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• W2116746932 abstract "We consider the p -Laplacian boundary value problem (1) − ( ϕ p ( u ′ ( x ) ) ) ′ = f ( x , u ( x ) , u ′ ( x ) ) , a.e. x ∈ ( 0 , 1 ) , (2) c 00 u ( 0 ) = c 01 u ′ ( 0 ) , c 10 u ( 1 ) = c 11 u ′ ( 1 ) , where p > 1 is a fixed number, ϕ 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 , ψ ± ( x ) ϕ p ( s ) − E ( x , s , t ) ⩽ f ( x , s , t ) ⩽ Ψ ± ( x ) ϕ p ( s ) + E ( x , s , t ) , ± s ⩾ 0 , where ψ ± , Ψ ± ∈ L 1 ( 0 , 1 ) , and E has the form E ( x , s , t ) = ζ ( x ) e ( | s | + | t | ) , with ζ ∈ L 1 ( 0 , 1 ) , ζ ⩾ 0 , e ⩾ 0 and lim r → ∞ e ( r ) r 1 − p = 0 . This condition allows the nonlinearity in (1) to behave differently as u → ± ∞ . Such a nonlinearity is often termed jumping . Related to (1) , (2) is the problem (3) − ( ϕ p ( u ′ ) ′ ) = a ϕ p ( u + ) − b ϕ p ( u − ) + λ ϕ p ( u ) , in ( 0 , 1 ) , together with (2) , where a , b ∈ L 1 ( 0 , 1 ) , λ ∈ R , and u ± ( x ) = max { ± u ( x ) , 0 } for x ∈ [ 0 , 1 ] . This problem is ‘positively-homogeneous’ and jumping. Values of λ 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 ‘Fučík 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." @default.
• W2116746932 created "2016-06-24" @default.
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• W2116746932 date "2006-07-01" @default.
• W2116746932 modified "2023-09-25" @default.
• W2116746932 title "p-Laplacian problems with jumping nonlinearities" @default.
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• W2116746932 doi "https://doi.org/10.1016/j.jde.2005.08.016" @default.
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