This paper deals with the class of Schrödinger–Kirchhoff-type biharmonic problems

where Δ2 denotes the biharmonic operator, and f ∈ C(ℝN × ℝ, ℝ) satisfies the Ambrosetti–Rabinowitz-type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.

The least Steklov eigenvalue d1 for the biharmonic operatorin bounded domains gives a bound for the positivity preserving property for the hingedplate problem, appears as a norm of a suitable trace operator, and gives the optimalconstant to estimate the L2-norm of harmonic functions. Theseapplications suggest to address the problem of minimizing d1in suitable classes of domains. We survey the existing results and conjectures about thistopic; in particular, the existence of a convex domain of fixed measure minimizingd1 is known, although the optimal shape is still unknown. Weperform several numerical experiments which strongly suggest that the optimal planar shapeis the regular pentagon. We prove the existence of a domain minimizingd1 also among convex domains having fixed perimeter andpresent some numerical results supporting the conjecture that, among planar domains, thedisk is the minimizer.

Let $D$ be a connected bounded domain in ${{\mathbb{R}}^{n}}$. Let $0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $ be the eigenvalues of the following Dirichlet problem:

$$\left\{ \begin{align} & {{\Delta }^{2}}u(x)\,+\,V(x)u(x)\,=\,\mu \rho (x)u(x),x\in \,D \\ & u{{|}_{\partial D}}\,=\,\frac{\partial u}{\partial n}\,{{|}_{\partial D}}\,=\,0, \\ \end{align} \right.$$

where $V(x)$ is a nonnegative potential, and $\rho (x)\,\in \,C(\overset{-}{\mathop{D}}\,)$ is positive. We prove the following inequalities:

$$\begin{align} & {{\mu }_{k+1}}\le \frac{1}{k}\sum\limits_{i=1}^{k}{\mu i+}{{[\frac{8(n+2)}{{{n}^{2}}}{{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}]}^{1/2}}\times \frac{1}{k}{{\sum\limits_{i=1}^{k}{[{{\mu }_{i}}({{\mu }_{k+1}}-{{\mu }_{i}})]}}^{1/2}}, \\ & \frac{{{n}^{2}}{{k}^{2}}}{8(n+2)}\le {{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}[\sum\limits_{i=1}^{k}{\frac{\mu _{i}^{1/2}}{{{\mu }_{k+1}}-{{\mu }_{i}}}}]\times \sum\limits_{i=1}^{k}{\mu _{i}^{1/2}}. \\ \end{align}$$

We study the biharmonic equation Δ2u = u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn, n ≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.