A class of semilinear elliptic equations on groups of polynomial growth

In this paper, we study the semilinear elliptic equation -Delta u + a(x)vertical bar u vertical bar(p-2) u - b(x)vertical bar u vertical bar(q-2) u = 0 on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N >= 1, where 2 <= p < q < +infinity. We first prove the existence of positive solutions to the above equation with constant coefficients (a) over bar, (b) over bar. Then we establish a decomposition of Palais-Smale sequences for the functional with variable coefficients a(x), b(x), which tend to the constants (a) over bar, (b) over bar at infinity. (c) 2023 Elsevier Inc. All rights reserved.