Multiple solutions and profile description for a nonlinear Schrödinger–Bopp–Podolsky–Proca system on a manifold

Abstract

We prove a multiplicity result for

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{2}\Delta _g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\ -\Delta _g \phi +a^{2}\Delta _g^{2} \phi + m^2 \phi =4\pi u^{2} \end{array}\right. } \text { in }M, \end{aligned}$$

where (M, g) is a smooth and compact 3-dimensional Riemannian manifold without boundary, $p\in (4,6)$, $a,m,q\ne 0$, $\varepsilon >0$ small enough. The proof of this result relies on Lusternik–Schnirellman category. We also provide a profile description for low energy solutions.

Appendices

Appendix A. Ekeland principle

In this Appendix we want to show that (4.12) holds.

To this end, we proceed as in [3, Lemma 5.4], applying the Ekeland Principle (see [10, Chapter 4]) as follows:

for every $\theta ,\iota >0$ and $u\in J_\varepsilon ^{m_\varepsilon +\theta /2}$, there exists $u_\iota \in {\mathcal {N}}_\varepsilon$ such that

$$\begin{aligned} J_\varepsilon (u_\iota )<J_\varepsilon (u), \quad \Vert u_\iota -u\Vert _\varepsilon<\iota , \quad J_\varepsilon (u_\iota )<J_\varepsilon (v)+\frac{\theta }{\iota }\Vert u_\iota -u\Vert _\varepsilon <\iota \text { for all }v\in {\mathcal {N}}_\varepsilon . \end{aligned}$$

Thus, for every k, taking $\theta =4\delta _k$ and $\iota =4\sqrt{\delta _k}$, there exists ${\tilde{u}}_k\in {\mathcal {N}}_{\varepsilon _{k}}$ such that

$$\begin{aligned} J_{\varepsilon _{k}}({\tilde{u}}_k)< & {} J_{\varepsilon _{k}}(u_k), \quad \Vert {\tilde{u}}_k - u_k\Vert _{\varepsilon _{k}}<4\sqrt{\delta _k}, \nonumber \\ J_{\varepsilon _{k}}({\tilde{u}}_k)< & {} J_{\varepsilon _{k}}(v)+\sqrt{\delta _k}\Vert {\tilde{u}}_k - v\Vert _{\varepsilon _{k}} \text { for all } v\in {\mathcal {N}}_{\varepsilon _{k}}. \end{aligned}$$

(A.1)

The boundedness of $\{\Vert u_k\Vert _{\varepsilon _{k}}\}$ and (A.1) implies that $\{\Vert {\tilde{u}}_k\Vert _{\varepsilon _{k}}\}$ is bounded too.

Observe, moreover, that, for every $\xi \in T_{{\tilde{u}}_k}{\mathcal {N}}_{\varepsilon _{k}}$, there exists a smooth curve $\gamma :[a,b]\rightarrow {\mathcal {N}}_{\varepsilon _{k}}$, with $a<0<b$, such that

$$\begin{aligned} \gamma (0)={\tilde{u}}_k \text { and } \gamma '(0)=\xi \end{aligned}$$

(see e.g. [1]).

Let $\{t_n\}\subset {\mathbb {R}}$ such that $t_n \rightarrow 0$.

Since

$$\begin{aligned} J_{\varepsilon _{k}}(\gamma (t_n)) = J_{\varepsilon _{k}}(\gamma (0)) + J_{\varepsilon _{k}}'(\gamma (0))[\gamma '(0)]t_n +O(t_n^2) = J_{\varepsilon _{k}}({\tilde{u}}_k) + J_{\varepsilon _{k}}'({\tilde{u}}_k)[\xi ] t_n +O(t_n^2) \end{aligned}$$

and

$$\begin{aligned} \Vert {\tilde{u}}_k - \gamma (t_n)\Vert _{\varepsilon _{k}} = \Vert \gamma (0) - \gamma (t_n)\Vert _{\varepsilon _{k}} = \Vert \gamma '(0)t_n +O(t_n^2)\Vert _{\varepsilon _{k}} = \Vert \xi t_n +O(t_n^2)\Vert _{\varepsilon _{k}}, \end{aligned}$$

by the Ekeland Variational Principle, we get

$$\begin{aligned} \sqrt{\delta _k}> \frac{J_{\varepsilon _{k}}({\tilde{u}}_k)-J_{\varepsilon _{k}}(\gamma (t_n))}{\Vert {\tilde{u}}_k - \gamma (t_n)\Vert _{\varepsilon _{k}}} = \frac{t_n}{|t_n|} \frac{J_{\varepsilon _{k}}'({\tilde{u}}_k)[\xi ]+O(t_n)}{\Vert \xi +O(t_n)\Vert _{\varepsilon _{k}}}. \end{aligned}$$

