The Fredholm alternative for the p-Laplacian: Bifurcation from infinity, existence and multiplicity

Drábek, Pavel; Takáč, Peter; Ulm, Michael; Girg, Petr

doi:10.1512/iumj.2004.53.2396

Cited by 29 publications

(29 citation statements)

References 14 publications

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“…This extension also fills the gap left open in several results on global bifurcations from (±∞, μ 1 ) for problem (1.1) obtained in Drábek et al [16,Section 5] and in Drábek, Girg, and Takáč [15,Section 3] as well.…”

Section: Introductionsupporting

confidence: 69%

“…His hypothesis [14, Eq. (14.43), p. 191] has been verified in Drábek et al [16,Theorem 4.1] for the special case h(x, u; λ) ≡ f (x) independent from u and λ, for bifurcations from infinity at λ = μ 1 . Extending a new asymptotic technique developed recently in Drábek et al [16,Theorem 4.1] and Takáč [34,Section 5] and [35,Section 6], we are able to verify Drábek's hypothesis [14,Eq.…”

Section: Introductionmentioning

confidence: 83%

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Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

Girg

Takáč

2008

Ann. Henri Poincaré

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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.

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Section: Introductionsupporting

confidence: 69%

Section: Introductionmentioning

confidence: 83%

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

Girg

Takáč

2008

Ann. Henri Poincaré

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“…In order to utilise the main bifurcation theorem from [7] we must verify that nontrivial solutions (λ, u) ∈ C are uniformly bounded in L ∞ ( ) provided that u 1,p is sufficiently small. To achieve this, we implement Moser iterations [15], and follow a style of proof from Proposition 1.2 in Guedda and Veron [10].…”

Section: Stage 2 Of Proofmentioning

confidence: 99%

“…Next we apply an asymptotic estimate from Theorem 4.1 in Drábek, Girg, Takác and Ulm [7]. In fact, only part of the estimate is required, and it is presented explicitly here for the convenience of the reader.…”

Section: Stage 3 Of Proofmentioning

confidence: 99%

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Asymptotic Bifurcation Results for Quasilinear Elliptic Operators

Chabrowski¹,

Drábek²,

Tonkes³

2005

Glasgow Math. J.

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Abstract.We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.2000 Mathematics Subject Classification. 47J15, 35J60, 35J65, 47J10.

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Ground state alternative for p-Laplacian with potential term

2006

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Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given byIn the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q ′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every ψ ∈ C ∞ 0 (Ω) satisfying ψv dx = 0 there exists a constant C > 0 such that C −1 W |u| p dx ≤ Q(u) + C uψ dx p for all u ∈ C ∞ 0 (Ω). As a consequence, we prove positivity properties for the quasilinear operator Q ′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

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The Fredholm alternative for the p-Laplacian: Bifurcation from infinity, existence and multiplicity

Cited by 29 publications

References 14 publications

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

Asymptotic Bifurcation Results for Quasilinear Elliptic Operators

Ground state alternative for p-Laplacian with potential term

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