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} -\Delta u + (-\Delta)^{s} u+ a(x)\ u =f(x,u) & \hbox{in $\Omega$,} u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$} \end{array}% \right. \end{equation*} where $s \in (0,1)$, $\Omega \subset \mathbb{R}^{n}$ 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 $\lambda_{1} \leqslant 0$ and $\lambda_{1} > 0$, where $\lambda_{1}$ denotes the first eigenvalue of $-\Delta + (-\Delta)^{s} +a(x)$. 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 $L^{\infty}$-boundedness of weak solutions and then establish up to $C^{2, \alpha}$-regularity up to boundary.