INFINITELY MANY SOLUTIONS FOR PROBLEMS IN FRACTIONAL ORLICZ-SOBOLEV SPACES

We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrodinger Phi-Laplace equation (-Delta)(Phi)u + V(x)phi(u)= xi(x)f(u) in R-d, where Phi(t) = integral(t)(0) phi(s) ds is an N-function, Delta(Phi) is the Phi-Laplacian operator, V : R-d -> R is a continuous function, xi is a function with sign -changing on Rd and the nonlinearity f is sublinear as vertical bar u vertical bar -> infinity. During the study of our problem, we deal with a new compact embedding theorem for the Orlicz Sobolev spaces. We also study the existence and multiplicity of solutions to the general fractional Phi-Laplacian equations of Kirchhoff type {M(integral(2d Phi()(R)u(x) - u(y)/K(vertical bar x - y vertical bar)) dxdy/N(vertical bar x - y vertical bar))(-Delta)(Phi)(K, N)u = f(x, u) in Omega, in R-d \ Omega. where Omega is an open bounded subset of R-d with smooth boundary partial derivative Omega, d > 2, and M : R-0(+)-> R+ is a continuous function and f : Omega x R -> R is a Caratheodory function. The proofs rely essentially on the fountain theorem and the genus theory.