Existence and regularity of weak solutions for mixed local and nonlocal semilinear elliptic equations

We study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left{% \begin{array}{ll} -Δu + (-Δ)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $\mathrm{\Omega }$,} u=0 & \hbox{in ${\mathbb{R}}^n\mathrm{\backslash }\mathrm{\Omega }$} \end{array}% \right. \end{equation*} where $s\in (0,1)$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain, the coefficient $a$ is a function of $x$ and the subcritical nonlinearity $f(x,u)$ has superlinear growth at zero and infinity. We show the existence of a non-trivial weak solution by Linking Theorem and Mountain Pass Theorem respectively for ${\lambda }_1⩽0$ and ${\lambda }_1>0$, where ${\lambda }_1$ denotes the first eigenvalue of $-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s+a(x)$. In particular, adding a symmetric condition to $f$, we obtain infinitely many solutions via Fountain Theorem. Moreover, for the regularity part, we first prove the $L^{\mathrm{\infty }}$-boundedness of weak solutions and then establish up to $C^{2,\alpha }$-regularity up to boundary.