Nonlinear elliptic problem of 2-\(q\)-Laplacian type with asymmetric nonlinearities (Q2877658)

Let \(\Omega \subset \mathbb R^N\) be a bounded domain with smooth boundary, \(N>2\), \(1<\min (q,r)\), \(\max(q,r)<2<\theta <2N/(N-2)\), \(b>0\). Denote by \(\{ \lambda_k\} \) the decreasing sequence of eigenvalues of the Laplacian for \(\Omega \). Let \(\lambda_k <a<\lambda_{k+1}\). It is shown that for \(\lambda >0\), \(\mu >0\) small enough the problem \(-\Delta u-\mu \nabla \cdot (|\nabla u|^{q-2} \nabla u) =- \lambda |u|^{r-2}u+au +b(u^+)^{\theta -1}\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), has at least three nontrivial solutions.