Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

Abstract

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problemHere, 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 from infinity and nodal solutions of quasilinear problems without the signum condition

“…We also note that the results of Theorem 3.1 can be easily extended to the high dimensional case by the use of the results of [8,15]. Using this fact and a proof similar to that of Lemma 3.1 of [9], we can also obtain similar results of Corollaries 3.1 and 3.2 for the high dimensional case.…”

On global bifurcation for the nonlinear Steklov problems

“…There are various sufficient conditions available on g for the existence of a bifurcation point of (1.3). For Dirichlet boundary condition, g = 1 [19,28,37], g ∈ L r (Ω) with r > N 2 [8], g ∈ L ∞ (R N ) [20]. There are a few works deals with h of the form λf (x)r(φ) with continuous r satisfying r(0) = 0 and certain growth condition at zero and at infinity, see for g, f in Hölder continuous spaces [43], in certain Lebesgue spaces [27], in Lorentz spaces [7,36].…”

“…Later, Rabinowitz [41,Theorem 1.3], extended this result by exhibiting a continuum of nontrivial solutions of (1.4) bifurcating from (λ, 0) which is either unbounded in R × X or meets at (λ * , 0), where µ = λ * −1 is an eigenvalue of L. Further, if µ has multiplicity one, then this continuum decompose into two subcontinua of nontrivial solutions of (1.4), see [3,17,18,41,42]. For p = 2, the Leray-Schauder degree is extended for certain maps between X to X ′ [11,44] and then an analogue of Rabinowitz result is proved for the first eigenvalue of A = λG, see [19,20,28,37].…”

A p(x)-Laplacian extension of the Díaz-Saa inequality and some applications

“…To verify inequality (1.5), in § 2 below (theorem 2.5), we slightly modify the method used in [12,13,22]. Our proof inequality (1.5) is based on the convexity of the restriction of the functional W to the cone…”

“…To verify ineq. (1.5), in Section 2 below (Theorem 2.4) we slightly modify the method used in [12,13,22]. Our proof of ineq.…”