Existence of multiple solutions of critical quasilinear elliptic Neumann problems (Q1582450)

Of concern is the existence of multiple solutions of \[ -\Delta_pu=\lambda a(x)\left|u\right|^{p-2}u+b(x)\left|u\right|^{p^*-2}u+h(x) \quad \text{on }\Omega,\qquad {{\partial u}\over{\partial n}}|_{\partial\Omega}\equiv 0. \] The following setting is assumed: \(\Omega\) is a smooth bounded domain in \({\mathbb R}^N\), \(1<p<N\), \(p^*:={{Np}\over{N-p}}\), \(a\in L^\infty(\Omega)\), \(b\in C(\overline \Omega)\), \(a^+\), \(a^-\), \(b^+\) and \(b^-\not\equiv 0\), and \(h\in L^{p/(p-1)}(\Omega)\). Typically, the existence of two nontrivial (positive) solutions is established under additional conditions. E.g., assume that \(\lambda\neq 0\), \(\int_\Omega a=0\) and \(\int_\Omega b>0\), then the above problem has two nontrivial solutions for small \(\lambda\) and \(h\), and it has two positive solutions, if \(h\equiv 0\), \(p\geq 2\) and \(\lambda\) is small. Proofs rely on a generalized Ekeland variational principle, which is derived in Section 3, and the concentrated compactness principle.