Asymptotic Bifurcation Results for Quasilinear Elliptic Operators

Abstract

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.

A-priori bounds and asymptotics on the eigenvalues in bifurcation problems for perturbed self-adjoint operators

“…In particular, due to the simplicity of the first eigenvalue of (3.6), this statement holds for n = 1 and thus essentially yields the result of [3] cited in the introduction.…”

“…Unfortunately, the coefficient of the leading term in (3.13) differs for a factor μ 0 1 from the value given in formula (3.5) of [3]. We are unable to find the miscalculation explaining this discrepancy.…”

“…We are unaware of further contributions to the subject except for the recent paper of Chabrowski, Drábek and Tonkes [3], who have refined and, in a sense, generalized the result just mentioned in the case of bifurcation problems from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. Precisely, in [3] the authors consider the existence of positive solutions in the Sobolev space W…”

“…In [3], the emphasis is on the quasilinear case (s = 2) and on the critical exponent s * for the power-type nonlinearity F μ , but the result clearly holds true a fortiori when s * is replaced by a non-critical exponent t, s < t < s * (in which case proofs become simpler as the relevant operators gain compactness). If we restrict our attention to the semilinear/subcritical case (s = 2, 2 < t < 2 * ), we have t − 2 as exponent in the asymptotic formula (1.8); and since in this case F μ satisfies the condition 3) -or more precisely, with its counterpart for the eigenvalues μ R ≡ λ R −1 of (1.4) -we see that the result of [3] not only agrees with, but indeed substantially improves, ours in that it yields an asymptotic formula for the eigenvalues μ along the bifurcating branch C.…”

Existence and Bifurcation Results for Fourth-Order Elliptic Equations Involving Two Critical Sobolev Exponents

“…Consequently, μ( ) = 0 or h For more details, we refer to [3,7]. The proof of the following lemma follows as an easy combination of Hölder's inequality with the Sobolev embeddings and it is omitted.…”