Consider the problem  ∆u = λe in B u = ∂u ∂n = 0 on ∂B, where B is the unit ball in R and λ is a parameter. Unlike the Gelfand problem the natural candidate u = −4 ln(|x|), for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. Dávila et al. in [5] use...

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turán numbers. In this paper, we determin...

We study the supergravity solutions describing non-extremal enhançons. There are two branches of solutions: a ‘shell branch’ connected to the extremal solution, and a ‘horizon branch’ which connects to the Schwarzschild black hole at large mass. We show that the shell branch solutions violate the weak energy condition, and are hence unphysical. We investigate linearized perturbations of the hor...

We consider the one-loop corrected geometry and thermodynamics of a rotating BTZ black hole by way of a dimensionally reduced dilaton model. The analysis begins with a comprehensive study of the non-extremal solution after which two different methods are invoked to study the extremal case. The first approach considers the extremal limit of the non-extremal calculations, whereas the second treat...

For a bounded linear operator on Hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎We study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎Also we show that any quasinilpotent operator $T$ has an hypernoncyclic vector‎, ‎and so $T$ has a nontrivial hyperinvariant subspace‎.

We discuss the thermodynamics of the N = 2, SU(N) gauge theory at large ’t Hooft coupling. The tool we use is the non-extremal deformation of the supergravity solution of Pilch and Warner (PW) [hep-th/0004063], dual to N = 4, SU(N) gauge theory softly broken to N = 2. We construct the exact non-extremal solution in fivedimensional gauged supergravity and further uplift it to ten dimensions. Tur...