Perturbation effects in nonlinear eigenvalue problems (original) (raw)

Decaying to zero bifurcation solution curve for some sublinear elliptic eigenvalue type problems

2016

By using some rearrangements technics, jointly with a study of the radially symmetric case, we prove that the bifurcation curve for some sublinear elliptic eigenvalue type problems converges to zero when the eigenvalue parameter converges to in nity. The result improves some previous work in the literature (dealing mainly with the one-dimensional case or imposing some restrictions to the spatial dimension of the open set) and has application in several contexts. AMS Classi cation: 35P30, 35B32,35J25, 35J10

A global bifurcation result for a class of semipositone elliptic systems

Journal of Mathematical Analysis and Applications, 2017

We consider a system of the form −Δu = λ (θ 1 v + + f (v)) in Ω; −Δv = λ (θ 2 u + + g (u)) in Ω; u = 0 = v on ∂Ω, ⎫ ⎬ ⎭ where s + def = max{s, 0}, θ 1 and θ 2 are fixed positive constants, λ ∈ R is the bifurcation parameter, and Ω ⊂ R N (N > 1) is a bounded domain with smooth boundary ∂Ω (a bounded open interval if N = 1). The nonlinearities f, g : R → R are continuous functions that are bounded from below, sublinear at infinity and have semipositone structure at the origin (f (0), g(0) < 0). We show that there are two disjoint unbounded connected components of the solution set and discuss the nodal properties of solutions on these components. Finally, as a consequence of these results, we infer the existence and multiplicity of solutions for λ in a neighborhood containing the simple eigenvalue of the associated eigenvalue problem.

Bifurcations for semilinear elliptic equations with convex nonlinearity

1999

Weinvestigatetheexactnumberofpositivesolutionsofthesemilinear Dirichlet boundary value problem u+f(u)=0 on a ball in Rnwhere f is a strictly convex C2 function on (0;1). For the one-dimensional case we classify all strictly convex C2 functions according to the shape of the bifurcation diagram. The exact number of positive solutions may be 2, 1, or 0, depending on the radius. This full classication is due

An elliptic system with bifurcation parameters on the boundary conditions

2009

In this paper we consider the elliptic system ∆u = a(x)u p v q , ∆v = b(x)u r v s in Ω, a smooth bounded domain, with boundary conditions ∂u ∂ν = λu, ∂v ∂ν = µv on ∂Ω. Here λ and µ are regarded as parameters and p, q > 1, r, s > 0 verify (p−1)(s−1) > qr. We consider the case where a(x) ≥ 0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω 0 , while b(x) > 0 in Ω. Our main results include existence of nonnegative nontrivial solutions in the range 0 < λ < λ 1 ≤ ∞, µ > 0, where λ 1 is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ, µ → 0, as λ → λ 1 < ∞ for fixed µ (respectively µ → ∞ for fixed λ) and when both λ, µ → ∞ in case λ 1 = ∞.

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem −Δpu = λ|u| p−2 u + h x, u(x); λ in Ω ; u = 0 on ∂Ω. (P) Here, Ω is a bounded domain in R N (N ≥ 1), Δpu def = div(|∇u| p−2 ∇u) denotes the Dirichlet p-Laplacian on W 1,p 0 (Ω), 1 < p < ∞, and λ ∈ R is a spectral parameter. Let μ1 denote the first (smallest) eigenvalue of −Δp. Under some natural hypotheses on the perturbation function h : Ω× R × R → R, we show that the trivial solution (0, μ1) ∈ E = W 1,p 0 (Ω) × R is a bifurcation point for problem (P) and, moreover, there are two distinct continua, Z + μ 1 and Z − μ 1 , consisting of nontrivial solutions (u, λ) ∈ E to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ1). The continua Z + μ 1 and Z − μ 1 are either both unbounded in E, or else their intersection Z + μ 1 ∩ Z − μ 1 contains also a point other than (0, μ1). For the semilinear problem (P) (i.e., for p = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union Z + μ 1 ∩ Z − μ 1 looks like (for p > 2) in an interesting particular case. Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer's original work.

Bifurcation of Nonlinear Equations: I. Steady State Bifurcation

2004

We prove in this article some general steady state bifurcation theorem for a class of nonlinear eigenvalue problems, in the case where algebraic multiplicity of the eigenvalues of the linearized problem is even. These theorems provide an addition to the classical Krasnoselskii and Rabinowitz bifurcation theorems, which require the algebraic multiplicity of the eigenvalues is odd. For this purpose, we prove a spectral theorem for completely continuous fields, which can be considered as a generalized version of the classical Jordan matrix theorem and the Fredholm theorem for compact operators. An application to a system of second order elliptic equations is given as well.