Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary

Abstract

Let (M, g) be a smooth compact, n dimensional Riemannian manifold, n = 3, 4 with smooth n − 1 dimensional boundary ∂ M. We search for the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that C ¹ stable critical points of the mean curvature of the boundary generates H ¹(M) solutions when the perturbation parameter $\varepsilon$ is sufficiently small.

Appendix: Technical lemmas

Lemma 5.1.

There exists $\varepsilon _{0}> 0$ and c > 0 such that, for any $\xi _{0} \in \partial M$ and for any $\varepsilon \in (0,\varepsilon _{0})$ it holds

$$\displaystyle{ \left \Vert \frac{\partial } {\partial y_{h}}Z_{\varepsilon,\xi (y)}^{l}\right \Vert _{ \varepsilon } = O\left (\frac{1} {\varepsilon } \right ),\ \ \left \Vert \frac{\partial } {\partial y_{h}}W_{\varepsilon,\xi (y)}\right \Vert _{\varepsilon } = O\left (\frac{1} {\varepsilon } \right ), }$$

(38)

for $h = 1,\ldots,n - 1$, $l = 1,\ldots,n$

Lemma 5.2.

Let us consider the functions

$$\displaystyle{\tilde{v}_{\varepsilon,\xi }(z) = \left \{\begin{array}{cl} \psi (W_{\varepsilon,\xi })\left (\varPsi _{\xi }^{\partial }(\varepsilon z)\right )&\text{ for }z \in D^{+}(R/\varepsilon )\\ \\ 0 &\text{ for }z \in \mathbb{R}^{3}\setminus D^{+}(R/\varepsilon ) \end{array} \right.}$$

where $D^{+}(r/\varepsilon ) = \left \{z = (\bar{z},z_{n}),\ \bar{z} \in \mathbb{R}^{n-1},\vert \bar{z}\vert <r/\varepsilon,\ 0 \leq z_{n} <R/\varepsilon )\right \}$ . Then there exists a constant c > 0 such that

$$\displaystyle{\|\tilde{v}_{\varepsilon,\xi }(z)\|_{L^{2^{{\ast}}}(\mathbb{R}_{+}^{n})} \leq c\varepsilon ^{2}.}$$

Furthermore, take a sequence $\varepsilon _{n} \rightarrow 0$ , up to subsequences, $\left \{ \frac{1} {\varepsilon _{n}^{2}}\tilde{v}_{\varepsilon _{n},\xi }\right \}_{n}$ converges weakly in $L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$ as $\varepsilon$ goes to 0 to a function $\gamma \in D^{1,2}(\mathbb{R}^{3})$ . The function γ solves, in a weak sense, the equation

$$\displaystyle{ -\varDelta \gamma = qU^{2}\text{ in }\mathbb{R}_{ +}^{n} }$$

(39)

Proof.

We prove the Lemma for Problem (1), being the Problem (2) completely analogous. By definition of $\tilde{v}_{\varepsilon,\xi }(z)$ and by (1) we have, for all $z \in D^{+}(r/\varepsilon )$,

$$\displaystyle\begin{array}{rcl} & & -\sum _{ij}\partial _{j}\left (\vert g_{\xi }(\varepsilon z)\vert ^{1/2}g_{\xi }^{ij}(\varepsilon z)\partial _{ i}\tilde{v}_{\varepsilon,\xi }(z)\right ) = \\ & & \qquad \qquad =\varepsilon ^{2}\vert g_{\xi }(\varepsilon z)\vert ^{1/2}\left \{qU^{2}(z)\chi _{ r}^{2}(\varepsilon z) -\left [1 + q^{2}U^{2}(z)\chi _{ R}^{2}(\varepsilon z)\right ]\tilde{v}_{\varepsilon,\xi }(z)\right \}{}\end{array}$$

(40)

