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

Abstract

In this paper, we consider the following coupled gradient-type quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{*{20}l} - {\text{div}} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &{}{\hbox { in }}\Omega ,\\ - {\text{div}} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left( x, u, v\right) &{}{\hbox { in }}\Omega ,\\ u = v = 0 &{}{\hbox { on }}\partial \Omega , \end{array} \right. \end{aligned}$$ - 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 $$\Omega$$ Ω is an open bounded domain in $${\mathbb {R}}^N$$ R N , $$N\ge 2$$ N ≥ 2 . We suppose that some $$\mathcal {C}^{1}$$ C 1 –Carathéodory functions $$A, B:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ A , B : Ω × R × R N → R exist such that $$a(x,t,\xi ) = \nabla _{\xi } A(x,t,\xi )$$ a ( x , t , ξ ) = ∇ ξ A ( x , t , ξ ) , $$A_t(x,t,\xi ) = \frac{\partial A}{\partial t} (x,t,\xi )$$ A t ( x , t , ξ ) = ∂ A ∂ t ( x , t , ξ ) , $$b(x,t,\xi ) = \nabla _{\xi } B(x,t,\xi )$$ b ( x , t , ξ ) = ∇ ξ B ( x , t , ξ ) , $$B_t(x,t,\xi ) =\frac{\partial B}{\partial t}(x,t,\xi )$$ B t ( x , t , ξ ) = ∂ B ∂ t ( x , t , ξ ) , and that $$G_u(x, u, v)$$ G u ( x , u , v ) , $$G_v(x, u, v)$$ G v ( x , u , v ) are the partial derivatives of a $$\mathcal {C}^{1}$$ C 1 –Carathéodory nonlinearity $$G:\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ G : Ω × R × R → R . Roughly speaking, we assume that $$A(x,t,\xi )$$ A ( x , t , ξ ) grows at least as $$(1+|t|^{s_1p_1})|\xi |^{p_1}$$ ( 1 + | t | s 1 p 1 ) | ξ | p 1 , $$p_1 > 1$$ p 1 > 1 , $$s_1 \ge 0$$ s 1 ≥ 0 , while $$B(x,t,\xi )$$ B ( x , t , ξ ) grows as $$(1+|t|^{s_2p_2})|\xi |^{p_2}$$ ( 1 + | t | s 2 p 2 ) | ξ | p 2 , $$p_2 > 1$$ p 2 > 1 , $$s_2 \ge 0$$ s 2 ≥ 0 , and that G(x, u, v) can also have a supercritical growth related to $$s_1$$ s 1 and $$s_2$$ 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.