MULTIPLE SOLUTIONS FOR GENERALIZED BIHARMONIC EQUATIONS WITH SINGULAR POTENTIAL AND TWO PARAMETERS

We investigate a more general nonlinear biharmonic equation Delta(2)u - beta Delta(p)u + V-lambda(x)u = f(x, u) in R-N, where Delta(2) := Delta(Delta) is the biharmonic operator, N >= 1, lambda > 0 and beta is an element of R are parameters, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u) with p >= 2. Differently from previous works on biharmonic problems, we replace Laplacian with p-Laplacian, and suppose that V(x) = lambda a(x) - b(x) with lambda > 0 and b(x) can be singular at the origin, in particular we allow beta to be a real number. Under suitable conditions on V-lambda(x) and f (x, u), the multiplicity of solutions is obtained for lambda > 0 sufficiently large. Our analysis is based on variational methods as well as the Gagliardo-Nirenberg inequality.