\((p, q)\) systems with critical terms in \(\mathbb{R}^N\)
The authors consider the following system in \(\mathbb{R}^N\) \[ \begin{cases} -\text{div}(A(|\nabla u|\nabla u))+B(|u|)u-\sigma \frac{|u|^{p-2}u}{|x|^p}=\lambda H_u(x,u,v)+\frac{\alpha}{p^*}|v|^\beta |u|^{\alpha-2}u,\\ \\ -\text{div}(A(|\nabla v|\nabla v))+B(|v|)v-\sigma \frac{|v|^{p-2}v}{|x|^p}=\lambda H_v(x,u,v)+\frac{\beta}{p^*}|u|^\alpha|v|^{\beta -2}v, \end{cases} \] where \(A,B:\mathbb{R}_+\rightarrow \mathbb{R}_+\) are \(C^1\) strictly increasing functions, \(H_u,H_v\) are the partial derivatives of a Carathéodory function \(H\), \(\lambda,\sigma\) are real parameters, \(p\in (1,N)\), and \(\alpha,\beta\in (1,\infty)\) satisfy \(\alpha+\beta=p^*\), where \(p^*\) is the critical exponent for the Hardy-Sobolev embedding \(D^{1,p}(\mathbb{R}^N) \hookrightarrow L^s(\mathbb{R}^N,|x|^{-s})\). Some structure conditions are assumed on the functions \(A,B\), which are fulfilled, for instance, when the system is driven by general \((p,q)\)-elliptic operators. In particular, the function \(A\) is assumed to satisfy a weaker growth condition than the usual ones. Finally, a subcritical growth is imposed on the function \(H\). Under these assumptions, the authors prove that, if \(\sigma\) is less than a certain (explicitly computed) positive constant, then there exists a positive constant \(\lambda(\sigma)\) such that, for all \(\lambda\geq \lambda(\sigma)\), the system admits a solution \((u_{\sigma,\lambda},v_{\sigma,\lambda})\), with \(u_{\sigma,\lambda},v_{\sigma,\lambda}\neq 0\), such that \((u_{\sigma,\lambda},v_{\sigma,\lambda}) \rightarrow (0,0)\), as \(\lambda \rightarrow \infty\), strongly in the space \((W^{1,m}(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n))^2\), where \(m\in (1,p^*)\) is one of the exponents involved in the growth condition imposed on \(A\). In the special case \(\sigma=0\), the authors prove that the same conclusion holds under milder conditions on \(B\). The method of proof is variational and based on the mountain pass lemma and concentration-compactness principles.
scientific article; zbMATH DE number 6983309
scientific article