EXISTENCE OF WEAK SOLUTIONS FOR THE KIRCHHOFF-TYPE EQUATION WITH MIXED BOUNDARY CONDITIONS IN VARIABLE EXPONENT SOBOLEV SPACE

Keywords: Kirchhoﬀ-type equation, p(·)-Laplacian, mean curvature operator, mixed boundary value problem, variable exponent Sobolev space

In this paper, we consider the Kirchhoﬀ-type equation for a class of

nonlinear operators containing p(·)-Laplacian and mean curvature oper-

ator with mixed boundary conditions. More precisely, we are concerned

with the problem under the Dirichlet condition on a part of the boundary

and the Steklov boundary condition on an another part of the boundary.

We show the existence of one, two and infinitely many nontrivial weak

solutions of the equation according to the conditions on given functions.