Apriori decay estimates for Hardy–Sobolev–Maz’ya equations

Abstract

In this article we establish sharp decay estimates of solutions to the Euler–Lagrange equation corresponding to the Hardy–Sobolev–Maz’ya inequality with cylindrical weights on both sides,

$$\begin{aligned} -{\text {div}}\left( |y|^{-r} |\nabla u|^{p-2}\nabla u\right) = f(x,u), \, \text { for } u \in D^{1,p}({\mathbb {R}}^n, |y|^{-r}) \end{aligned}$$

where, $p \in (1,n)$, $n \ge 2$, $1 \le k < n$, ${\mathbb {R}}^n= {\mathbb {R}}^k \times {\mathbb {R}}^{n-k}$, $x = (y,z) \in {\mathbb {R}}^k \times {\mathbb {R}}^{n-k}$, $f : {\mathbb {R}}^n\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a Caratheodory function satisfying the bounds $|f(x,t)| \le \Lambda |t|^{p_{r,s}^*- 1} |y|^{-s}$ for some $\Lambda > 0$, $p_{r,s}^*\in (p, p^*]$ denotes the critical exponent $p_{r,s}^*= \frac{p(n-s)}{n-p-r}$ with parameters $r, s < k$ when $k \ge 2$ (additionally assume $r \in (-(p-1),\min (1, n-p))$ when $k=1$). Using the sharp asymptotics we establish existence of positive solutions of a Brézis-Nirenberg problem involving lower order perturbations of Hardy–Sobolev equation in a smooth bounded domain containing origin. We show that the problem

$$\begin{aligned} -\Delta _p u = \frac{|u|^{p_s^*- 2}u}{|y|^s} + \rho (u), u \ge 0 \text { in } \Omega \end{aligned}$$

for $u \in W_{0}^{1,p}(\Omega )$ has a solution for a class of $C^1$ odd functions $\rho$ when $n > p^2$ and $\displaystyle \int _{{\mathbb {R}}^n} {\tilde{\rho }} \left( |x|^{-\frac{n-p}{p-1}}\right) \,dx > 0$ where, $\displaystyle {\tilde{\rho }}(t) = \int _{0}^t \rho (r)\,dr$.

Appendix

Calculations for the test function in Theorem-1.1:

Note that for the test function $\displaystyle \varphi = \eta ^p(v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}v$ with $\gamma > 0$ we have

$$\begin{aligned} \nabla \varphi&= p\eta ^{p-1}(v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}v \nabla \eta + \frac{\gamma - 1}{2}\eta ^p(v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}2v^2\nabla v + \eta ^p(v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}\nabla v \end{aligned}$$

(6.1)

$$\begin{aligned}&= p\eta ^{p-1}(v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}v \nabla \eta + \eta ^p(v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}\left[ (\gamma - 1)v^2 + (v^2 + \epsilon ^2)\right] \nabla v \end{aligned}$$

(6.2)

$$\begin{aligned}&= p\eta ^{p-1}(v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}v \nabla \eta + \eta ^p(v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}\left[ \gamma v^2 + \epsilon ^2\right] \nabla v. \end{aligned}$$

(6.3)

Therefore we have

$$\begin{aligned} |\nabla v|^{p-2} \nabla v \cdot \nabla \varphi&= p\eta ^{p-1}|\nabla v|^{p-2}(\nabla v \cdot \nabla \eta ) v (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}\nonumber \\&\quad + \eta ^p|\nabla v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}\left[ \gamma v^2 + \epsilon ^2\right] . \end{aligned}$$

(6.4)

We apply Young’s inequality to the first term in the RHS of the expression (6.4) with indices $\frac{1}{p'} + \frac{1}{p} = 1$ with an arbitrary $\delta > 0$,

$$\begin{aligned} p(\eta ^{p-1}|\nabla v|^{p-2}\nabla v) \cdot (v\nabla \eta ) (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}&\le \left[ \delta \eta ^p |\nabla v|^p\nonumber \right. \\&\left. \quad + C_p\delta ^{-(p-1)}|\nabla \eta |^p|v|^p\right] (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}}. \end{aligned}$$

