The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \textup{(P)}{-div⁡(A⁢(x,u)⁢|∇⁡u|p1-2⁢∇⁡u)+1p1⁢Au⁢(x,u)⁢|∇⁡u|p1=Gu⁢(x,u,v)in Ω,-div⁡(B⁢(x,v)⁢|∇⁡v|p2-2⁢∇⁡v)+1p2⁢Bv⁢(x,v)⁢|∇⁡v|p2=Gv⁢(x,u,v)in Ω,u=v=0on ∂⁡Ω,\left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1% }-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&\displaystyle=G_{u}(% x,u,v)&&\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{% p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&\displaystyle=G_{v}(x,u,v)&&\displaystyle% \phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} is an open bounded domain, p1{p_{1}}, p2>1{p_{2}>1} and A⁢(x,u){A(x,u)}, B⁢(x,v){B(x,v)} are 𝒞1{\mathcal{C}^{1}}-Carathéodory functions on Ω×ℝ{\Omega\times\mathbb{R}} with partial derivatives Au⁢(x,u){A_{u}(x,u)}, respectively Bv⁢(x,v){B_{v}(x,v)}, while Gu⁢(x,u,v){G_{u}(x,u,v)}, Gv⁢(x,u,v){G_{v}(x,u,v)} are given Carathéodory maps defined on Ω×ℝ×ℝ{\Omega\times\mathbb{R}\times\mathbb{R}} which are partial derivatives of a function G⁢(x,u,v){G(x,u,v)}. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional 𝒥{{\mathcal{J}}}, related to problem (P), admits at least one critical point in the “right” Banach space X . Moreover, if 𝒥{{\mathcal{J}}} is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition, a “good” decomposition of the Banach space X and suitable generalizations of the Ambrosetti–Rabinowitz Mountain Pass Theorems.

中文翻译：

---

一类梯度型耦合拟线性椭圆系统的存在性和多重性结果

其中Ω⊂ℝN{\ Omega \ subset \ mathbb {R} ^ {N}}是一个开放边界域，p1 {p_ {1}}，p2> 1 {p_ {2}> 1}和A⁢（x， u）{A（x，u）}，B⁢（x，v）{B（x，v）}是𝒞1{\ mathcal {C} ^ {1}}-Ω×ℝ{\ Omega \ times \ mathbb {R}}的偏导数Au⁢（x，u）{A_ {u}（x，u）}，分别为Bv⁢（x，v）{B_ {v}（x，v）}，而Gu⁢（x，u，v）{G_ {u}（x，u，v）}，Gv⁢（x，u，v）{G_ {v}（x，u，v）}被赋予Carathéodory映射在Ω×ℝ×ℝ{\ Omega \ times \ mathbb {R} \ times \ mathbb {R}}上，它们是函数G⁢（x，u，v）{G（x，u，v）}的偏导数。我们证明，即使系数使变分方法更加困难，在与问题（P）相关的适当假设函数𝒥{{\ mathcal {J}}}下，“正确” Banach空间中也至少包含一个临界点X 。此外，如果𝒥{{\ mathcal {J}}}是偶数，则（P）具有无限多个弱有界解。证据，