Positive solutions to elliptic equations in unbounded cylinder

Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains

Non-existence of Positive Solutions of Fully Nonlinear Elliptic Equations in Unbounded Domains

In this paper we consider fully nonlinear elliptic operators of the form F (x, u,Du,D2u). Our aim is to prove that, under suitable assumptions on F , the only nonnegative viscosity super-solution u of F (x, u,Du,D2u) = 0 in an unbounded domain Ω is u ≡ 0. We show that this uniqueness result holds for the class of nonnegative super-solutions u satisfying inf x∈Ω u(x) + 1 dist(x, ∂Ω) = 0, and the...

Research Article Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

We will show that under suitable conditions on f and h, there exists a positive number λ∗ such that the nonhomogeneous elliptic equation −Δu + u = λ( f (x,u) + h(x)) in Ω, u ∈ H 0 (Ω), N ≥ 2, has at least two positive solutions if λ ∈ (0,λ∗), a unique positive solution if λ = λ∗ , and no positive solution if λ > λ∗ , where Ω is the entire space or an exterior domain or an unbounded cylinder dom...

Positive Solutions of Elliptic Equations withSingular

In this paper, a nonlinear elliptic boundary value problem with singular nonlinearity is studied, where L is a uniformly elliptic operator, is a bounded domain in R N , N 2, 0 may take the value 0 on @, and f(x; s) is possibly singular near s = 0. Some results regarding the existence of positive solutions for the problem are given under a set of hypotheses that make neither monotonicity nor str...

Positive Solutions of Quasilinear Elliptic Equations

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and t...