Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

In this paper, we consider the following coupled gradient-type quasilinear elliptic system {[ - div ( a(x, u, ∇ u) ) + A_t (x, u, ∇ u) = G_u(x, u, v) in Ω ,; - div ( b(x, v, ∇ v) ) + B_t(x, v, ∇ v) = G_v( x, u, v) in Ω ,; u = v = 0 on ∂Ω , ]. where Ω is an open bounded domain in ℝ^N , N≥ 2 . We suppose that some 𝒞^1 –Carathéodory functions A, B:Ω×ℝ×ℝ^N→ℝ exist such that a(x,t,ξ ) = ∇ _ξ A(x,t,ξ ) , A_t(x,t,ξ ) = ∂ A/∂ t (x,t,ξ ) , b(x,t,ξ ) = ∇ _ξ B(x,t,ξ ) , B_t(x,t,ξ ) =∂ B/∂ t(x,t,ξ ) , and that G_u(x, u, v) , G_v(x, u, v) are the partial derivatives of a 𝒞^1 –Carathéodory nonlinearity G:Ω×ℝ×ℝ→ℝ . Roughly speaking, we assume that A(x,t,ξ ) grows at least as (1+|t|^s_1p_1)|ξ |^p_1 , p_1 > 1 , s_1 ≥ 0 , while B(x,t,ξ ) grows as (1+|t|^s_2p_2)|ξ |^p_2 , p_2 > 1 , s_2 ≥ 0 , and that G ( x , u , v ) can also have a supercritical growth related to s_1 and s_2 . Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.