Multiplicity of solutions for two-point boundary value problems with asymptotically asymmetric nonlinearities

Sadyrbaev, Felix

doi:10.1016/0362-546x(95)00113-a

Cited by 13 publications

(17 citation statements)

References 9 publications

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“…The literature is very rich of contributions also in this direction. Among the various possible references, we quote the book by Krasnosel'skiȋ-Perov-Povolotskiȋ-Zabreȋko [28] and the more recent generalizations attained by Sadyrbaev [46] and by , where more regularity is required on the asymptotically asymmetric nonlinearities. An interesting result is Vol.…”

Section: Introductionmentioning

confidence: 99%

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

Dalbono

Zanolin

2007

MedJM

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Section: Introductionmentioning

confidence: 99%

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

Dalbono

Zanolin

2007

MedJM

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“…For the case p = 2, results similar to Theorem 6.4 have been obtained in [22], using a global bifurcation proof similar to the above proof of Theorem 6.4, and in [25], using a Prüfer angle proof.…”

Section: Global Bifurcation and Multiplicity Resultsmentioning

confidence: 68%

p-Laplacian problems with jumping nonlinearities

Rynne

2006

Journal of Differential Equations

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We consider the p-Laplacian boundary value problemc 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 ,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.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 halfeigenfunctions.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.

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“…For the separated problem (with L ¼ L 0 ), results similar to Theorem 5.3 have been obtained in [33], while results similar to Theorem 5.4 have been obtained in many papers, including [28] (with a; b constant) or [33]. For the periodic problem, results similar to Theorem 5.4 have been obtained in [25] with L ¼ L 0 and f ; a; b independent of t:…”

Section: Solvability Conditionsmentioning

confidence: 78%

Half-eigenvalues of periodic Sturm–Liouville problems

Binding

Rynne

2004

Journal of Differential Equations

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We consider the nonlinear Sturm-Liouville problem Àðpu 0 Þ 0 þ qu ¼ au þ À bu À þ lu; in ð0; 2pÞ; ð1Þ uð0Þ ¼ uð2pÞ; ðpuÞ 0 ð0Þ ¼ ðpuÞ 0 ð2pÞ; ð2Þ where 1=p; qAL 1 ð0; 2pÞ; with p40 a.e. on ð0; 2pÞ; a; bAL 1 ð0; 2pÞ; l is a real parameter, and u 7 ðtÞ ¼ maxf7uðtÞ; 0g for tA½0; 2p: Values of l for which (1)-(2) has a non-trivial solution u will be called half-eigenvalues while the corresponding solutions u will be called halfeigenfunctions. The set of half-eigenvalues will be denoted by S H : 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 associated with S H : These properties yield results on the existence and non-existence of solutions of the problem(together with (2)), where hAL 1 ð0; 2pÞ; f : ½0; 2p Â R-R is a Carathe´odory function, and the limits aðtÞ :¼ limx-N

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Multiplicity of solutions for two-point boundary value problems with asymptotically asymmetric nonlinearities

Cited by 13 publications

References 9 publications

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

p-Laplacian problems with jumping nonlinearities

Half-eigenvalues of periodic Sturm–Liouville problems

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