On the Existence of Disjoint Infinite Solution Sets for a Critical Hardy-Sobolev-Maz'ya Equation

Résumé

In this work, we study the following Hardy--Sobolev--Maz’ya equation involving critical growth:\[ \left\{ \begin{array}{ll} -\Delta u - \lambda \dfrac{u}{\vert y\vert ^2} = \dfrac{\vert u\vert ^{2^{*}(s)-2}u}{\vert y\vert ^s} + \mu \vert u\vert ^{q-2}u, & \text{in } \Omega, \\ u = 0, & \text{on } \partial \Omega, \end{array} \right. \] where \(\Omega \subset \mathbb{R}^N\) is a bounded domain containing a point \(x^0 = (0, z^0) \in \mathbb{R}^k \times \mathbb{R}^{N-k}\), with \(2 \leq k < N\), \(x = (y, z)\), \(0 \leq s < 2\), and \(2^{*}(s) = \dfrac{2(N - s)}{N - 2}\). We assume \(0 \leq \lambda < \overline{\lambda} := \dfrac{(k - 2)^2}{4}\) for \(k > 2\), and \(\lambda = 0\) when \(k = 2\), with parameters satisfying \(1 < q < 2\), \(\mu > 0\), and \(N > \dfrac{s + 2 + 2q}{q - 1}\). Using an approximating argument, local Pohozaev-type identities, and variational methods, including the Fountain Theorem and its dual version, we establish the existence of two disjoint and infinite sets of solutions under these assumptions.

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