A C0 interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh

2013 ◽  
Vol 30 (3) ◽  
pp. 838-861 ◽  
Author(s):  
Sebastian Franz ◽  
Hans-Görg Roos ◽  
Andreas Wachtel
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Penalty Method ◽  
Singularly Perturbed ◽  
Interior Penalty ◽  
Order Elliptic Problem ◽  
Interior Penalty Method

scholarly journals Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Edcarlos D. Silva ◽  
Marcos L. M. Carvalho ◽  
Claudiney Goulart
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Energy Functional ◽  
Critical Nonlinearities ◽  
Asymptotically Periodic ◽  
Whole Space ◽  
Continuous Potential ◽  
Order Elliptic Problem

<p style='text-indent:20px;'>It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by <inline-formula><tex-math id="M2">\begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \alpha &gt; 0, \beta \in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.</p>

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