The Picone inequality (identity) is a well-known tool with wide applications in PDEs, see, e.g., [¹]. In my paper with M. Tanaka [²], Proposition 8, we found one form of the Picone inequality which appears to be quite useful for studying problmes with the $(p,q)$-Laplacian. In this post, I would like to provide a slight generalization of Proposition 8 from [²].

Theorem. Let $1 < q < p < \infty$ and $\alpha, \beta > 0$. Assume that $u>0$ and $\varphi \geq 0$ are some differentiable functions in a domain $\Omega$. Then

\[|\nabla u|^{p-2} \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\alpha C} |\nabla \varphi|^p\]

and

\[|\nabla u|^{q-2} \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\beta}|\nabla (\varphi^{p/q})|^q,\]

where $C = 1$ if $p \leq q+1$, and $C= \frac{(q-1)^{p-2} (p-q)}{(p-2)^{p-2}}$ if $p \geq q+1$.

In particular, if $\mu>0$, then

\[(|\nabla u|^{p-2} + \mu |\nabla u|^{q-2}) \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\alpha C} |\nabla \varphi|^p + \frac{\mu}{\beta}|\nabla (\varphi^{p/q})|^q.\]

---

Proof. First, by standard calculations we get

\begin{align} \notag & |\nabla u|^{p-2} \nabla u \nabla \left( \frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}} \right)\newline \notag & = p |\nabla u|^{p-2} \nabla u \nabla \varphi \frac{\varphi^{p-1}}{\alpha u^{p-1} + \beta u^{q-1}} - |\nabla u|^{p} \varphi^p \frac{\alpha (p-1) u^{p-2} + \beta (q-1) u^{q-2}}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^2}\newline \label{eq:picone:1} & \leq p |\nabla u|^{p-1} |\nabla \varphi| \frac{\varphi^{p-1}}{\alpha u^{p-1} + \beta u^{q-1}} - |\nabla u|^{p} \varphi^p \frac{\alpha (p-1) u^{p-2} + \beta (q-1) u^{q-2}}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^2}. \end{align}

Applying to the first term Young’s inequality

\[ab =\frac{a}{\rho^{\frac{p-1}{p}}}\,\rho^{\frac{p-1}{p}} b \leq \frac{|a|^p}{p\rho^{p-1}} + \frac{\rho (p-1) |b|^{\frac{p}{p-1}}}{p}\]

with $a= |\nabla \varphi|$, $b= |\nabla u|^{p-1} \frac{\varphi^{p-1}}{\alpha u^{p-1} + \beta u^{q-1}}$, and any $\rho>0$, we obtain

\begin{align} \notag \eqref{eq:picone:1} & \leq \frac{|\nabla \varphi|^p}{\rho^{p-1}} + \frac{\rho(p-1) |\nabla u|^{p} \varphi^p}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^{\frac{p}{p-1}}} - |\nabla u|^{p} \varphi^p \frac{\alpha (p-1) u^{p-2} + \beta (q-1) u^{q-2}}{\left(\alpha u^{p-1} + \beta u^{q-1}\right)^2}\newline \notag & = \frac{|\nabla \varphi|^p}{\rho^{p-1}} + \frac{\rho(p-1) |\nabla u|^{p} \varphi^p}{\left( \alpha u^{p-1} + \beta u^{q-1} \right)^{\frac{p}{p-1}}} \left[ 1 - \frac{\alpha (p-1) + \beta (q-1) u^{q-p}}{\rho(p-1) \left(\alpha + \beta u^{q-p}\right)^{\frac{p-2}{p-1}}} \right]. \end{align}

Analysing the function

\[g(u) = \frac{\alpha (p-1) + \beta (q-1) u^{q-p}}{\rho(p-1) \left(\alpha + \beta u^{q-p}\right)^{\frac{p-2}{p-1}}},\]

we see that $g(u) \geq 1$ for all $u > 0$ in the following cases:

\begin{align} \notag &1)\quad p \leq q+1 \quad \text{and} \quad \rho \leq \alpha^\frac{1}{p-1},\newline \notag &2) \quad p \geq q+1 \quad \text{and} \quad \rho \leq \frac{(q-1)^\frac{p-2}{p-1}(p-q)^\frac{1}{p-1}}{(p-2)^\frac{p-2}{p-1}}\alpha^\frac{1}{p-1}. \end{align}

Under the assumptions 1) or 2) we get

\[|\nabla u|^{p-2} \nabla u \nabla \left( \frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}} \right) \leq \frac{|\nabla \varphi|^p}{\rho^{p-1}}.\]

Thus, taking $\rho$ as the maximal admissible value in the cases 1) and 2), we obtain the constat $C_1$.

Arguing in the similar way, for $\psi=\varphi^{p/q}$ (note that $p/q>1$ and $\varphi^p=\psi^q$) we obtain

\begin{align} \notag & |\nabla u|^{q-2} \nabla u \nabla \left( \frac{\psi^q}{\alpha u^{p-1} + \beta u^{q-1}} \right) \newline \notag &\le \frac{|\nabla \psi|^q}{\rho^{q-1}} + \frac{\rho(q-1) |\nabla u|^{q} \psi^q}{\left( \alpha u^{p-1} + \beta u^{q-1} \right)^{\frac{q}{q-1}}} \left[ 1 - \frac{\alpha (p-1) u^{p-q} + \beta (q-1)}{\rho(q-1) \left(\alpha u^{p-q} + \beta \right)^{\frac{q-2}{q-1}}} \right]. \end{align}

Analysing the function

\[g(u) = \frac{\alpha (p-1) u^{p-q} + \beta (q-1)}{\rho(q-1) \left(\alpha u^{p-q} + \beta \right)^{\frac{q-2}{q-1}}},\]

we see that $g(u) \geq 1$ for all $u > 0$ provided $\rho \leq \beta^\frac{1}{q-1}$. Thus, taking $\rho = \beta^\frac{1}{q-1}$, we get

\[|\nabla u|^{q-2} \nabla u \nabla \left( \frac{\psi^q}{\alpha u^{p-1} + \beta u^{q-1}} \right) \le \frac{|\nabla \psi|^q}{\rho^{q-1}} = \frac{|\nabla \psi|^q}{\beta}.\]

Combining the above inequalities, we complete the proof.

---

Bibliography