Nontrivial solutions for a class of gradient-type quasilinear elliptic systems - Candela, Anna Maria; Sportelli, Caterina (Topological Methods in Nonlinear Analysis, 2022)
The aim of this paper is to investigate the existence of weak bounded solutions of the gradient-type quasilinear elliptic system equationaPP cases - div ( a_i(x, u_i, abla u_i) ) + A_i, t (x, u_i, abla u_i) = G_i(x, ) &3cm in Ω for i{1,…,m}, = 0 &3cm on Ω, cases equation with m≥ 2 and =(u_1,…, u_m), where ΩR^N is an open bounded domain and some functions A_iΩ×R×R^N→R, i{1,…,m}, and GΩ×R^m→R exist such that a_i(x,t,ξ) = abla_ξ A_i(x,t,ξ), A_i, t (x,t,ξ) = A_it (x,t,ξ), and G_i(x,) = Gu_i(x,). We prove that, under suitable hypotheses, the functional J related to problem (P) is C^1 on a ``good’’ Banach space X and satisfies the weak Cerami-Palais-Smale condition. Then, generalized versions of the Mountain Pass Theorems allow us to prove the existence of at least one critical point and, if additionally J is even, of infinitely many critical points.