Multiple solvability of certain elliptic problems with critical nonlinearity exponents

It is proved that the problem $$\mathop {\sum\nolimits_{i = 1}^v {\nabla _i (|\nabla u|^{p - 2} \nabla _i u)^ - |u|^{p * - 1} u + \lambda |u|^{p - 2} u = 0 in \Omega .} }\limits_{n = 0 on \partial \Omega .}$$ where Ω ?R ^N a singly-connected region with an “odd” boundary, N > p, and p^* = Np/(N ? p) is a critical Sobolev exponent, has, under the appropriate conditions on λ, q, and N, no less than (2N+2) nontrivial solutions in \(\mathop W\limits^0 _{p^1 } (\Omega )\) .