Considering the left and right limits as $t_n\rightarrow 0$ we can conclude that

$$\begin{aligned} |J_{\varepsilon _{k}}'({\tilde{u}}_k)[\xi ]|\le \sqrt{\delta _k} \Vert \xi \Vert _{\varepsilon _{k}} \text { for all } \xi \in T_{{\tilde{u}}_k}{\mathcal {N}}_{\varepsilon _{k}}. \end{aligned}$$

(A.2)

Let now $\varphi \in H^1(M)$ be arbitrary.

Since ${\tilde{u}}_k\in {\mathcal {N}}_{\varepsilon _{k}}$, by (iii) in Lemma 3.1,

$$\begin{aligned} N'_{\varepsilon _{k}}({\tilde{u}}_k)[{\tilde{u}}_k] = -2 \Vert {\tilde{u}}_k\Vert _{\varepsilon _{k}}^2 -(p-4)|{\tilde{u}}_k^+|_{p,\varepsilon _{k}}^p \le -C <0. \end{aligned}$$

(A.3)

Then ${\tilde{u}}_k\notin T_{{\tilde{u}}_k} {\mathcal {N}}_{\varepsilon _{k}}$. Thus there exists $\lambda ,\mu \in {\mathbb {R}}$ and $\xi \in T_{{\tilde{u}}_k} {\mathcal {N}}_{\varepsilon _{k}}$ such that $\varphi =\lambda \xi + \mu {\tilde{u}}_k$.

Observe that, by (A.3), there exists $C\in (0,1)$ such that for all $u\in {\mathcal {N}}_{\varepsilon _{k}}, \xi \in T_u {\mathcal {N}}_{\varepsilon _{k}}$, $|\langle \xi ,u\rangle _{\varepsilon _{k}}|\le C\Vert \xi \Vert _{\varepsilon _{k}} \Vert u\Vert _{\varepsilon _{k}}$. Then a straightforward calculation shows that there exists $C>0$ such that $\Vert \lambda \xi \Vert _{\varepsilon _k} \le C\Vert \varphi \Vert _{\varepsilon _k}$.

Then, by (A.2), we get that for every $\varphi \in H^1(M)$

$$\begin{aligned} |J_{\varepsilon _{k}}'({\tilde{u}}_k)[\varphi ]| = |\lambda J_{\varepsilon _{k}}'({\tilde{u}}_k)[\xi ]| \le \sqrt{\delta _k} \Vert \lambda \xi \Vert _{\varepsilon _{k}} \le C \sqrt{\delta _k} \Vert \varphi \Vert _{\varepsilon _{k}}. \end{aligned}$$

(A.4)

Hence, we can conclude. Indeed, since

$$\begin{aligned} |J_{\varepsilon _{k}}'(u_k)[\varphi ]| \le |J_{\varepsilon _{k}}'(u_k)[\varphi ]-J_{\varepsilon _{k}}'({\tilde{u}}_k)[\varphi ]| +|J_{\varepsilon _{k}}'({\tilde{u}}_k)[\varphi ]|, \end{aligned}$$

by (A.4), it is enough to prove that

$$\begin{aligned} |J_{\varepsilon _{k}}'(u_k)[\varphi ]-J_{\varepsilon _{k}}'({\tilde{u}}_k)[\varphi ]| \le C\sqrt{\delta _{k}}\Vert \varphi \Vert _{\varepsilon _{k}}. \end{aligned}$$

(A.5)

