On nonhomogeneous elliptic problems involving the Hardy potential and critical Sobolev exponent (Q527058)

In this paper, the authors establish the existence of at least two weak solutions for the problem \[ -\Delta u-\mu \frac{u}{|x|^2}=\lambda u+|u|^{\frac{4}{N-2}}u+f(x),\quad\text{in}\quad\Omega, \] \[ u=0,\quad\text{on}\quad\partial \Omega, \] where \(\Omega\) is a bounded open set in \(\mathbb{R}^N\), with \(N\geq 3\), \(\mu\in [0,+\infty[\), \(\lambda \in [0,\lambda_1[\), and \(f\in H_0^1(\Omega)^{-1}\). Here, \(\lambda_1\) is the first eigenvalue of the Laplacian on \(H_0^1(\Omega)\). The authors prove that if \(\mu\) is less than a (explicitly computed) constant, depending only on \(N\), and \(\|f\|_{H_0^1(\Omega)^{-1}}\) is less than an (explicitly computed) constant, depending on \(\mu\), \(\lambda\), \(N\) and \(\Omega\), then the energy functional \(I\) associated to the problem and restricted to two suitable subsets of the corresponding Nehari manifold, admits global minima. Next, the authors are able to prove that these global minima are critical points of \(I\), one of which being a local minimum of \(I\). The proofs follow variational methods and involve the Ekeland variational principle, the Palais-Smale condition, and the mountain pass theorem.