Existence of nontrivial solution for fourth-order semilinear Δγ-Laplace equation RN

In this paper, we study existence of nontrivial solutions for a fourth-order semilinear Delta(gamma)-Laplace equation in R-N Delta(2)(gamma)u - Delta(gamma)u + lambda b(x)u = f(x, u), x is an element of R-N, u is an element of S-gamma(2)(R-N), where lambda > 0 is a parameter and Delta(gamma) is the subelliptic operator of the type Delta(gamma):- Sigma(N)(j=1)partial derivative(xj()r2 partial derivative xj), partial derivative xj :- partial derivative/partial derivative x(j) gamma -(gamma 1(,) gamma(2), ..., gamma(N)), Delta(2)(gamma) :- Delta(gamma)(Delta(gamma)). Under some suitable assumptions on b(x) and f (x, (zeta) over bar), we obtain the existence of nontrivial solution for lambda large enough.