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.
Nonlinear critical problems for the biharmonic operator with Hardy potential
D'AMBROSIO, Lorenzo;IANNELLI, Enrico
2015-01-01
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
