Positive superharmonic solutions to semi-linear elliptic eigenvalue problems

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 or...

Asymptotic Expansion of Solutions to Nonlinear Elliptic Eigenvalue Problems

We consider the nonlinear eigenvalue problem −∆u+ g(u) = λ sinu in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ RN (N ≥ 2) is an appropriately smooth bounded domain and λ > 0 is a parameter. It is known that if λ 1, then the corresponding solution uλ is almost flat and almost equal to π inside Ω. We establish an asymptotic expansion of uλ(x) (x ∈ Ω) when λ 1, which is explicitly represented by g.

Positive Super-solutions to Semi-linear Second-order Non-divergence Type Elliptic Equations in Exterior Domains

We study the problem of the existence and non-existence of positive super-solutions to a semi-linear second-order non-divergence type elliptic equation ∑N i,j=1 aij(x) ∂2u ∂xi∂xj + up = 0, −∞ < p < ∞, with measurable coefficients in exterior domains of RN . We prove that in a “generic” situation there is one critical value of p that separates the existence region from nonexistence. We reveal th...

The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in R, n ≥ 3, satisfying the nonlinear elliptic inequality 0 ≤ −∆u ≤ cδΩ(x)u in Ω, where c > 0, α ≥ 0 and p > 0 are constants, and δΩ(x) is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of posi...

Symmetry and Monotonicity Properties for Positive Solutions of Semi-linear Elliptic Pde's