Morse indices of solutions for super-linear elliptic PDE’s

Hajlaoui, Hatem; Harrabi, Abdellaziz; Mtiri, Foued

doi:10.1016/j.na.2015.01.002

Cited by 11 publications

(18 citation statements)

References 20 publications

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“…demanding that f = f odd,+ has the symmetry f (x, −u) = −f (x, u)) equation (2) has infinitely many stationary solutions. This is well known, see for example [10,26,40,49,50], although this can also be deduced from theorem A directly. Now theorem A implies that if all the stationary solutions are hyperbolic (a condition which can be ensured by adding a small perturbation), for such anti-symmetric nonlinearities f = f odd,+ there exist infinitely many travelling wave solutions of (RDE) with any given nonzero wave speed.…”

Section: Resultsmentioning

confidence: 76%

“…By the regularity results from section 3 we know that U and V form continuous curves in X 0 , hence t → L(t) is continuous. Hence L(t) L(X 0 ,L 2 (Ω)) C uniformly for t in a neighbourhood of t * , and consequently w satisfies an inequality of the form (26).…”

Section: Continuation For An Integro-differential Inequalitymentioning

confidence: 97%

“…For any given k ∈ Z, a classical result from Bahri and Lions (see [10], but also [26,27,40,49,50]) then gives us a priori bounds on the L ∞ norm of rest points Z with a given morse index m f (z) k. In light of (50) this gives L ∞ bounds on Z with a given index µ f (Z) k. Thus S k (X 0 , f ) is finite for each k, i.e. X 0 is finitely generating for f .…”

Section: Furthermorementioning

confidence: 99%

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A Floer Homology Approach to Traveling Waves in Reaction-Diffusion Equations on Cylinders

Bakker¹,

Berg²,

Vandervorst³

2018

SIAM J. Appl. Dyn. Syst.

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Travelling waves form a prominent feature in the dynamics of scalar reactiondiffusion equations on unbounded cylinders. The travelling waves can be identified with the bounded solutions of the elliptic PDEwhere c = 0 is the wavespeed, Ω ⊂ R d is a bounded domain, ∆ is the Laplacian on Ω, and B denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of (1).Restrictions on the nonlinearity f are kept to a minimum, for instance, any nonlinearity exhibiting polynomial growth in u can be considered. In particular, the set of bounded solutions of the travelling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity f , thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of (1), thus obtaining existence and multiplicity results for travelling wave solutions of reaction-diffusion equations on unbounded cylinders.

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Section: Resultsmentioning

confidence: 76%

Section: Continuation For An Integro-differential Inequalitymentioning

confidence: 97%

Section: Furthermorementioning

confidence: 99%

See 1 more Smart Citation

A Floer Homology Approach to Traveling Waves in Reaction-Diffusion Equations on Cylinders

Bakker¹,

Berg²,

Vandervorst³

2018

SIAM J. Appl. Dyn. Syst.

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“…As in [20], we shall employ a cut-off function with compact support to derive a variant of the Pohozaev identity. This device allows us to avoid the spherical integrals raised in [21], which are very difficult to control, especially for the polyharmonic situations.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…This device allows us to avoid the spherical integrals raised in [21], which are very difficult to control, especially for the polyharmonic situations. For , the Pohozaev identity is similar to [7, 8, 20, 22]. …”

Section: Auxiliary Resultsmentioning

confidence: 99%

Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

2017

Self Cite

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Under some assumptions on the nonlinearity f, we will study the nonexistence of nontrivial stable solutions or solutions which are stable outside a compact set of for the following semilinear higher-order problem: with . The main methods used are the integral estimates and the Pohozaev identity. Many classes of nonlinearity will be considered; even the sign-changing nonlinearity, which has an adequate subcritical growth at zero as for example , where , , , . More precisely, we shall revise the nonexistence theorem of Berestycki and Lions (Arch. Ration. Mech. Anal. 82:313-345, 1983) in the class of smooth finite Morse index solutions as the well known work of Bahri and Lions (Commun. Pure Appl. Math. 45:1205-1215, 1992). Also, the case when is a nonnegative function will be studied under a large subcritical growth assumption at zero, for example or , and . Extensions to solutions which are merely stable are discussed in the case of supercritical growth with .

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Transversality Theory with Applications to Differential Equations

Motreanu

Motreanu²

2016

Essays in Mathematics and Its Applications

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Morse indices of solutions for super-linear elliptic PDE’s

Cited by 11 publications

References 20 publications

A Floer Homology Approach to Traveling Waves in Reaction-Diffusion Equations on Cylinders

A Floer Homology Approach to Traveling Waves in Reaction-Diffusion Equations on Cylinders

Classification of stable solutions for non-homogeneous higher-order elliptic PDEs

Transversality Theory with Applications to Differential Equations

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