A variational approach to a quasilinear elliptic problem involving the p-Laplacian and nonlinear boundary condition

Using the technique of Brown and Wu [K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326-1336], we present a note on the paper [T.F. Wu, A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential, Electron. J. Differential Equations 131 (2006) 1-15] by Wu. Indeed, we extend the multiplicity results for a class of semilinear problems to the quasilinear elliptic problems of the form: -Delta(p)u + m(x)vertical bar u vertical bar(p-2)u = lambda a(x)vertical bar u vertical bar(p-2)u, x is an element of Omega, vertical bar del u vertical bar(p-2)partial derivative u/partial derivative n = b(x)vertical bar u vertical bar(r-2)u, Here Delta(p) denotes the p-Laplacian operator defined by ,Delta(p)z = div(vertical bar del z vertical bar(p-2)del z), 1 < q < p < r < p*(p* = pN/N-p if N > p, p* = infinity if N <= p), Omega subset of R-N is a bounded domain with smooth boundary, partial derivative/partial derivative n is the outer normal derivative, lambda is an element of R \ {0}, the weight m(x) is a bounded on function with parallel to m parallel to(infinity) > 0 and a(x), b(x) are continuous functions which change sign in (Omega) over bar. (c) 2009 Elsevier Ltd. All rights reserved.