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DCDS-S

Existence and nonexistence results for positive solutions to a linearly perturbed critical growth biharmonic problem under Steklov boundary conditions, are determined. Furthermore, by investigating the critical dimensions for this problem, a Sobolev inequality with remainder terms, of both interior and boundary type, is deduced.

CPAA

The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality:
$$
\left( \frac{N-1}{2} \right)^{2(k -l)} :=\displaystyle \inf_{ u \in C^{\infty}_{0}(\mathbb{H}^{N} ) \setminus \{0\}} \qquad \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} },
$$
where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.

CPAA

We study elliptic problems at critical growth under Steklov
boundary conditions in bounded domains. For a second order problem we prove
existence of nontrivial nodal solutions. These are obtained by
combining a suitable linking argument with fine estimates on the
concentration of Sobolev minimizers on the boundary. When the domain is the
unit ball, we obtain a multiplicity result by taking advantage of the explicit form
of the Steklov eigenfunctions. We also partially extend the results in the
ball to the case of fourth order Steklov boundary value problems.

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