Uniqueness and sign properties of minimizers in a quasilinear indefinite problem

Let $1<q<p$ and $a\in C(\bar{\mathrm{\Omega }})$ be sign-changing, where $\mathrm{\Omega }$ is a bounded and smooth domain of ${\mathbb{R}}^N$. We show that the functional $$I_q(u):={\int }_{\mathrm{\Omega }}(\frac{1}{p}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p-\frac{1}{q}a(x)\mathrm{\mid }u{\mathrm{\mid }}^q),$$ has exactly one nonnegative minimizer $U_q$ (in $W_0^{1,p}(\mathrm{\Omega })$ or $W^{1,p}(\mathrm{\Omega })$). In addition, we prove that $U_q$ is the only possible $\mathit{\text{positive}}$ solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_q$ is positive, which also guarantees that minimizers of $I_q$ do not change sign. Several of these results are new even for $p=2$.