(6.5)

Therefore from (6.4) and (6.5) we have

$$\begin{aligned}&|\nabla v|^{p-2} \nabla v \cdot \nabla \varphi \nonumber \\&\quad \ge \eta ^p|\nabla v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}\left[ \gamma v^2 + \epsilon ^2\right] - \left[ \delta \eta ^p |\nabla v|^p + C_p\delta ^{-(p-1)}|\nabla \eta |^p|v|^p\right] (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}} \end{aligned}$$

(6.6)

$$\begin{aligned}&= \eta ^p|\nabla v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}\left[ \gamma v^2 + \epsilon ^2 - \delta (v^2 + \epsilon ^2)\right] - C_p\delta ^{-(p-1)} |\nabla \eta |^p|v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}} \end{aligned}$$

(6.7)

$$\begin{aligned}&\ge \frac{1}{2} \eta ^p|\nabla v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}}\left[ \gamma v^2 + \epsilon ^2 \right] - C_\gamma |\nabla \eta |^p|v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}} \end{aligned}$$

(6.8)

where, in (6.7) we chose $\delta = \frac{1}{2}\min (\gamma , 1)$ and $C_\gamma = C_p \delta ^{-(p-1)}$.

Now, consider the function $\displaystyle w = (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2p}}v$ then we note that

$$\begin{aligned} \nabla w&= \frac{\gamma - 1}{p}(v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2p}-1}v^2 \nabla v + (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2p}} \nabla v\nonumber \\&= (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2p}-1} \left[ \frac{p + \gamma - 1}{p} v^2 + \epsilon ^2\right] \nabla v. \end{aligned}$$

(6.9)

Then note that

$$\begin{aligned} |\nabla w|^p&= (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}-p} \left[ \frac{p + \gamma - 1}{p} v^2 + \epsilon ^2\right] ^p |\nabla v|^p \end{aligned}$$

(6.10)

$$\begin{aligned}&\le \max \left( \frac{p + \gamma -1}{p}, 1\right) ^p (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}} |\nabla v|^p \end{aligned}$$

(6.11)

$$\begin{aligned}&\le \max \left( \frac{p + \gamma -1}{p}, 1\right) ^p (v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}} \max \left( \frac{1}{\gamma },1\right) (\gamma v^2 + \epsilon ^2) |\nabla v|^p. \end{aligned}$$

(6.12)

Therefore, from (6.10) we have

$$\begin{aligned} c(\gamma )^{-1}\eta ^p|\nabla w|^p&\le \frac{1}{2}(v^2 + \epsilon ^2)^{\frac{\gamma - 3}{2}} \left[ \gamma v^2 + \epsilon ^2\right] \eta ^p |\nabla v|^p \end{aligned}$$

(6.13)

where,

$$\begin{aligned} c(\gamma ) = 2\max \left( \frac{p + \gamma -1}{p}, 1\right) ^p \max \left( \frac{1}{\gamma },1\right) \end{aligned}$$

(6.14)

and combining with (6.8) we get

$$\begin{aligned} |\nabla v|^{p-2} \nabla v \cdot \nabla \varphi&\ge c(\gamma )^{-1}\eta ^p|\nabla w|^p - C_\gamma |\nabla \eta |^p|v|^p (v^2 + \epsilon ^2)^{\frac{\gamma - 1}{2}} \end{aligned}$$

(6.15)

$$\begin{aligned}&\ge \frac{c(\gamma )^{-1}}{2^{p-1}}|\nabla (\eta w)|^p - C_\gamma ' |\nabla \eta |^p|w|^p \end{aligned}$$

(6.16)

where, $C_\gamma ' = c(\gamma )^{-1} + C_\gamma$.