Dirichlet problem in a quarter plane for the generalized laplace equation

For a second-order partial differential equation we investigate a boundary value problem with data on the entireboundary in the positive quadrant. The considered equation contains the Riemann-Liouville fractional derivative with respectto the variable y and becomes the Laplace equation if the order of fractional derivative is tend to two. For the solution aregiven the integral representation and asymptotic properties. The existence of the regular solution is proven. The uniquenesstheorem is proven in the class of functions that have continuous partial derivatives of first order with respect to x and order $\beta$ − 1 with respect to y and a fractional integral of order 2 − $\beta$ wich vanish at infinity. Acknowledgements The work is supported within the framework of the state assignments of the Ministry of Education and Science of the Russian Federation (project No FEGS-2020-0001).