On a singular and nonhomogeneous \(N\)-Laplacian equation involving critical growth (Q534743)

The following quasilinear elliptic equation \[ -\triangle_N u:=-\text{div}(|\nabla u|^{N-2}\nabla u)=|x|^{-a}f(x,u) +h(x), x \in \Omega, \;u\in W^{1,N}_0(\Omega), \;a\in [0,N), \] is considered, where \(\Omega \subset \mathbb{R}^N \;(N\geq 3)\) is a smooth bounded domain and \(h(x)\not\equiv 0\) is in the dual space of \(W^{1,N}_0(\Omega)\). By using the Ekeland's variational principle and the mountain pass theorem, the authors prove that, if the norm of \(h(x)\) is small enough, the above problem possesses at least two solutions which have negative energy and positive energy, respectively. Here the nonlinear term \(f(x,u)\) may have critical growth in the sense of \(-\triangle_N u\) in \(\mathbb{R}^N\). Moreover, the authors also prove that the two solutions for the problem are nonnegative (nonpositive) if \(h(x)\geq 0\) (\(h(x)\leq 0\), respectively).