Degree and global bifurcation for elliptic equations with multivalued unilateral conditions

Eisner, Jan; Kučera, Milan; Väth, Martin

doi:10.1016/j.na.2005.07.021

Cited by 16 publications

(27 citation statements)

References 12 publications

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Section: Introductionmentioning

confidence: 87%

“…Only in the scalar case (N = 1) and for the classical Laplacian (p 1 = 2), a special case of our main result (under more restrictive growth conditions and only for a more special operator) was obtained in [12]. Moreover, the idea of the compactness proof of [12] can only be used in Hilbert spaces. We will therefore prove in Appendix A a compactness theorem which is of independent interest and which generalizes Schauder's theorem on the compactness of adjoint operators to the multivalued setting.…”

Section: Introductionmentioning

confidence: 87%

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Continuity, compactness, and degree theory for operators in systems involving p-Laplacians and inclusions

Väth

2008

Journal of Differential Equations

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The study of weak solutions for systems of nonlinear partial differential equations of elliptic type with inclusions leads to a multivalued operator of superposition type in Sobolev spaces. We show that, under natural assumptions, this operator has the properties which allow to apply degree theory (fixed point index) for multivalued maps. More precisely, this operator is upper semicontinuous and compact with nonempty convex compact values. For the particular case of systems involving p-Laplacians, we show that there is a homeomorphism transforming the whole system to a situation for which a fixed point index is available. . 1 The paper was written in the framework of a Heisenberg fellowship (Az. VA 206/1-2); financial support by the DFG is gratefully acknowledged. The author wants to thank D. Werner for valuable remarks and suggestions and a referee for pointing out the approach (7.7).

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 87%

Continuity, compactness, and degree theory for operators in systems involving p-Laplacians and inclusions

Väth

2008

Journal of Differential Equations

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“…Since this is not a closed subset of R and the involved operators have typically no continuous extension to the closure, the classical Rabinowitz technique cannot directly be employed for the proof. We point out that all these generalizations will be applied for the main bifurcation result of the forthcoming paper [9].…”

Section: A Rabinowitz Type Bifurcation Theoremmentioning

confidence: 99%

“…5. (That result has already been applied in [9].) However, as mentioned above, an alternative and, in a sense, more general approach is related with the theory of "epi" maps which we discuss in Sect.…”

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confidence: 91%

Global solution branches and a topological implicit function theorem

Väth

2006

Annali di Matematica

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We show some abstract (purely set-topological) principles which allow to prove the existence of global solution branches. The results apply either for the locally compact situation and then allow to prove global bifurcation results of Rabinowitz type, or they apply for a locally connected situation and allow to prove global branches of arbitrarily small perturbations without any compactness hypotheses. As two applications, we obtain a generalization of the Rabinowitz theorem for bifurcation from an interval and an implicit function type theorem for nondifferentiable functions.

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“…To explain these relations it is necessary to start our exposition with the evolution system (2), (5), but in fact we will consider the stationary problem corresponding to (2), (3) with d changing along a curve σ in R 2 + . More precisely, we will consider a continuous mapping σ = [σ 1 , σ 2 ] : R + → R 2 + and the problem…”

Section: Introductionmentioning

confidence: 99%

Global bifurcation for quasivariational inequalities of reaction–diffusion type

Kučera¹

2008

Journal of Mathematical Analysis and Applications

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We consider a reaction-diffusion system with implicit unilateral boundary conditions introduced by U. Mosco. We show that global continua of stationary spatially nonhomogeneous solutions bifurcate in the domain of parameters where bifurcation in the case of classical boundary conditions is excluded. The problem is formulated as a quasivariational inequality and the proof is based on the Leray-Schauder degree.

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Degree and global bifurcation for elliptic equations with multivalued unilateral conditions

Cited by 16 publications

References 12 publications

Continuity, compactness, and degree theory for operators in systems involving p-Laplacians and inclusions

Continuity, compactness, and degree theory for operators in systems involving p-Laplacians and inclusions

Global solution branches and a topological implicit function theorem

Global bifurcation for quasivariational inequalities of reaction–diffusion type

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