Andrei Grecu
University of Craiova, Craiova, Romania

For each bounded and open set $\Omega\subset\mathbb{R}^N$ ($N\geq 2$) with smooth boundary denoted by $\partial\Omega$ and each real number $p\in(1,\infty)$ we analyse the torsion problem of the $p$-Bilaplacian, namely $\Delta(|\Delta u|^{p-2}\Delta u)=1$ in $\Omega$ with $u=\Delta u=0$ on $\partial\Omega$. Firstly, we show that for each $p\in(1,\infty)$ the problem has a unique weak solution $u_p$. Secondly, we prove that $u_p$ converges uniformly, as $p\rightarrow\infty$, in $C^1(\overline\Omega)$ to a certain function, say $v_2$, which is exactly the unique solution of the problem $-\Delta u=1$ in $\Omega$ with $u=0$ on $\partial\Omega$. Next, we show that each solution $u_p$ is also a solution for the minimization problem $${\cal T}(p;\Omega):=\inf_{u\in {\cal X}_p(\Omega)\setminus\{0\}}\frac{\frac{1}{|\Omega|}\int_\Omega|\Delta u|^p\;dx}{\left(\frac{1}{|\Omega|}\int_\Omega u\;dx\right)^p}\,,$$ where ${\cal X}_p(\Omega):=\{u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega):\;u(x)\geq 0,\; a.e.\;x\in\Omega \}$ Further, we show that the function $(1,\infty)\ni p\mapsto{\cal T}(p;\Omega)$ is strictly increasing provided that $\Omega$ is a convex and bounded open set for which $|\Omega|^{-1}\int_\Omega v_2\;dx$ is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant ${\cal T}(p;\Omega )$ when $|\Omega|^{-1}\int_\Omega v_2\;dx$ is small. That last variational characterization fails to hold true when $|\Omega|^{-1}\int_\Omega v_2\;dx>1$.