A NONLINEAR EIGENVALUE PROBLEM WITH p(x)-GROWTH AND GENERALIZED ROBIN BOUNDARY VALUE CONDITION

We are concerned with the study of the following nonlinear eigenvalue problem with Robin boundary condition {-div (a(x, del u)) = lambda b(x, u) in Omega partial derivative A/partial derivative n + beta(x)c(x, u) = 0 on partial derivative Omega. The abstract setting involves Sobolev spaces with variable exponent. The main result of the present paper establishes a sufficient condition for the existence of an unbounded sequence of eigenvalues. Our arguments strongly rely on the Lusternik-Schnirelmann principle. Finally, we focus to the following particular case, which is a p(x)-Laplacian problem with several variable exponents: {-div (a(0)(x)vertical bar del u vertical bar(p(x)-2)del u) = lambda b(0)(x)vertical bar u vertical bar(q(x)-2)u in Omega vertical bar del u vertical bar(P(x)-2)partial derivative U/partial derivative n + beta(x)vertical bar u vertical bar(r(x)-2)u = 0 on partial derivative Omega. Combining variational arguments, we establish several properties of the eigenvalues family of this nonhomogeneous Robin problem.