Multiple solutions for mixed local and nonlocal elliptic equations

In this article, we establish some multiplicity results for a mixed local and nonlocal semi-linear elliptic equation driven by the superposition of Brownian and Lévy processes, taking the form \(\begin{aligned} \left\{ \begin{array}{ll} - \Delta u + (-\Delta )^s u=\lambda |u|^{p-2}u+ g(x,u) & \quad \hbox {in } \Omega , \\ u=0 & \quad \hbox {in } {\mathbb {R}}^n\backslash \Omega . \\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+{(-\mathrm{\Delta })}^{s}u=\lambda {|u|}^{p-2}u+g(x,u)\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{in}{\mathbb{R}}^n\backslash \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$ Under quite general assumptions, the existence of at least five weak solutions is proved for any bounded domain \(\Omega \) $\mathrm{\Omega }$ . Furthermore, in view of suitable regularity results, classical solutions with more features are investigated. More precisely, on the one side, based on the method of the descending flow, we obtain at least six classical solutions: two positive solutions, two negative solutions and two sign-changing solutions, and five of which possess determined energy signs. On the other side, the existence of at least six classical solutions with definite energy signs is established by the Nehari manifolds. In this case, we show that four solutions have constant signs, and one solution is sign-changing.