Mashiyev, R. A.2024-04-242024-04-2420112193-567X2191-4281https://doi.org/10.1007/s13369-011-0141-xhttps://hdl.handle.net/11468/14944In this article, we study the following nonlinear Neumann boundary value problem {(partial derivative u/partial derivative v) (-div(vertical bar del u vertical bar p(x)-2 del u) + a(x)vertical bar u vertical bar p(x)-2u = lambda f(x, u)) (on x is an element of partial derivative Omega) (in x is an element of Omega) where Omega subset of R-N (N >= 3), Omega is a bounded smooth domain and, p is an element of C ((Omega) over bar) with inf(x is an element of(Omega) over bar) p(x) > N, f : R -> R is a continuous function, and nu is the unit normal exterior vector on partial derivative Omega and a is an element of L-8(Omega), with ess infa(x) = a(0) > 0 and lambda > 0 is a real number. We first deal with the case that f (t) = b vertical bar t vertical bar(q-2)t - d vertical bar t vertical bar(s-2)t, t is an element of R, where b and d are positive constants. Then we deal with the case that f (x, t) = vertical bar t vertical bar(q(x)-2)t - vertical bar t vertical bar(s(x)-2)t, x is an element of Omega, t is an element of R, where q, s is an element of C ((Omega) over bar). Using the direct Ricceri variational principle, we establish the existence of at least three weak solutions of this problem in weighted-variable-exponent Sobolev space W-a(1,p(x)) (Omega).eninfo:eu-repo/semantics/closedAccessP(X)-LaplacianVariable Exponent Lebesgue And Sobolev SpacesRicceri Variational PrincipleArticle36815591567WOS:0002982942000082-s2.0-8375519587710.1007/s13369-011-0141-xQ1Q3