### 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, | c_{j 0} | + | c_{j 1} | > 0. The function f : [ 0, 1 ] × R^{2} ? R is a Carathéodory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R^{2},{A formula is presented} where ?_{±}, ?_{±} ? L^{1} ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L^{1} ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and lim_{r ? 8} e ( r ) r^{1 - 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 ? L^{1} ( 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 language | English |
---|---|

Pages (from-to) | 501-524 |

Number of pages | 24 |

Journal | Journal of Differential Equations |

Volume | 226 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Jul 2006 |

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### Keywords

- Half-eigenvalues
- Jumping nonlinearity
- p-Laplacian

### Cite this

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*Journal of Differential Equations*, vol. 226, no. 2, pp. 501-524. https://doi.org/10.1016/j.jde.2005.08.016

**p-Laplacian problems with jumping nonlinearities.** / Rynne, Bryan P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - p-Laplacian problems with jumping nonlinearities

AU - Rynne, Bryan P.

PY - 2006/7/15

Y1 - 2006/7/15

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.

KW - Half-eigenvalues

KW - Jumping nonlinearity

KW - p-Laplacian

UR - http://www.scopus.com/inward/record.url?scp=33646502375&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2005.08.016

DO - 10.1016/j.jde.2005.08.016

M3 - Article

VL - 226

SP - 501

EP - 524

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -