Affiliation:
1. School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu, 730030, China
2. The Department of Basic Courses, Army Logistics University, Chongqing, 430030, China
3. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
Abstract
<abstract><p>In this paper, making use of a new non-smooth variational approach established by Moameni<sup>[<xref ref-type="bibr" rid="b13">13</xref>,<xref ref-type="bibr" rid="b14">14</xref>,<xref ref-type="bibr" rid="b15">15</xref>,<xref ref-type="bibr" rid="b16">16</xref>]</sup>, we establish the existence of solutions to the following mixed local and nonlocal elliptic problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+(-\Delta)^s u = \mu g(x,u)+b(x), &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u\geq0,\; \; \; \; \; &x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u = 0,\; \; \; \; \; &x\in\mathbb{R}^{N}\setminus\Omega, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \Omega \subset \mathbb{R}^{N} $ is a bounded smooth domain, $ (-\Delta)^{s} $ is the restricted fractional Laplacian, $ \mu > 0 $, $ 0 < s < 1 $, $ N > 2s $, $ g $ satisfies some growth condition and $ b(x)\in L^m(\Omega) $ for $ m\geq 2 $. The interesting feature of our work is that we show that the nonlocal operator has an important influence in the existence of solutions to the above equation since $ g $ has new growth condition.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
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