Cylindrical Solutions for a Critical Grushin-Type Equation via Local Pohozaev Identities

Abstract

We study a critical Grushin-type equation by applying Lyapunov-Schmidt reduction method and obtain the existence of infinitely many positive multi-bubbling solutions with cylindrical symmetry. Particularly, since the concentration points here can be saddle points of some function related to the potential, we locate the concentration points of these solutions by local Pohozaev identities rather than estimating the derivatives of the reduced functional as usual.

Appendix.: Basic estimates

In this section, we give four basic estimates, where Lemma A.1 will run through the whole paper.

Lemma A. 1

(Lemma B.1, [25]) Let α ≥ 1,β ≥ 1 and i≠j. Then for any $0<\vartheta \le \min \limits \{\alpha ,\beta \}$, there is a constant C > 0 such that

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{(1+|y|+|z-\xi_{i}|)^{\alpha}}\frac{1}{(1+|y|+|z-\xi_{j}|)^{\beta}}\\ &\le&\frac{C}{|\xi_{i}-\xi_{j}|^{\vartheta}}\bigg[\frac{1}{(1+|y|+|z-\xi_{i}|)^{\alpha+\beta-\vartheta}} +\frac{1}{(1+|y|+|z-\xi_{j}|)^{\alpha+\beta-\vartheta}}\bigg]. \end{array} $$

Lemma A. 2

(Lemma B.2, [25]) Let N ≥ 5 and $k\in \left [\frac {N+1}{2},N-1\right )$. Then for any constant 0 < 𝜗 < N − 2, there is a constant C > 0 such that for all $x=(y,z)\in \mathbb R^{k}\times \mathbb R^{N-k}$,

$$ {\int}_{\mathbb{R}^{N}}\frac{1}{|\tilde x-x|^{N-2}}\frac{1}{|\tilde y|(1+|\tilde y|+|\tilde z-\xi|)^{1+\vartheta}}\text{d}\tilde x\le\frac{C}{(1+| y|+|z-\xi|)^{\vartheta}}. $$

Lemma A. 3

Let N ≥ 5 and $k\in \left [\frac {N+1}{2},N-1\right )$. Then there is a constant C > 0 and a small constant ι > 0 such that for all $x=(y,z)\in \mathbb R^{k}\times \mathbb R^{N-k}$,

$$ \begin{array}{@{}rcl@{}} &&{\int}_{\mathbb{R}^{N}}\frac{1}{|\tilde x-x|^{N-2}}\frac{\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}^{2^{\star}-2}(\tilde x)}{|\tilde y|}\sum\limits_{i=1}^{m}\frac{1}{(1+\lambda|\tilde y|+\lambda|\tilde z-\xi_{i}|)^{\frac{N-2}{2}+\sigma}}\text{d}\tilde x\\ &\le&\sum\limits_{i=1}^{m}\frac{C}{(1+\lambda|y|+\lambda|z-\xi_{i}|)^{\frac{N-2}{2}+\sigma+\iota}},\quad\text{where }\sigma=\frac{N-4}{N-2}. \end{array} $$

Proof

The proof is similar to that of Lemma B.3 in [25], so we omit the details. □

Lemma A. 4

If N ≥ 5 and (K1)-(K2)-(K3) hold, then

$$ \begin{array}{@{}rcl@{}} \frac{\partial J(\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda})}{\partial\lambda}&=&m\bigg[-\frac{C_{1}}{\lambda^{3}}+\sum\limits_{i=2}^{m}\frac{C_{2}}{\lambda^{N-1}|\xi_{i}-\xi_{1}|^{N-2}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\bigg]\\ &=&m\bigg[-\frac{C_{1}}{\lambda^{3}}+\frac{C_{3}m^{N-2}}{\lambda^{N-1}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\bigg], \end{array} $$

where C₁,C₂,C₃ are some positive constants, and ε > 0 is a small constant.

