| Abstract: |
| We prove multiplicity of solutions for the mixed local-nonlocal elliptic equation of the form
$\begin{eqnarray*}
\begin{split}
-\Delta_pu+(-\Delta)_p^s u &= \frac{\lambda}{u^{\gamma}}+u^r \text { in } \Omega, \
u > 0 \text{ in } \Omega,\
u = 0 \text { in }\mathbb{R}^n \backslash \Omega;
\end{split}
\end{eqnarray*}$
where
$\begin{equation*}
(-\Delta )_p^s u(x)= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}} d y.
\end{equation*}$
Under the assumptions that $\Omega$ is a smooth bounded domain
in $\mathbb{R}^{n}$, $1$ |
|