This follows observing that

$$\begin{aligned} |J_{\varepsilon _{k}}'(u_k)[\varphi ]-J_{\varepsilon _{k}}'({\tilde{u}}_k)[\varphi ]|&= \Big |\frac{1}{\varepsilon _{k}^3}\Big [\varepsilon _{k}^2\int _M \nabla _{g} (u_k-{\tilde{u}}_k)\nabla _{g}\varphi d\mu _{g} +\omega \int _M (u_k-{\tilde{u}}_k)\varphi d\mu _{g}\\&\quad +q^2 \int _M (\phi _{u_k}u_k-\phi _{{\tilde{u}}_k}{\tilde{u}}_k)\varphi d\mu _{g}\\&\quad -\int _M (|u_k^+|^{p-2}u_k^+ - |{\tilde{u}}_k^+|^{p-2}{\tilde{u}}_k^+)\varphi d\mu _g\Big ] \Big |. \end{aligned}$$

Then, by Hölder inequality and (A.1),

$$\begin{aligned} \Big |\frac{1}{\varepsilon _{k}^3}\Big [\varepsilon _{k}^2\int _M \nabla _{g} (u_k-{\tilde{u}}_k)\nabla _{g}\varphi d\mu _{g} +\omega \int _M (u_k-{\tilde{u}}_k)\varphi d\mu _{g} \Big |\le & {} \Vert u_k-{\tilde{u}}_k\Vert _{\varepsilon _{k}} \Vert \varphi \Vert _{\varepsilon _{k}}\\< & {} 4 \sqrt{\delta _{k}} \Vert \varphi \Vert _{\varepsilon _{k}}. \end{aligned}$$

Moreover

$$\begin{aligned} \frac{1}{\varepsilon _{k}^3}\Big |\int _M (\phi _{u_k}u_k-\phi _{{\tilde{u}}_k}{\tilde{u}}_k)\varphi d\mu _{g}\Big |\le & {} \frac{1}{\varepsilon _{k}^3}\Big |\int _M \phi _{u_k} (u_k-{\tilde{u}}_k)\varphi d\mu _{g}\Big |\nonumber \\&+\frac{1}{\varepsilon _{k}^3}\Big |\int _M (\phi _{u_k}-\phi _{{\tilde{u}}_k}){\tilde{u}}_k \varphi d\mu _{g}\Big |. \end{aligned}$$

(A.6)

Considering the first term in the right hand side of (A.6), by Lemma 2.1, Sobolev embedding $H^2(M)\subset C^0(M)$, Hölder inequality, (2.1), the boundedness of $\{\Vert u_k\Vert _{\varepsilon _{k}}\}$, and (A.1),

$$\begin{aligned} \frac{1}{\varepsilon _{k}^3}\Big |\int _M \phi _{u_k}(u_k-{\tilde{u}}_k)\varphi d\mu _{g}\Big |\le & {} |\phi _{u_k}|_{C^0} |u_k-{\tilde{u}}_k|_{2,\varepsilon _{k}} |\varphi |_{2,\varepsilon _{k}} \\\le & {} C | u_k |_2^2 \Vert u_k-{\tilde{u}}_k\Vert _{\varepsilon _{k}} \Vert \varphi \Vert _{\varepsilon _{k}} \le C \varepsilon _{k}^3 \sqrt{\delta _{k}} \Vert \varphi \Vert _{\varepsilon _{k}}. \end{aligned}$$

To estimate the second term in the right hand side of (A.6), first observe that

$$\begin{aligned} -\Delta _g (\phi _{u_k}-\phi _{{\tilde{u}}_k}) + a^2 \Delta _g^2 (\phi _{u_k}-\phi _{{\tilde{u}}_k}) + (\phi _{u_k}-\phi _{{\tilde{u}}_k}) {=} 4\pi (u_k^2{-}{\tilde{u}}_k^2). \end{aligned}$$