Proof

Although the proof is similar to that of Lemma A.4 in [20], we give the details for readers’ convenience. By Eq. (1.12), we have

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial J(\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda})}{\partial\lambda}=\int\nabla\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}\nabla\frac{\partial\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}-\int K(x)\frac{\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial\bar {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\\ &=&\frac{\partial J({\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda})}{\partial\lambda}+\int\bigg[|\nabla\eta|^{2}{\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}+\eta\nabla\eta\nabla {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\\ &&+\eta {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}\nabla\eta\nabla\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}+(\eta^{2}-1)\nabla {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}\nabla\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\bigg]\\ &&-\int K(x)(\eta^{2^{\star}}-1)\frac{{\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\\ &:=&\frac{\partial J({\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda})}{\partial\lambda}+\tilde J_{1}-\tilde J_{2}. \end{array} $$

Firstly, we show $\tilde J_{1}=O(\frac {m}{\lambda ^{3+\varepsilon }})$. In terms of the symmetry, Eqs. (2.28) and (2.10), we get

$$ \begin{array}{@{}rcl@{}} &&\Big|\int|\nabla\eta|^{2}{\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\Big| \le\frac{Cm}{\lambda}\int|\nabla\eta|^{2}\Big(U_{\xi_{1},\lambda}^{2}+U_{\xi_{1},\lambda}\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\Big)\\ &\le&\frac{Cm}{\lambda}\bigg[\frac{1}{\lambda^{N-2}}+\frac{1}{\lambda^{2}}\sum\limits_{i=2}^{m}\frac{1}{(\lambda|\xi_{i}-\xi_{1}|)^{N-4-\varepsilon}}\bigg]\le\frac{Cm}{\lambda^{3+\varepsilon}}. \end{array} $$

Similarly, we find that the other three terms in $\tilde J_{1}$ also have the same estimates as above.

Secondly, we show $\tilde J_{2}=O(\frac {m}{\lambda ^{3+\varepsilon }})$. Retaining the symmetry, the estimates of P₀₂ in Lemma 2.5, (K1), Eqs. (2.28) and (2.8), we deduce that

$$ \begin{array}{@{}rcl@{}} &&\Big|\int K(x)(\eta^{2^{\star}}-1)\frac{{\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\Big|\\ &\le&\frac{Cm}{\lambda^{2+\varepsilon}}\int\frac{(\eta^{2^{\star}}-1)\lambda^{\frac{N+2}{2}}}{\lambda|y|(1+\lambda|y|+\lambda|z-\xi_{1}|)^{\frac{N}{2}+\sigma}}\sum\limits_{i=1}^{m}\frac{\lambda^{\frac{N-2}{2}}}{(1+\lambda|y|+\lambda|z-\xi_{i}|)^{N-2}}\\ &\le&\frac{Cm}{\lambda^{2+\varepsilon}}\bigg[\frac{1}{\lambda}+\frac{1}{\lambda}\sum\limits_{i=2}^{m}\frac{1}{(\lambda|\xi_{i}-\xi_{1}|)^{\sigma}}\bigg]\le\frac{Cm}{\lambda^{3+\varepsilon}}. \end{array} $$

Thirdly, we estimate $\frac {\partial J({\Upsilon }_{\bar r,\bar z^{\prime \prime },\lambda })}{\partial \lambda }$, which can be disintegrated as follows:

$$ \begin{array}{@{}rcl@{}} \frac{\partial J({\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda})}{\partial\lambda}&=&\int\big(1-K(x)\big)\frac{{\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda}\\ &&-\int\frac{{\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}^{2^{\star}-1}-{\sum}_{i=1}^{m}U_{\xi_{i},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial {\Upsilon}_{\bar r,\bar z^{\prime\prime},\lambda}}{\partial\lambda} :=\tilde J_{3}-\tilde J_{4}. \end{array} $$

Next we shall compute $\tilde J_{3}$. By symmetry, (K1), (K2), (K3), Eqs. (1.11) and (2.10), we obtain that

$$ \begin{array}{@{}rcl@{}} \tilde J_{3}&=&m\bigg[\int\big(1-K(x)\big)\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial U_{\xi_{1},\lambda}}{\partial\lambda}+O\Big(\int\frac{1-K(x)}{\lambda}\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\Big)\\ &&\quad+O\Big(\frac{1}{\lambda}\int\frac{U_{\xi_{1},\lambda}}{|y|}\Big(\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\Big)^{2^{\star}-1}\Big)\bigg] =m\bigg[-\frac{C_{1}}{\lambda^{3}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\bigg]. \end{array} $$

