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))+At(x,u,∇u)=Gu(x,u,v)inΩ,-div(b(x,v,∇v))+Bt(x,v,∇v)=Gv(x,u,v)inΩ,u=v=0on∂Ω,where Ω is an open bounded domain in R^N, N≥ 2. We suppose that some C¹–Carathéodory functions A, B: Ω × R× R^N→ R exist such that a(x, t, ξ) = ∇ _ξA(x, t, ξ) , At(x,t,ξ)=∂A∂t(x,t,ξ), b(x, t, ξ) = ∇ _ξB(x, t, ξ) , Bt(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 C¹–Carathéodory nonlinearity G: Ω × R× R→ R. Roughly speaking, we assume that A(x, t, ξ) grows at least as (1+|t|s1p1)|ξ|p1, p₁> 1 , s₁≥ 0 , while B(x, t, ξ) grows as (1+|t|s2p2)|ξ|p2, p₂> 1 , s₂≥ 0 , and that G(x, u, v) can also have a supercritical growth related to s₁ and s₂. 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.