1 results
EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 102 / Issue 3 / June 2017
- Published online by Cambridge University Press:
- 09 September 2016, pp. 392-404
- Print publication:
- June 2017
-
- Article
-
- You have access
- Export citation
-
We study the existence of a weak solution of a nonlocal problem
where
$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$
${\mathcal{L}}_{k}$ 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.