Indeed, as the arguments of P₀₂ in Lemma 2.5 and by Eq. (2.10), we have

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{\lambda}\int\frac{U_{\xi_{1},\lambda}}{|y|}\Big(\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\Big)^{2^{\star}-1}\\ &\le&\frac{C}{\lambda^{2+\varepsilon}}\int\frac{\lambda^{\frac{N-2}{2}}}{|y|(1+\lambda|y|+\lambda|z-\xi_{1}|)^{N-2}}\sum\limits_{i=2}^{m}\frac{\lambda^{\frac{N}{2}}}{(1+\lambda|y|+\lambda|z-\xi_{i}|)^{\frac{N}{2}+\sigma}}\\ &\le&\frac{C}{\lambda^{2+\varepsilon}}\sum\limits_{i=2}^{m}\frac{1}{(\lambda|\xi_{i}-\xi_{1}|)^{\frac{N-2}{2}}}\le\frac{C}{\lambda^{3+\varepsilon}}. \end{array} $$

By Taylor expansion, (K1), Eqs. (1.11) and (2.10), we get

$$ \begin{array}{@{}rcl@{}} &&\int\frac{|1-K(x)|}{\lambda}\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\\ &\le&{\int}_{\big\{(y,z^{\prime},z^{\prime\prime})\in \mathbb{R}^{N}:|(r,z^{\prime\prime})-(r_{0}, z_{0}^{\prime\prime})|\le\lambda^{-\frac{1+\varepsilon}{2}}\big\}}\frac{C}{\lambda^{2+\varepsilon}}\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\\ &&+{\int}_{\big\{(y,z^{\prime},z^{\prime\prime})\in \mathbb{R}^{N}:|(r,z^{\prime\prime})-(r_{0}, z_{0}^{\prime\prime})|>\lambda^{-\frac{1+\varepsilon}{2}}\big\}}\frac{C}{\lambda}\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\\ &\le&\frac{C}{\lambda^{2+\varepsilon}}\sum\limits_{i=2}^{m}\frac{1}{(\lambda|\xi_{i}-\xi_{1}|)^{N-2}}+\frac{C}{\lambda^{2-\varepsilon}}\sum\limits_{i=2}^{m}\frac{1}{(\lambda|\xi_{i}-\xi_{1}|)^{N-3-\varepsilon}}\le\frac{C}{\lambda^{3+\varepsilon}}. \end{array} $$

Denoting

$$D_{\lambda,\varepsilon}:=\big\{(y,z^{\prime},z^{\prime\prime})\in \mathbb{R}^{N}:|(r,z^{\prime\prime})-(r_{0},z_{0}^{\prime\prime})|\le\lambda^{-\frac{1-\varepsilon}{2}}\big\}$$

and

$$\bar D_{\lambda,\varepsilon}:=\big\{(y,z^{\prime},z^{\prime\prime})\in \mathbb{R}^{N}:|(|z^{\prime}+\xi_{1}^{\prime}|,z^{\prime\prime})-(r_{0},z_{0}^{\prime\prime})|\le\lambda^{-\frac{1-\varepsilon}{2}}\big\},$$

