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

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

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...

Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

We consider the elliptic problem −Δu+ u= b(x)|u|p−2u+ h(x) in Ω, u∈H1 0 (Ω), where 2 < p < (2N/(N − 2)) (N ≥ 3), 2 < p <∞ (N = 2), Ω is a smooth unbounded domain in RN , b(x) ∈ C(Ω), and h(x) ∈ H−1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x)→ b∞ > 0 as |x| →∞ and b(x)≥ c for some suitable constants ...

EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS OF Rn

This paper deals with a class of nonlinear elliptic equations in an unbounded domain D of Rn, n≥ 3, with a nonempty compact boundary, where the nonlinear term satisfies some appropriate conditions related to a certain Kato class K∞(D). Our purpose is to give some existence results and asymptotic behaviour for positive solutions by using the Green function approach and the Schauder fixed point t...

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Ω ⊂ R be a ball or an annulus and f : R → R absolutely continuous, superlinear, subcritical, and such that f(0) = 0. We prove that the least energy nodal solution of −∆u = f(u), u ∈ H 0 (Ω), is not radial. We also prove that Fučik eigenfunctions, i. e. solutions u ∈ H 0 (Ω) of −∆u = λu − μu−, with eigenvalue (λ, μ) on the first nontrivial curve of the Fučik spectrum, are not radial. A relat...