Uniqueness and positivity issues in a quasilinear indefinite problem

We consider the problem $$\begin{aligned} (P_\lambda )\quad -\Delta _{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\ge 0\quad \text{ in } \Omega \end{aligned}$$ under Dirichlet or Neumann boundary conditions. Here $$\Omega $$ is a smooth bounded domain of $${\mathbb {R}}^{N}$$ ( $$N\ge 1$$ ), $$\lambda \in {\mathbb {R}}$$ , $$10$$ ). In particular, at most . Under some condition on a, above uniqueness result fails values >0$$ as we obtain, besides ground state solution, second also provide q such these become analyze formation cores generic solution.