Elliptic p-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by -Delta(p1)u(1) = -vertical bar u(1)vertical bar(p1-2)u(1) in Omega, -Delta(p2)u(2) = -vertical bar u(2)vertical bar(p2-2)u(2) in Omega, vertical bar del u(1)vertical bar(p1-2)del(u1) center dot nu = g(1)(x, u(1), u(2)) on partial derivative Omega, vertical bar del u(2)vertical bar(p2-2)del(u2) center dot nu = g(2)(x, u(1), u(2)) on partial derivative Omega, where nu(x) is the outer unit normal of Omega at x is an element of partial derivative Omega, Delta(pi) denotes the pi-Laplacian and gi: partial derivative Omega x R x R. Rare Caratheodory functions that satisfy general growth and structure conditions for i = 1, 2. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of g(i) near zero related to the first eigenvalue of the p(i)-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, (g(1), g(2)) = del g with a smooth function (s(1), s(2)) bar right arrow g(x, s(1), s(2)). By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the p(i)-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution. (c) 2024 The Author(s). Published by Elsevier Inc.