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Existence and global analytic bifurcation for singular biharmonic equation with Navier boundary condition


Authors: Jacques Giacomoni, Guillaume Warnault and S. Prashanth
Journal: Proc. Amer. Math. Soc. 145 (2017), 151-164
MSC (2010): Primary 35J40, 35B09, 35J75, 35J91; Secondary 35B40, 35B32
DOI: https://doi.org/10.1090/proc/13179
Published electronically: July 22, 2016
MathSciNet review: 3565368
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Abstract: Let $ \Omega $ be a bounded smooth domain in $ \mathbb{R}^N$, $ N\geq 2,$ and let $ \rho $ denote the distance to the boundary function. Let $ K$ be a positive function such that $ K =O(\rho ^{-\beta })$ near $ \partial \Omega $ for some $ \beta \geq 0$. We consider the following fourth order singular elliptic problem (for $ \alpha >0$):

$\displaystyle \displaystyle \left \{\begin {array}{ll} & \Delta ^2 u = K(x)u^{-... ...artial \Omega }=0, \,\Delta u\vert _{\partial \Omega } = 0. \end{array}\right .$ ($ P$)

We show the existence of a unique $ C^2(\overline {\Omega })$ solution to the above problem when $ 0<\alpha + \beta <2$. We also show the nonexistence of such $ C^2$ solutions when $ K \sim \rho ^{-\beta }$ near $ \partial \Omega $ with $ \alpha + \beta \geq 2$.

We then consider an associated bifurcation problem involving a nonlinearity of the type $ K u^{-\alpha }+\lambda f(u)$ where $ f$ is taken to be super-linear at infinity. We show the existence of a global (in $ \lambda $) path-connected analytic branch of solutions to this bifurcation problem when again $ \alpha +\beta <2$.


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Additional Information

Jacques Giacomoni
Affiliation: Laboratoire de Mathématiques et de leurs Applications-Pau (UMR CNRS 5142) Bat. IPRA, Avenue de l’Université, F-64013 Pau, France
Email: jacques.giacomoni@univ-pau.fr

Guillaume Warnault
Affiliation: Laboratoire de Mathématiques et de leurs Applications-Pau (UMR CNRS 5142) Bat. IPRA, Avenue de l’Université, F-64013 Pau, France
Email: guillaume.warnault@univ-pau.fr

S. Prashanth
Affiliation: TIFR-Centre For Applicable Mathematics, Post Bag No. 6503, Sharada Nagar, GKVK Post Office, Bangalore 560065, India
Email: pras@math.tifrbng.res.in

DOI: https://doi.org/10.1090/proc/13179
Received by editor(s): November 10, 2015
Received by editor(s) in revised form: March 1, 2016
Published electronically: July 22, 2016
Additional Notes: All the authors of this work were supported by IFCAM
Communicated by: Catherine Sulem
Article copyright: © Copyright 2016 American Mathematical Society