Then, using also Sobolev and Hölder inequalities, (2.1), (A.1), and the boundedness of $\{\Vert u_k\Vert _{\varepsilon _{k}}\}$ and $\{\Vert {\tilde{u}}_k\Vert _{\varepsilon _{k}}\}$,

$$\begin{aligned} \Vert \phi _{u_k}-\phi _{{\tilde{u}}_k}\Vert _{H^2}^2&= 4\pi \int _M (\phi _{u_k}-\phi _{{\tilde{u}}_k}) (u_k^2-{\tilde{u}}_k^2) d\mu _{g} \le C \Vert \phi _{u_k}-\phi _{{\tilde{u}}_k}\Vert _{H^2} \int _M |u_k^2-{\tilde{u}}_k^2|d\mu _g\\&\le C \Vert \phi _{u_k}-\phi _{{\tilde{u}}_k}\Vert _{H^2} |u_k-{\tilde{u}}_k|_2 |u_k+{\tilde{u}}_k|_2 \le C \varepsilon _{k}^3 \Vert \phi _{u_k}-\phi _{{\tilde{u}}_k}\Vert _{H^2} \Vert u_k-{\tilde{u}}_k\Vert _{\varepsilon _{k}}. \end{aligned}$$

Thus, using also Sobolev imbeddings, Hölder inequality, and (A.1),

$$\begin{aligned} \frac{1}{\varepsilon _{k}^3}\Big |\int _M (\phi _{u_k}-\phi _{{\tilde{u}}_k}) {\tilde{u}}_k \varphi d\mu _{g}\Big |&\le \frac{1}{\varepsilon _{k}^3} |\phi _{u_k}-\phi _{{\tilde{u}}_k}|_{C^0} |{\tilde{u}}_k|_2 |\varphi |_2 \le \frac{C}{\varepsilon _{k}^3} \Vert \phi _{u_k}-\phi _{{\tilde{u}}_k}\Vert _{H^2} |{\tilde{u}}_k|_2 |\varphi |_2\\&\le C \varepsilon _{k}^3 \Vert u_k-{\tilde{u}}_k\Vert _{\varepsilon _{k}} |{\tilde{u}}_k|_{2,\varepsilon _{k}} |\varphi |_{2,\varepsilon _{k}} \le C \varepsilon _{k}^3 \sqrt{\delta _{k}} \Vert \varphi \Vert _{\varepsilon _{k}}. \end{aligned}$$

Finally, by Lagrange Theorem, Hölder inequality, (2.1), boundedness of $\{\Vert u_k\Vert _{\varepsilon _{k}}\}$ and $\{\Vert {\tilde{u}}_k\Vert _{\varepsilon _{k}}\}$, (A.1), we have

$$\begin{aligned}&\Big |\frac{1}{\varepsilon _{k}^3}\int _M (|u_k^+|^{p-2}u_k^+ - |{\tilde{u}}_k^+|^{p-2}{\tilde{u}}_k^+)\varphi d\mu _g \Big | \\&\quad \le \frac{p-1}{\varepsilon _{k}^3} \int _M |\theta _k u_k^+ +(1-\theta _k ) {\tilde{u}}_k^+|^{p-2} |u_k^+ - {\tilde{u}}_k^+| |\varphi | d\mu _g\\&\quad \le C (|u_k^+|_{p,\varepsilon _{k}}^{p-2} + |{\tilde{u}}_k^+|_{p,\varepsilon _{k}}^{p-2}) |u_k^+ - {\tilde{u}}_k^+|_{p,\varepsilon _{k}} |\varphi |_{p,\varepsilon _{k}}\\&\le C \sqrt{\delta _{k}} \Vert \varphi \Vert _{\varepsilon _{k}}, \end{aligned}$$

completing the proof of (A.5).

Appendix B. Bootstrap argument

In this section, through a classical bootstrap argument, we prove that $\{{\bar{v}}_{j}^{1}\} \subset C^{2}(B(0,R/2))$ and that it is bounded in $C^{2}(B(0,R/2))$.

Let $R>0$.

Observe that, arguing as in (5.1), for j large,

$$\begin{aligned} B(0,R) \subset B\Bigg (0,\frac{r}{4\varepsilon _{j}}\Bigg ) \subset B\Bigg (-\frac{Q_{\varepsilon _{j}}^1}{\varepsilon _{j}},\frac{r}{2\varepsilon _{j}}\Bigg ). \end{aligned}$$

(B.1)

Thus, in B(0, R), since $u_{\varepsilon _{j}}$ is a solution of (2.3), we have

$$\begin{aligned} \begin{aligned}&-\varepsilon _{j}^2 (\Delta _g u_{\varepsilon _{j}}) \big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big ) \\&\quad = - u_{\varepsilon _{j}} \big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big ) + (u_{\varepsilon _{j}})^{p-1}\big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big ) \\&\qquad - \phi _{u_{\varepsilon _j}} \big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big ) u_{\varepsilon _{j}} \big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big ). \end{aligned} \end{aligned}$$

