A multiplicity result for perturbed symmetric quasilinear elliptic systems (Q1848307)

In [\textit{M. Squassina}, Existence of multiple solutions for quasilinear diagonal elliptic systems, Electron. J. Differ. Equ. 14, 1-12 (1999; Zbl 0921.35052)] it was recently shown that diagonal quasilinear elliptic systems of the type \[ -\sum^n_{i,j=1} D_j(a^k_{ij}(x, u)D_i u_k)+{1\over 2} \sum^n_{i,j=1}\sum^N_{h=1} D_{S_k} a^h_{ij}(x,u) D_i u_h D_j u_h= D_{S_k} G(x,u)\text{ in }\Omega,\tag{1} \] \(1\leq k\leq N\), where \(\Omega\subset \mathbb{R}^n\) is open and bounded, possess a sequence \((u^m)\) of weak solutions in \(H^1_0(\Omega, \mathbb{R}^N)\) under suitable assumptions, including symmetry on \(G\) and the coefficients \(a^h_{ij}\). In order to prove this result, the author looked for critical points of the functional \(f_0: H^1_0(\Omega, \mathbb{R}^N)\to \mathbb{R}\) defined by \[ f_0(u)= {1\over 2} \int_{\Omega} \sum^n_{i,j=1} \sum^N_{h=1} a^h_{ij}(x, u) D_i u_h D_j u_h dx- \int_\Omega G(x,u) dx. \] In this paper the authors study the effects of dropping the symmetry of system (1) and show that for each \(\varphi\in L^2(\Omega, \mathbb{R}^N)\) the perturbed problem \[ -\sum^n_{i,j=1} D_j(a^k_{ij}(x, u)D_i u_k)+{1\over 2} \sum^n_{i,j=1}\sum^N_{h=1} D_{S_k} a^h_{ij}(x, u) D_iu_h D_ju_h= D_{S_k}G(x,u)+ \varphi_k\text{ in }\Omega, \] still has infinitely many weak solutions. In order to prove this result an adaptation of the perturbation argument given in [\textit{P. H. Rabinowitz}, Trans. Am. Math. Soc. 272, 753-769 (1982; Zbl 0589.35004)] is used.