Multiplicity of solutions for a nonlinear boundary value problem in the upper half-space

We obtain multiple solutions for the nonlinear boundary value problem -Delta u - 1/2 (x . del u) = lambda a(x)vertical bar u vertical bar(q-2)u, in R-+(N), partial derivative u/partial derivative v = b(x')vertical bar u vertical bar(p-2)u, on partial derivative R-+(N), where R-+(N) = {(x ',x(N)): x' is an element of RN -1, x(N) > 0} is the upper half-space, N >= 3, lambda > 0 is a parameter, 1 < q < 2 < p <= 2(*) = 2(N - 1)/(N - 2). The potentials a and b satisfy mild conditions which allow( )us to use variational methods. In some results, they can be indefinite in sign. (C) 2020 Elsevier Inc. All rights reserved.