A global bifurcation result for a class of semipositone elliptic systems (original) (raw)

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.

Degree and global bifurcation for elliptic equations with multivalued unilateral conditions

Nonlinear Analysis: Theory, Methods & Applications, 2006

A bifurcation problem for an elliptic multivalued boundary value problem with a real parameter is considered. The existence of global bifurcation between two eigenvalues of a certain type of the Laplacian is proved. For a class of abstract inclusions with compact multivalued mappings in a Hilbert space, it is shown how the degree can be determined near the eigenvalues of a particular type of an associated linear single-valued problem, and the jump of the degree is proved. As a consequence, global bifurcation for such abstract inclusions is obtained. The weak formulation of the boundary value problem mentioned is a particular case. ᭧

Bifurcation points of a degenerate elliptic boundary-value problem

Rendiconti Lincei - Matematica e Applicazioni, 2006

Partial differential equations.-Bifurcation points of a degenerate elliptic boundaryvalue problem, by GILLES EVÉQUOZ and CHARLES A. STUART, communicated on 12 May 2006. ABSTRACT.-We consider the nonlinear elliptic eigenvalue problem −∇ • {A(x)∇u(x)} = λf (u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω is a bounded open subset of R N and f ∈ C 1 (R) with f (0) = 0 and f (0) = 1. The ellipticity is degenerate in the sense that 0 ∈ Ω and A(x) > 0 for x = 0 but lim x→0 A(x)/|x| 2 = 1. We show that there is vertical bifurcation at all points λ in the interval (N 2 /4, ∞). Bifurcation also occurs at any eigenvalues of the linearized problem that are below N 2 /4. Our treatment is based on recent results concerning the bifurcation points of equations with nonlinearities that are Hadamard differentiable, but not Fréchet differentiable.

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

A bifurcation problem governed by the boundary condition I

2007

We deal with positive solutions of ∆u = a(x)u p in a bounded smooth domain Ω ⊂ R N subject to the boundary condition ∂u/∂ν = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ1.