Details

Autor(en) / Beteiligte

• Drábek, P.
• Girg, P.
• Takáč, P.
• Ulm, M.

Titel

The Fredholm Alternative for the p-Laplacian: Bifurcation from Infinity, Existence and Multiplicity

Ist Teil von

• Indiana University mathematics journal, 2004-01, Vol.53 (2), p.433-482

Ort / Verlag

Bloomington, IN: Department of Mathematics INDIANA UNIVERSITY

Erscheinungsjahr

2004

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• This work is concerned with the existence and multiplicity of weak solutions $u\epsilon W^{1,p}_0(\Omega )$ to the quasilinear elliptic problem $(P)\qquad \begin{cases} -\Delta _pu=\lambda |u|^{p-2}u+f(x) \qquad \text{in} \Omega;\\ u=0 \qquad \qquad \qquad \qquad \qquad \text{on}\partial \Omega \end{cases}$, with the spectral parameter λ ∈ ℝ near the (simple) principal eigenvalue λ1 of the positive Dirichlet p-Laplacian −Δp in a bounded domain Ω ⊂ ℝN, for 1 < p < ∞. Here, Δpu ≡ div(|∇u|p−2∇u) and f ∈ L∞(Ω) is a given function. A priori bounds on the solutions are obtained from a rather precise description of possible "large solutions" investigated by bifurcations from infinity. They take the form u = t−1(φ1 + v⊺) as t → 0, t ∈ ℝ \ {0}, where φ1 stands for the (positive) eigenfunction associated with λ1, and v⊺ is a relatively small perturbation of φ1 which is orthogonal to φ1. We also allow λ and f to vary with t → 0. Our method is based on the linearization of Δp near φ1. As a result of our asymptotic formula for λ depending on t and f, with the integral ʃΩ fφ1 dx playing a major role, we are able to obtain a number of new results for problem (P). Some of these results for p ≠ 2 are quite different from the linear case p = 2.