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 semilinear 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.