Multiplicity Results for a (p1(x),p2(x))-Laplacian Equation via Variational Methods

We prove the existence and multiplicity of nontrivial weak solutions for the following (p(1)(x),p(2)(x)) Laplacian equation involving variable exponents: {-div(|del u|(p)(1)(x)-2 del u)-div(|del u|(p)(2)(x)-2 del u)+|u|(p2(x)-2)u=lambda h(x,u),in Omega, u=0,on partial derivative Omega. Using Ricceri's variational principle, we show the existence of at least three weak solutions for the problem. We also apply the variational method and genus theory to establish the existence of infinitely many solutions. Then, we prove the closedness of the set of eigenfunctions, such that p(x) equivalent to p.