Let us consider the Dirichlet problem {(-Δ)mu=|u| pα-2u/|x|α+λu in Ω D βu|∂Ω = 0 for |β|≤m-1 where Ω ⊂ ℝn is a bounded open set containing the origin, n>2m, 0<α<2m and pα = 2(n-α)/(n-2m). We find that, when n ≥ 4m, this problem has a solution for any 0<λ<Λ m,1 where Λm,1 is the first Dirichlet eigenvalue of (-Δ)m in Ω, while, when 2m<n<4m, the solution exists if λ is sufficiently close toΛm,1, and we show that these space dimensions are critical in the sense of Pucci-Serrin and Grunau. Moreover, we find corresponding existence and nonexistence results for the Navier problem, i.e. with boundary conditions Δju| ∂Ω = 0 for 0 ≤ j ≤ m-1. To achieve our existence results it is crucial to study the behaviour of the radial positive solutions (whose analytic expression is not known) of the limit problem (-Δ) mu = upα-1|x|-α in the whole space ℝn.

This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the C-infinity-well-posedness which is also necessary if the space dimension is equal to one. The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix.

In this paper we study the problem \[ \begin{cases}{\mathcal L}_\mu[u]:=\Delta^2u -\mu\frac{u}{|x|^4}=\lambda u +|u|^{2^*-2}u\quad\hbox{in\ }\Omega\\ u=\frac{\partial u}{\partial n}=0\quad\hbox{on\ }\partial\Omega\end{cases} \] where \Omega ⊂ R^n is a bounded open set containing the origin, n ≥ 5 and 2^∗ = 2n/(n − 4). We find that this problem is critical (in the sense of Pucci–Serrin and Grunau) depending on the value of μ ∈ [0, μ), μ being the best constant in Rellich inequality. To achieve our existence results it is crucial to study the behavior of the radial solutions (whose analytic expression is not known) of the limit problem Lμ u = u^(2* −1) in the whole space R^n . On the other hand, our non–existence results depend on a suitable Pohozaev-type identity, which in turn relies on some weighted Hardy–Rellich inequalities.

We consider the Cauchy problem for homogeneous linear third order weakly hyperbolic equations with time depending coefficients. We study the relation between the regularity of the coefficients and the Gevrey class in which the Cauchy problem is well-posed.