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

Abstract

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.