In this article, we investigate the existence, uniqueness and regularity of weak solutions to the following semilinear mixed local and nonlocal elliptic operators \(\begin{aligned} \left\{ \begin{array}{rl} -\Delta u+(-\Delta )^s u=h(u)f, & \quad x\in \Omega ,\\ u\geqslant 0,~~~~~~~& \quad x\in \Omega ,\\ u=0,~~~~~~~& \quad x\in {\mathbb {R}}^{N}\setminus \Omega , \end{array} \right. \end{aligned}\) where \(0<s<1\) , \(\Omega \subset {\mathbb {R}}^N(N\geqslant 3)\) is a bounded \({\mathcal {C}}^{1,1}\) domain, \((-\Delta )^s\) is the restricted fractional Laplace operator, h(s) is a continuous function that behaves as \(s^{-\gamma _1}\) near zero and as \(s^{-\gamma _2}\) at infinity with \(\gamma _1, \gamma _2\geqslant 0\) . \(f\in L^m(\Omega )(m\geqslant 1)\) is a nonnegative function, or has a growth of negative powers of eigenfunction \(\phi \) near the boundary \(\partial \Omega \) , where \(\phi \) is the first positive eigenfunction to the mixed local and nonlocal eigenvalue problem. A distinguished feature of this paper is that we show that the existence and the regularity of the solutions are influenced by the competition between the nonlocal term \((-\Delta )^s\) , the behavior of h at infinity (or zero) and the summability of the datum f. Additionally, we prove that \((-\Delta )^s\) and the behavior of h at infinity have regularizing effect. Moreover, we establish a threshold for m \((f\in L^m(\Omega ))\) for the boundedness of the solutions. We explain how the regularity of the datum f and the behavior of the nonlinearly of h, when \(0\leqslant \gamma _2\leqslant 1\) , effect the important properties of the solution. We also show when \(\gamma _2>1\) , this does not effect the regularly. Our study includes more general nonlinear h and data f than all the previous results of mixed local and nonlocal operators.