(B.2)

But, since

$$\begin{aligned} (\Delta _g u_{\varepsilon _{j}}) \big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big )= & {} \frac{1}{\varepsilon _{j}^2} g_{P^{1}}^{il}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z) \partial _{il} {\bar{v}}_j^1 (z) \\&+ \frac{1}{\varepsilon _{j}^2 |g_{P^1}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)|^{1/2}} \partial _{l}\\&\left( g_{P^{1}}^{il}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}\cdot )|g_{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}\cdot )|^{1/2}\right) (z) \partial _{i} {\bar{v}}_j^1(z), \end{aligned}$$

(B.2) reads as

$$\begin{aligned} -g_{P^{1}}^{il}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z) \partial _{il} {\bar{v}}_j^1 (z) = f_j(z), \end{aligned}$$

(B.3)

where

$$\begin{aligned} f_j(z)&:= \frac{1}{|g_{P^1}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)|^{1/2}} \partial _{l}\left( g_{P^{1}}^{il}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}\cdot )|g_{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}\cdot )|^{1/2}\right) (z) \partial _{i} {\bar{v}}_j^1(z) - {\bar{v}}_j^1(z) \\&\quad - {\bar{\phi }}_j(z) {\bar{v}}_j^1(z) + ({\bar{v}}_j^1)^{p-1}(z) \end{aligned}$$

and ${\bar{\phi }}_j(z):=\phi _{u_{\varepsilon _j}} \big (\exp _{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)\big )$.

Let us show that ${\bar{v}}_j^1 \in C^{0,\alpha }(B(0,R))$ for some $\alpha \in (0,1)$.

Let us consider equation (B.3) in B(0, R) and let $q_0:=6$. Since $p \in (4, 6)$, then $q_0/(p-1)\in (6/5,2)$. Thus

$$\begin{aligned} \min \left\{ 2, \frac{q_0}{p-1} \right\} = \frac{q_0}{p-1} \end{aligned}$$

and so, using the boundedness of $\{{\bar{v}}_j^1\}$ in $H^1({\mathbb {R}}^3)$ and of $\{\Vert u_{\varepsilon _{j}}\Vert _{\varepsilon _{j}}\}$, (B.1), and Lemma 2.1, $f_j \in L^{q_0/(p-1)} (B(0,R))$ and

$$\begin{aligned} \left( \int _{B (0, R)} |{\bar{\phi }}_j {\bar{v}}_j^1|^{q_0/(p-1)} dz\right) ^{(p-1)/q_0}&\le C\left( \int _{B (0, R)} |{\bar{\phi }}_j {\bar{v}}_j^1|^2 dz\right) ^{1/2} \le C |{\bar{\phi }}_j |_{L^\infty (B (0, R))}\\&\le C |\phi _{u_{\varepsilon _{j}}} (\exp _{P^1}(Q_{\varepsilon _{j}}^1+\varepsilon _{j}\cdot )) |_{L^\infty (B (-Q_{\varepsilon _{j}}^1/\varepsilon _{j}, r/(2\varepsilon _{j})))}\\&\le C |\phi _{u_{\varepsilon _{j}}} (\exp _{P^1}(\cdot )) |_{L^\infty (B (0, r/2))} \\&\le C |\phi _{u_{\varepsilon _{j}}} |_{\infty } \le C \Vert \phi _{u_{\varepsilon _{j}}} \Vert _{H^2} \\&\le C |u_{\varepsilon _{j}} |_{2}^2 \le C \varepsilon _{j}^3. \end{aligned}$$

Hence, by (B.3),

$$\begin{aligned} | \Delta {\bar{v}}_j^1 |_{L^{q_0/(p-1)} (B(0,R))} \le C | f_j |_{L^{q_0/(p-1)} (B(0,R))} \le C, \end{aligned}$$

and so, by a classical interpolation inequality, ${\bar{v}}_j^1 \in W^{2, q_0/(p-1)} (B(0,R))$ and

$$\begin{aligned} \Vert {\bar{v}}_j^1\Vert _{W^{2, 6/(p-1)} (B(0,R))} \le C. \end{aligned}$$

If $q_0/(p-1)>3/2$, namely if

$$\begin{aligned} (p - 1) - 4 < 0, \end{aligned}$$

we get that ${\bar{v}}_j^1$ is continuous and, by the previous arguments, $\{{\bar{v}}_j^1\}$ is bounded in $C^{0,\alpha }(B(0,R))$ for some $\alpha \in (0,1)$.

