Positive solutions to a linearly perturbed critical growth biharmonic problem
Elvise Berchio Filippo Gazzola
Discrete & Continuous Dynamical Systems - S 2011, 4(4): 809-823 doi: 10.3934/dcdss.2011.4.809
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.
keywords: critical exponent Biharmonic equations Steklov boundary conditions.
Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms
Elvise Berchio Debdip Ganguly
Communications on Pure & Applied Analysis 2016, 15(5): 1871-1892 doi: 10.3934/cpaa.2016020
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.
keywords: hyperbolic space Higher order Poincaré inequalities Poincaré-Hardy inequalities
Nodal solutions to critical growth elliptic problems under Steklov boundary conditions
Elvise Berchio Filippo Gazzola Dario Pierotti
Communications on Pure & Applied Analysis 2009, 8(2): 533-557 doi: 10.3934/cpaa.2009.8.533
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.
keywords: nodal solutions Steklov Critical growth

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