Lemma. Let $\Omega \subset \mathbb{R}^N$ be a bounded open set, $N \geq 1$. Let $1 \leq q \leq p \leq \gamma$, and $\gamma \leq p^*$ if $N>p$ and $\gamma<\infty$ if $N < p$. Then there exists $C=C(\Omega,q,p,\gamma)$ such that for any $u \in W_0^{1,p}(\Omega)$ the following inequality is satisfied:

$\left(\int_\Omega |u|^q \, dx \right)^{\gamma-p} \left(\int_\Omega |u|^\gamma \, dx \right)^{p-q} \leq C \left(\int_\Omega |\nabla u|^p \, dx \right)^{\gamma-q}.$

In fact, this inequality easily follows from the general Sobolev inequality just by applying the latter one to each term on the lhs of the former one. (I’m almost sure there could be some investigation of an inequality of this type in the literature, but I was not able to find it.)

1) It is just beautiful that the exponents are permuted)

2) If one considers the optimization problem for the constant $C$, then it is possible to see that a corresponding minimizer of

$\inf_{u \in W_0^{1,p}(\Omega) \setminus \{0\}} \frac{\left(\int_\Omega |\nabla u|^p \, dx \right)^{\gamma-q}}{\left(\int_\Omega |u|^q \, dx \right)^{\gamma-p} \left(\int_\Omega |u|^\gamma \, dx \right)^{p-q}}$

exists and satisfies the convex-concave equation

$-\Delta_p u = \lambda |u|^{q-2} u + |u|^{\gamma-2} u.$

In fact, such a minimization problem was explicitly considered in connection with the above equation in [1].

# Bibliography

1. Il’yasov, Y. (2005). On nonlocal existence results for elliptic equations with convex–concave nonlinearities. Nonlinear Analysis: Theory, Methods & Applications, 61(1-2), 211-236.