Existence of solutions to elliptic equation with mixed local and nonlocal operators

Abstract

<abstract><p>In this paper, making use of a new non-smooth variational approach established by Moameni^{[<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>]}, 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), &amp;x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u\geq0,\; \; \; \; \; &amp;x\in\Omega,\\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; u = 0,\; \; \; \; \; &amp;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 &gt; 0 $, $ 0 &lt; s &lt; 1 $, $ N &gt; 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>