Lazer-mckenna type problem involving mixed local and nonlocal elliptic operators

In this article, we investigate the existence, uniqueness and regularity of weak solutions to the following semilinear mixed local and nonlocal elliptic operators {-Delta u + ( - Delta)( s) u = h(u)f, x is an element of Omega , u >= 0, x is an element of Omega, u = 0 , x is an element of R-N \ Q , where 0 < s < 1, Omega subset of R-N(N 3) is a bounded 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)>= 0. f is an element of L (m) (Q)( m >= 1) is a nonnegative function, or has a growth of negative powers of eigenfunction phi near the boundary partial derivative Omega, where phi is the first positive eigenfunction to the mixed local and nonlo cal 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 is an element of L- m (Omega)) for the boundedness of the solutions. We explain how the regularity of the datum f and the behavior of the non- linearly of h , when 0 <= gamma (2) <= 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 nonlo cal operators.