We prove the existence of weak solutions of the homogeneous Dirichlet problem related toa class of nonlinear elliptic equations whose prototype is\begin{multline*}\label{modello}\sum_{|\al|=2}\Da \Big[|\DD^2 u |^{p-2}\Da u\Big] -\sum_{|\al|=1}\Da \Big[|\Du u |^{q-2}\Da u\Big]%\\%\hfill+ u \Big[ |\DD^1 u|^q + |\DD^2 u| ^p \Big] = f \end{multline*}where $\Om$ is an open bounded subset of $\R^N$ ($ N\geq 3$) with sufficiently smooth boundary, $u: \Om \rightarrow \R$ is the unknown function, $\DD^h u=\Big\{\Da u: |\al|=h \Big\}$, $|\DD^hu|=\big[\sum\limits_{|\al|=h}|\Da u|^2\big]^{\frac{1}{2}}$, for $h=1,2$, numbers $p$, $q\in [2, N[$ and$f \in L^{1}(\Om).

Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions.

CIRMI, Giuseppa Rita;D'ASERO, Salvatore;LEONARDI, Salvatore
2014

Abstract

We prove the existence of weak solutions of the homogeneous Dirichlet problem related toa class of nonlinear elliptic equations whose prototype is\begin{multline*}\label{modello}\sum_{|\al|=2}\Da \Big[|\DD^2 u |^{p-2}\Da u\Big] -\sum_{|\al|=1}\Da \Big[|\Du u |^{q-2}\Da u\Big]%\\%\hfill+ u \Big[ |\DD^1 u|^q + |\DD^2 u| ^p \Big] = f \end{multline*}where $\Om$ is an open bounded subset of $\R^N$ ($ N\geq 3$) with sufficiently smooth boundary, $u: \Om \rightarrow \R$ is the unknown function, $\DD^h u=\Big\{\Da u: |\al|=h \Big\}$, $|\DD^hu|=\big[\sum\limits_{|\al|=h}|\Da u|^2\big]^{\frac{1}{2}}$, for $h=1,2$, numbers $p$, $q\in [2, N[$ and$f \in L^{1}(\Om).
Higher order equations; Measure data
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11769/16766
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