On a nonresonance condition between the first and the second eigenvalues for the p‐Laplacian

Abstract

Abstract. We are concerned with the existence of solution for the Dirichlet problem x,u) lies in some sense between the first and the second eigenvalues of the p-Laplacian p . Extensions to more general operators which are (p − 1)-homogeneous at infinity are also considered.2000 Mathematics Subject Classification. 35J65.

The Beginning of the Fučik Spectrum for the p-Laplacian

“…It improves the result of [3] where :=; and where the strict inequalities in (F) are imposed on # \ (x),…”

The behavior of the best Sobolev trace constant and extremals in thin domains

“…See [1,6,7,13,16,17,21], etc. These eigenvalue problems are far from being completely understood and any new information that one can give could be helpful in the understanding of nonlinear phenomena.…”

Nonresonance conditions for generalised ϕ-Laplacian problems with jumping nonlinearities

“…(B) There exists an odd, increasing homeomorphism φ : R → R such that, for a.e. x ∈ [0, 1], and all (s, t) ∈ R 2 ,…”

“…However, to introduce some terminology and to motivate our discussion of the general ψ -Laplacian problem, we will give a brief description of the p-Laplacian results in the following paragraph. Nonresonance conditions for the φ-Laplacian have also been considered in many recent papers, see for example, [1,2,8,[11][12][13], and the survey [14], and the references therein. Nonresonance conditions of the type we consider here do not appear to have been discussed for the general ψ -Laplacian, except for a partial result, using eigenvalues, in the paper [1].…”

“…Nonresonance conditions for jumping f , with general L 1 coefficients ξ ± , Ξ ± , are obtained in [15] in terms of half-eigenvalues of the p-Laplacian; other types of nonresonance conditions are described in [15,Section 7], and in the references therein. Nonresonance conditions for non-jumping f , in terms of eigenvalues, are described in, for example, [1,10]. When f is jumping, but the functions ξ ± , Ξ ± are constants, nonresonance conditions have been given in terms of the so-called 'Fučík spectrum', see the references in [15].…”