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Existence and non-existence results for variational higher order elliptic systems

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  • Let $α ∈ \mathbb{N}$, $α ≥ 1$ and $(-Δ)^{α} = -Δ((-Δ)^{α-1})$ be the polyharmonic operator. We prove existence and non-existence results for the following Hamiltonian systems of polyharmonic equations under Dirichlet boundary conditions

    $\begin{cases}\begin{aligned}(-Δ)^{α} u = H_v(u, v) \\(-Δ)^{α} v = H_u(u, v) \\\end{aligned} \text{ in } Ω \subset \mathbb{R}^N \\\frac{\partial^{r} u}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω \\\frac{\partial^{r} v}{\partial ν^{r}} = 0, \, r = 0, \dots, α-1, \text{ on } \partial Ω\end{cases}$

    where $Ω$ is a sufficiently smooth bounded domain, $N >2α$, $ν$ is the outward pointing normal to $\partial Ω$ and the Hamiltonian $H ∈ C^1 (\mathbb{R}^2; \mathbb{R})$ satisfies suitable growth assumptions.

    Mathematics Subject Classification: 35J48, 35J58, 35J50.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  The light grey region represents values of $p, q$ for which we prove existence of solutions to 14, whereas the dark grey region is the domain of non-existence given by Corollary 2. The curve $l_1$ is $\frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2\alpha}{N}$, whereas $l_2$ is given by $pq = 1$

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