆ R (N ≥ 2) is an unbounded domain, and L is a second-order elliptic operator. We mainly confine ourselves to the cases F (x, u) = W (x)u with real p and W (x) a real valued function on G, and F (x, u) = g(u) with g : R→ R continuous and g(0) = 0. The operator L = H − V is of Schrödinger type, namely V = V (x) is a real potential and H = −∆ or more generally H = −∇ · a · ∇ is a second order elliptic operator in the divergence form, with a symmetric measurable uniformly elliptic matrix a = a(x). Also of interest are general classes of non-linear operators H, as well as the case when H is in non-divergence form. Nonlinear equations of the types (1) are ubiquitous in natural sciences. They arise in a variety of important phenomena in quantum mechanics (Euclidean scalar field equations), physics (nonlinear optics, laser propagation), astrophysics (stellar structure), population ecology and population genetics (logistic type equations), etc. The phenomena of existence/nonexistence and qualitative properties of positive solutions to nonlinear elliptic and parabolic equations has been an important topic of the theory of modern PDE’s since the pioneering papers by Fujita (60’s), Serrin (70’s), Gidas and Spruck (80’s). Since the late 80’s positive solutions of semilinear equations were intensively studied using two major techniques. The first one is a reduction to an ODE’s technique (Gidas and Spruck, Ni, Serrin). The second one relies on testfunction methods (Brezis, Nirenberg, Pohozaev). While there were major successes in both approaches, they have some limitations, the main one being severe restrictions on the classes of coefficients and domains under consideration, as well as the lack of a cogent explanation of the phenomena involved. The research will focus upon the study of existence and nonexistence of positive solutions for elliptic equations (1), on general domains and under general assumptions on the coefficients, with general nonlinearities, expanding the ideas of [15, 16]. The goal is to obtain a complete characterization of the relation between the existence of positive solutions to equation (1) and the geometry of the domain and the behaviour of coefficients at infinity.