On the Optimality of the Assumptions Used to Prove the Existence and Symmetry of Minimizers of Some Fractional Constrained Variational Problems
 Hichem Hajaiej
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Abstract
In this paper, we discuss the optimality of the assumptions used, in a previous paper, to prove existence and symmetry of minimizers of the fractional constrained variational problem: $$\inf \;\left\{\frac{1}{2} \int\nabla_s u^2  \int F(x, u):\;u\in H^s (\mathbb{R}^N) \mbox{ and } \int u^2 = c^2\right\},$$ where c is a prescribed number. More precisely, we will show that if one of the conditions, used to prove that all minimizers of the above constrained variational problem, are radial and radially decreasing for all c, do not hold true, then there are several interesting situations:
There is no minimizer at all.
The infimum is achieved but no minimizer is radial.
For some values of c there is no minimizer. For large values, the minimizer is radial and radially decreasing.
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 Hajaiej, H.: Variational problems related to some fractional kinetic equations. JMAA (accepted for publication)
 Hajaiej, H.: Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity (2012, preprint)
 Hajaiej, H., Molinet, L., Ozawa, T., Wang, B.: Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized Boson equations (2012, preprint)
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 Title
 On the Optimality of the Assumptions Used to Prove the Existence and Symmetry of Minimizers of Some Fractional Constrained Variational Problems
 Journal

Annales Henri Poincaré
Volume 14, Issue 5 , pp 14251433
 Cover Date
 20130701
 DOI
 10.1007/s000230120212x
 Print ISSN
 14240637
 Online ISSN
 14240661
 Publisher
 SP Birkhäuser Verlag Basel
 Additional Links
 Topics
 Authors

 Hichem Hajaiej ^{(1)}
 Author Affiliations

 1. Department of Mathematics, King Saud University, 11451, Riyadh, Kingdom of Saudi Arabia