p-Laplacian problems with jumping nonlinearities

Research output: Contribution to journalArticle

Abstract

We consider the p-Laplacian boundary value problem{A formula is presented}{A formula is presented} where p > 1 is a fixed number, f{symbol}p ( s ) = | s |p - 2 s, s ? R, and for each j = 0, 1, | cj 0 | + | cj 1 | > 0. The function f : [ 0, 1 ] × R2 ? R is a Carathéodory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R2,{A formula is presented} where ?±, ?± ? L1 ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L1 ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and limr ? 8 e ( r ) r1 - p = 0. This condition allows the nonlinearity in (1) to behave differently as u ? ± 8. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem{A formula is presented} together with (2), where a, b ? L1 ( 0, 1 ), ? ? R, and u± ( x ) = max { ± u ( x ), 0 } for x ? [ 0, 1 ]. This problem is 'positively-homogeneous' and jumping. Values of ? for which (2), (3) has a nontrivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and nonexistence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2). When the functions a and b are constant the set of half-eigenvalues is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. © 2005 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)501-524
Number of pages24
JournalJournal of Differential Equations
Volume226
Issue number2
DOIs
Publication statusPublished - 15 Jul 2006

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Jumping Nonlinearities
P-Laplacian
Eigenvalue
Global Bifurcation
Eigenfunctions
Nonlinearity
Multiplicity Results
Nontrivial Solution
Nonexistence
Solvability
Bifurcation
Boundary Value Problem

Keywords

  • Half-eigenvalues
  • Jumping nonlinearity
  • p-Laplacian

Cite this

@article{2c71964f407441b9986695ba335be27a,
title = "p-Laplacian problems with jumping nonlinearities",
abstract = "We consider the p-Laplacian boundary value problem{A formula is presented}{A formula is presented} where p > 1 is a fixed number, f{symbol}p ( s ) = | s |p - 2 s, s ? R, and for each j = 0, 1, | cj 0 | + | cj 1 | > 0. The function f : [ 0, 1 ] × R2 ? R is a Carath{\'e}odory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R2,{A formula is presented} where ?±, ?± ? L1 ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L1 ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and limr ? 8 e ( r ) r1 - p = 0. This condition allows the nonlinearity in (1) to behave differently as u ? ± 8. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem{A formula is presented} together with (2), where a, b ? L1 ( 0, 1 ), ? ? R, and u± ( x ) = max { ± u ( x ), 0 } for x ? [ 0, 1 ]. This problem is 'positively-homogeneous' and jumping. Values of ? for which (2), (3) has a nontrivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and nonexistence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2). When the functions a and b are constant the set of half-eigenvalues is closely related to the 'Fuc{\'i}k spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. {\circledC} 2005 Elsevier Inc. All rights reserved.",
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p-Laplacian problems with jumping nonlinearities. / Rynne, Bryan P.

In: Journal of Differential Equations, Vol. 226, No. 2, 15.07.2006, p. 501-524.

Research output: Contribution to journalArticle

TY - JOUR

T1 - p-Laplacian problems with jumping nonlinearities

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N2 - We consider the p-Laplacian boundary value problem{A formula is presented}{A formula is presented} where p > 1 is a fixed number, f{symbol}p ( s ) = | s |p - 2 s, s ? R, and for each j = 0, 1, | cj 0 | + | cj 1 | > 0. The function f : [ 0, 1 ] × R2 ? R is a Carathéodory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R2,{A formula is presented} where ?±, ?± ? L1 ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L1 ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and limr ? 8 e ( r ) r1 - p = 0. This condition allows the nonlinearity in (1) to behave differently as u ? ± 8. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem{A formula is presented} together with (2), where a, b ? L1 ( 0, 1 ), ? ? R, and u± ( x ) = max { ± u ( x ), 0 } for x ? [ 0, 1 ]. This problem is 'positively-homogeneous' and jumping. Values of ? for which (2), (3) has a nontrivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and nonexistence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2). When the functions a and b are constant the set of half-eigenvalues is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. © 2005 Elsevier Inc. All rights reserved.

AB - We consider the p-Laplacian boundary value problem{A formula is presented}{A formula is presented} where p > 1 is a fixed number, f{symbol}p ( s ) = | s |p - 2 s, s ? R, and for each j = 0, 1, | cj 0 | + | cj 1 | > 0. The function f : [ 0, 1 ] × R2 ? R is a Carathéodory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R2,{A formula is presented} where ?±, ?± ? L1 ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L1 ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and limr ? 8 e ( r ) r1 - p = 0. This condition allows the nonlinearity in (1) to behave differently as u ? ± 8. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem{A formula is presented} together with (2), where a, b ? L1 ( 0, 1 ), ? ? R, and u± ( x ) = max { ± u ( x ), 0 } for x ? [ 0, 1 ]. This problem is 'positively-homogeneous' and jumping. Values of ? for which (2), (3) has a nontrivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and nonexistence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2). When the functions a and b are constant the set of half-eigenvalues is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. © 2005 Elsevier Inc. All rights reserved.

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