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EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS
Published online by Cambridge University Press: 09 September 2016
Abstract
We study the existence of a weak solution of a nonlocal problem $$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$
${\mathcal{L}}_{k}$ is a general nonlocal integrodifferential operator of fractional type,
$\unicode[STIX]{x1D707}$ is a real parameter and
$\unicode[STIX]{x1D6FA}$ is an open bounded subset of
$\mathbb{R}^{n}$ (
$n>2s$, where
$s\in (0,1)$ is fixed) with Lipschitz boundary
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. Here
$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ and
$h:\mathbb{R}\rightarrow \mathbb{R}$ are functions satisfying suitable hypotheses.
MSC classification
- Type
- Research Article
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- Copyright
- © 2016 Australian Mathematical Publishing Association Inc.
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