If, instead, $q_0/(p-1) \le 3/2$, namely if

$$\begin{aligned} (p - 1) - 4 \ge 0, \end{aligned}$$

or, equivalently, $5 \le p < 6$, then $W^{2, q_0/(p-1)} (B(0,R))$ embeds in $L^{q_1} (B(0,R))$ with

$$\begin{aligned} q_1: = \frac{6}{(p - 1) - 4}. \end{aligned}$$

Then we consider

$$\begin{aligned} \min \left\{ 2, \frac{q_1}{p - 1} \right\} \end{aligned}$$

and we iterate the procedure.

So, at the nth step we take

$$\begin{aligned} q_n: = \frac{6}{(p - 1)^n - 4 \sum _{k = 0}^{n - 1} (p - 1)^k} = \frac{6}{(p - 1)^n - 4 \frac{(p - 1)^n - 1}{p - 2}} = \frac{6 (p - 2)}{(p - 6) (p - 1)^n + 4}, \end{aligned}$$

and we consider

$$\begin{aligned} \min \left\{ 2, \frac{q_n}{p - 1} \right\} . \end{aligned}$$

We can conclude if

$$\begin{aligned} \min \left\{ 2, \frac{q_n}{p - 1} \right\} >\frac{3}{2} \end{aligned}$$

which occurs in a finite number of steps since

$$\begin{aligned} \frac{q_n}{p - 1} > \frac{3}{2}, \end{aligned}$$

namely for

$$\begin{aligned} n > \frac{\log 4 - \log (6 - p)}{\log (p - 1)} - 1. \end{aligned}$$

Observe that, at each step, whenever

$$\begin{aligned} \min \left\{ 2, \frac{q_n}{p - 1} \right\} =\frac{q_n}{p - 1}>\frac{3}{2}, \end{aligned}$$

arguing as before we get that $\{{\bar{v}}_j^1\}$ is bounded in $C^{0,\alpha }(B(0,R))$ for some $\alpha \in (0,1)$.

Now, let us write (B.3) as

$$\begin{aligned} \begin{aligned}&-g_{P^{1}}^{il}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z) \partial _{il} {\bar{v}}_j^1 - \frac{1}{|g_{P^1}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}z)|^{1/2}} \partial _{l}\\&\qquad \left( g_{P^{1}}^{il}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}\cdot )|g_{P^{1}}(Q_{\varepsilon _{j}}^{1}+\varepsilon _{j}\cdot )|^{1/2}\right) (z) \partial _{i} {\bar{v}}_j^1\\&\quad +\omega {\bar{v}}_j^1 + q^2 {\bar{\phi }}_j(z) {\bar{v}}_j^1 = ({\bar{v}}_j^1)^{p-1}. \end{aligned} \end{aligned}$$

(B.4)

The continuity of ${\bar{v}}_j^1$ implies that the right hand side of (B.4) is in $L^2(B(0,R))$. In addition, also $|\nabla ({\bar{v}}_j^1)^{p-1}| \in L^2(B(0,R))$ and so the right hand side of (B.4) is in $H^1(B(0,R))$. Thus, [14, Theorem 8.10] implies that ${\bar{v}}_j^1\in W_\mathrm{loc}^{3,2}(B(0,2R/3))$ and so, by classical embeddings, ${\bar{v}}_j^1\in C^{1,\alpha }(B(0,2R/3))$ for some $\alpha \in (0,1)$. Then, repeating the procedure we get that ${\bar{v}}_j^1\in C^{2,\alpha }(B(0,R/2))$ for some $\alpha \in (0,1)$ and, by Schauder estimate [14, page 93],

$$\begin{aligned} \Vert {\bar{v}}_{j}^{1}\Vert _{C^{2,\alpha }(B(0,R/4))}\le C. \end{aligned}$$