On a nonlinear nonlocal problem of elliptic type

The solvability of a nonlinear nonlocal problem of the elliptic type that is a generalized Bitsadze–Samarskii-type problem is analyzed. Theorems on sufficient solvability conditions are stated. In particular, a nonlocal boundary value problem with $p$-Laplacian is studied. The results are illustrated by examples considered earlier in the linear theory (for $p=2$). The examples show that, in contrast to the linear case under the same “nice” nonlocal boundary conditions, for $p>2$, the problem can have one or several solutions, depending on the right-hand side.