and using Eq. (1.11), (K2), (K3), we can deduce that

$$ \begin{array}{@{}rcl@{}} &&\int\big(1-K(x)\big)\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial U_{\xi_{1},\lambda}}{\partial\lambda}\\ &=&-{\int}_{D_{\lambda,\varepsilon}}\bigg[\sum\limits_{i,j=1}^{N-k}\frac{1}{2}\frac{\partial^{2}K(x_{0})}{\partial z_{i}\partial z_{j}}(z_{i}-z_{0i})(z_{j}-z_{0j})+O\big(|x-x_{0}|^{3}\big)\bigg]\frac{1}{|y|}\frac{1}{2^{\star}}\frac{\partial U_{\xi_{1},\lambda}^{2^{\star}}}{\partial\lambda}\\ &&+{\int}_{D^{c}_{\lambda,\varepsilon}}\big(1-K(x)\big)\frac{U_{\xi_{1},\lambda}^{2^{\star}-1}}{|y|}\frac{\partial U_{\xi_{1},\lambda}}{\partial\lambda}\\ &=&-{\int}_{\bar D_{\lambda,\varepsilon}}\bigg[\sum\limits_{i,j=1}^{N-k}\frac{1}{2}\frac{\partial^{2}K(x_{0})}{\partial z_{i}\partial z_{j}}(z_{i}+\xi_{1i}-z_{0i})(z_{j}+\xi_{1j}-z_{0j})\\ &&+O\big(| x+(0,\xi_{1})-x_{0}|^{3}\big)\bigg]\frac{1}{|y|}\frac{1}{2^{\star}}\frac{\partial U_{0,\lambda}^{2^{\star}}}{\partial\lambda}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\\ &=&-\frac{\partial}{\partial\lambda}{\int}_{\mathbb{R}^{N}}\bigg[\sum\limits_{i,j=1}^{N-k}\frac{1}{2}\frac{\partial^{2}K(x_{0})}{\partial z_{i}\partial z_{j}}\Big(\frac{ z_{i}}{\lambda}+\xi_{1i}-z_{0i}\Big)\Big(\frac{ z_{j}}{\lambda}+\xi_{1j}-z_{0j}\Big)\\ &&+O\Big(\big|\frac{ x}{\lambda}+(0,\xi_{1})-x_{0}\big|^{3}\Big)\bigg]\frac{1}{|y|}\frac{U_{0,1}^{2^{\star}}}{2^{\star}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\\ &=&-\frac{\partial}{\partial\lambda}{\int}_{\mathbb{R}^{N}}\sum\limits_{i,j=1}^{N-k}\frac{1}{2}\frac{\partial^{2}K(x_{0})}{\partial z_{i}\partial z_{j}}\Big(\frac{ z_{i}}{\lambda}+\xi_{1i}-z_{0i}\Big)\Big(\frac{ z_{j}}{\lambda}+\xi_{1j}-z_{0j}\Big)\frac{1}{|y|}\frac{U_{0,1}^{2^{\star}}}{2^{\star}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\\ &=&-\frac{\partial}{\partial\lambda}{\int}_{\mathbb{R}^{N}}\sum\limits_{i=1}^{N-k}\frac{1}{2}\frac{\partial^{2}K(x_{0})}{\partial {z_{i}^{2}}}\frac{ {z_{i}^{2}}}{\lambda^{2}}\frac{1}{|y|}\frac{U_{0,1}^{2^{\star}}}{2^{\star}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\\ &=&\frac{1}{\lambda^{3}}\frac{\Delta K(x_{0})}{2^{\star} (N-k)}{\int}_{\mathbb{R}^{N}}\frac{|z|^{2}}{|y|}U_{0,1}^{2^{\star}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\\ &=&-\frac{C_{1}}{\lambda^{3}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big) \quad \text{ for some constant } C_{1}>0. \end{array} $$

Finally we estimate $\tilde J_{4}$. By symmetry, the estimates of P₀₂ in Lemma 2.5 and Eq. (2.10), we have

$$ \begin{array}{@{}rcl@{}} \tilde J_{4}&=&m\bigg[\int\frac{2^{\star}-1}{|y|}U_{\xi_{1},\lambda}^{2^{\star}-2}\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\frac{\partial U_{\xi_{1},\lambda}}{\partial\lambda}+O\Big(\frac{1}{\lambda}\int\frac{U_{\xi_{1},\lambda}}{|y|}\Big(\sum\limits_{i=2}^{m}U_{\xi_{i},\lambda}\Big)^{2^{\star}-1}\Big)\bigg]\\ &=&m\bigg[-\sum\limits_{i=2}^{m}\frac{C_{2}}{\lambda^{N-1}|\xi_{i}-\xi_{1}|^{N-2}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\bigg] =m\bigg[-\frac{C_{3}m^{N-2}}{\lambda^{N-1}}+O\Big(\frac{1}{\lambda^{3+\varepsilon}}\Big)\bigg] \end{array} $$

for some constants C₂ > 0 and C₃ > 0.

The proof is completed by combining the estimates of $\tilde J_{1},\ \tilde J_{2}, \ \tilde J_{3}$ and $\tilde J_{4}$. □