Nonresonance conditions for generalised φ-Laplacian problems with jumping nonlinearities

Research output: Contribution to journalArticle

Abstract

We consider the boundary value problem(0.1)- ? (x, u (x), u' (x))' = f (x, u (x), u' (x)), a.e. x ? (0, 1),(0.2)c00 u (0) = c01 u' (0), c10 u (1) = c11 u' (1), where | cj 0 | + | cj 1 | > 0, for each j = 0, 1, and ?, f : [0, 1] × R2 ? R are Carathéodory functions, with suitable additional properties. The differential operator generated by the left-hand side of (0.1), together with the boundary conditions (0.2), is a generalisation of the usual p-Laplacian, and also of the so-called f{symbol}-Laplacian (which corresponds to ? (x, s, t) = f{symbol} (t), with f{symbol} an odd, increasing homeomorphism). For the p-Laplacian problem (and more particularly, the semilinear case p = 2), 'nonresonance conditions' which ensure the solvability of the problem (0.1), (0.2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fucík spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on ? and f, we extend these conditions to the general problem (0.1), (0.2). © 2009 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)2364-2379
Number of pages16
JournalJournal of Differential Equations
Volume247
Issue number8
DOIs
Publication statusPublished - 15 Oct 2009

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Jumping Nonlinearities
Nonresonance
P-Laplacian
Eigenvalue
Homeomorphism
Growth Conditions
Semilinear
Solvability
Differential operator
Odd
Boundary Value Problem
Boundary conditions

Cite this

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title = "Nonresonance conditions for generalised φ-Laplacian problems with jumping nonlinearities",
abstract = "We consider the boundary value problem(0.1)- ? (x, u (x), u' (x))' = f (x, u (x), u' (x)), a.e. x ? (0, 1),(0.2)c00 u (0) = c01 u' (0), c10 u (1) = c11 u' (1), where | cj 0 | + | cj 1 | > 0, for each j = 0, 1, and ?, f : [0, 1] × R2 ? R are Carath{\'e}odory functions, with suitable additional properties. The differential operator generated by the left-hand side of (0.1), together with the boundary conditions (0.2), is a generalisation of the usual p-Laplacian, and also of the so-called f{symbol}-Laplacian (which corresponds to ? (x, s, t) = f{symbol} (t), with f{symbol} an odd, increasing homeomorphism). For the p-Laplacian problem (and more particularly, the semilinear case p = 2), 'nonresonance conditions' which ensure the solvability of the problem (0.1), (0.2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fuc{\'i}k spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on ? and f, we extend these conditions to the general problem (0.1), (0.2). {\circledC} 2009 Elsevier Inc. All rights reserved.",
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Nonresonance conditions for generalised φ-Laplacian problems with jumping nonlinearities. / Rynne, Bryan P.

In: Journal of Differential Equations, Vol. 247, No. 8, 15.10.2009, p. 2364-2379.

Research output: Contribution to journalArticle

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AB - We consider the boundary value problem(0.1)- ? (x, u (x), u' (x))' = f (x, u (x), u' (x)), a.e. x ? (0, 1),(0.2)c00 u (0) = c01 u' (0), c10 u (1) = c11 u' (1), where | cj 0 | + | cj 1 | > 0, for each j = 0, 1, and ?, f : [0, 1] × R2 ? R are Carathéodory functions, with suitable additional properties. The differential operator generated by the left-hand side of (0.1), together with the boundary conditions (0.2), is a generalisation of the usual p-Laplacian, and also of the so-called f{symbol}-Laplacian (which corresponds to ? (x, s, t) = f{symbol} (t), with f{symbol} an odd, increasing homeomorphism). For the p-Laplacian problem (and more particularly, the semilinear case p = 2), 'nonresonance conditions' which ensure the solvability of the problem (0.1), (0.2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fucík spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on ? and f, we extend these conditions to the general problem (0.1), (0.2). © 2009 Elsevier Inc. All rights reserved.

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