Bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity

Abstract

This paper deals with existence results for the following nonlinear problem with the Dirichlet p-Laplacian D p in a bounded domain OCR N :NÞ is a fixed number, h hðx; sÞ is a given function from O Â R into R; and lAR stands for a spectral parameter. We focus on l close to l 1 ; including the resonant case l ¼ l 1 : The nonlinearity h is assumed to be of Landesman-Lazer type, but we can deal with vanishing nonlinearities as well. Our asymptotic method substitutes the Lyapunov-Schmidt method in some sense. Unlike in the semilinear case p ¼ 2; our method can treat more general nonlinearities if pa2 (vanishing nonlinearities with very fast decay).

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

“…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.…”

Poincaré inequality and Palais–Smale condition for the p-Laplacian

“…topological) methods, cf. [6][7][8][9][10]13,17,18,[20][21][22]. As a new result, in analogy with the case p > 2 and c > 0, we show that the different form of the (P.S.)…”

Asymptotic bifurcation problems for quasilinear equations existence and multiplicity results

“…It can be actually proved that this cone has a nonempty interior with respect to the topology induced by L ∞ norm (see [7]). Let us also note that the weak solution belongs to C 1,α (Ω) with some α ∈ (0, 1) provided ∂Ω is smooth enough ( [8], [9]).…”

“…The results addressed in the following sections were obtained in papers Drábek, Girg, Takáč and Ulm [9] and Drábek, Girg and Takáč [8], and they rely on the bifurcation theory. In combination with the method of lower and upper solutions they lead to the following main results.…”