Boundary value problem for a nonlinear equation of mixed type

In this paper, we consider the boundary value problem for a nonlinear Lavrentiev–Bitsadze equation of mixed type∂2uˆ∂x2+(sgny)(1+uˆx2)∂2uˆ∂y2=0, whose coefficients depend on the first order derivatives of unknown function. Above y=0y=0 and below y=0y=0, the equation becomes the nonlinear elliptic equation and nonlinear hyperbolic equation respectively, this is different from the equation studied in Chen (2011) [9]. We prove the existence of solution to the problem by the method which can be used to study more difficult problems for nonlinear equations of mixed type arising in gas dynamics.