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On critical \(p\)-Laplacian systems. (English) Zbl 1372.35034

Summary: We consider the critical \(p\)-Laplacian system \[ \begin{cases} -\Delta_p u-\frac{\lambda a}{p}| u|^{a-2}u| v|^{b}=\mu_{1}| u|^{p^\ast-2}u+\frac{\alpha\gamma}{p^\ast}| u|^{\alpha-2}u| v|^\beta,\quad & x\in\Omega,\\ -\Delta_pv-\frac{\lambda b}{p}| u|^a| v |^{b-2}v=\mu_2| v|^{p^\ast-2}v+\frac{\beta\gamma}{p^\ast} | u|^{\alpha}| v|^{\beta-2}v,\quad & x\in\Omega,\\ u,v\text{ in }D_{0}^{1,p}(\Omega),&\end{cases} \] where \(\Delta_{p}u:=\operatorname{div}(| u|^{p-2} u)\) is the \(p\)-Laplacian operator defined on
\[ D^{1,p}(\mathbb{R}^{N}):=\left\{ u L^{p^\ast}(\mathbb{R}^N):| u| L^p(\mathbb{R}^N)\right\}, \]
endowed with the norm \(\| u\|_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}| u|^{p}\,dx)^{\frac{1}{p}}\), \(N\geq 3\), \(1<p<N\), \(\lambda,\mu_1,\mu_2\geq0\), \(\gamma\neq0\), \(a,b,\alpha,\beta>1\) satisfy \(a+b=p\), \(\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}\), the critical Sobolev exponent, \(\Omega\) is \(\mathbb{R}^{N}\) or a bounded domain in \(\mathbb{R}^{N}\) and \(D_{0}^{1,p}(\Omega)\) is the closure of \(C_{0}^{\infty}(\Omega)\) in \(D^{1,p}(\mathbb{R}^{N})\). Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

MSC:

35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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