Uniqueness and sign properties of minimizers in a quasilinear indefinite problem

Abstract

Let $1<q<p$ and $a\in C(\overline{\Omega })$ be sign-changing, where $\Omega$ is a bounded and smooth domain of ${\mathbb{R}}^N$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_q$ (in $W_0^{1,p}(\Omega )$ or $W^{1,p}(\Omega )$). In addition, we prove that $U_q$ is the only possible \textit{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 $p=2$.

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• Uniqueness and positivity issues in a quasilinear indefinite problem