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EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS

Part of: Nonlinear operators and their properties Elliptic equations and systems

Published online by Cambridge University Press:  09 September 2016

V. RAGHAVENDRA  and
RASMITA KAR
V. RAGHAVENDRA
Affiliation:
Department of Mathematics, LNMIIT, Jaipur-302031, India email vrag@iitk.ac.in
RASMITA KAR*
Affiliation:
Department of Mathematics, NIT Rourkela, Rourkela-769008, India email rasmitak6@gmail.com
*
✉rasmitak6@gmail.com
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Abstract

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We study the existence of a weak solution of a nonlocal problem

$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$
where ${\mathcal{L}}_{k}$ 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.

Keywords

demicontinuous operatorsnonlocal operator

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 47H99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 3 , June 2017 , pp. 392 - 404
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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