By (40), and remarking that $\tilde{v}_{\varepsilon,\xi }(z) \geq 0$ we have

$$\displaystyle\begin{array}{rcl} & & \|\tilde{v}_{\varepsilon,\xi }(z)\|_{D^{1,2}\left (D^{+}(r/\varepsilon )\right )}^{2} \leq C\int \limits _{ D^{+}(R/\varepsilon )}\vert g_{\xi }(\varepsilon z)\vert ^{1/2}g_{\xi }^{ij}(\varepsilon z)\partial _{ i}\tilde{v}_{\varepsilon,\xi }(z)\partial _{j}\tilde{v}_{\varepsilon,\xi }(z)dz {}\\ & & = C\varepsilon ^{2}\int \limits _{ D^{+}(R/\varepsilon )}\vert g_{\xi }(\varepsilon z)\vert ^{1/2}\left \{qU^{2}(z)\chi _{ R}^{2}(\varepsilon z)\tilde{v}_{\varepsilon,\xi }(z) -\left [1 + q^{2}U^{2}(z)\chi _{ R}^{2}(\varepsilon z)\right ]\tilde{v}_{\varepsilon,\xi }^{2}(z)\right \}dz {}\\ & & \leq C\varepsilon ^{2}\int \limits _{ D^{+}(R/\varepsilon )}\vert g_{\xi }(\varepsilon z)\vert ^{1/2}qU^{2}(z)\chi _{ R}^{2}(\varepsilon \vert z\vert )\tilde{v}_{\varepsilon,\xi }(z)dz {}\\ & & \leq C\varepsilon ^{2}\|\tilde{v}_{\varepsilon,\xi }(z)\|_{L^{2^{{\ast}}}\left (D^{+}(R/\varepsilon )\right )}\|U\|_{ L^{ \frac{4n} {n+2} }}^{2} \leq C\varepsilon ^{2}\|\tilde{v}_{\varepsilon,\xi }(z)\|_{D^{1,2}\left (D^{+}(R/\varepsilon )\right )} {}\\ \end{array}$$

Thus we have

$$\displaystyle{ \|\tilde{v}_{\varepsilon,\xi }(z)\|_{D^{1,2}\left (D^{+}(R/\varepsilon )\right )} \leq C\varepsilon ^{2}\text{ and }\vert \tilde{v}_{\varepsilon,\xi }(z)\vert _{L^{2^{{\ast}}}(\mathbb{R}_{ +}^{n})} \leq C\varepsilon ^{2}. }$$

(41)

By (41), if $\varepsilon _{n}$ is a sequence which goes to zero, the sequence $\left \{ \frac{1} {\varepsilon _{n}^{2}} \tilde{v}_{\varepsilon _{n},\xi }\right \}_{n}$ is bounded in $L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$. Then, up to subsequence, $\left \{ \frac{1} {\varepsilon _{n}^{2}} \tilde{v}_{\varepsilon _{n},\xi }\right \}_{n}$ converges to some $\tilde{\gamma }\in L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$ weakly in $L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$.

Moreover, by (40), for any $\varphi \in C_{0}^{\infty }(\mathbb{R}_{+}^{n})$, it holds

$$\displaystyle\begin{array}{rcl} & & \int \limits _{\text{supp }\varphi }\sum _{ij}\vert g_{\xi }(\varepsilon z)\vert ^{1/2}g_{\xi }^{ij}(\varepsilon z)\partial _{ i}\frac{\tilde{v}_{\varepsilon,\xi }(z)} {\varepsilon _{n}^{2}} \partial _{j}\varphi (z)dz = \\ & & \int \limits _{\text{supp }\varphi }\left \{qU^{2}(z)\chi _{ r}^{2}(\varepsilon \vert z\vert ) -\left [1 + q^{2}U^{2}(z)\chi _{ R}^{2}(\varepsilon z)\right ]\tilde{v}_{\varepsilon,\xi }(z)\right \}\vert g_{\xi }(\varepsilon z)\vert ^{1/2}\varphi (z)dz.{}\end{array}$$

(42)

Consider now the functions

$$\displaystyle{v_{\varepsilon,\xi }(z):=\psi (W_{\varepsilon,\xi })\left (\varPsi _{\xi }^{\partial }(\varepsilon z)\right )\chi _{ R}(\varepsilon z) =\tilde{ v}_{\varepsilon,\xi }(z)\chi _{r}(\varepsilon z)\text{ for }z \in \mathbb{R}_{+}^{n}.}$$

We have immediately that $v_{\varepsilon,\xi }(z)$ is bounded in $D^{1,2}(\mathbb{R}_{+}^{n})$, thus the sequence $\left \{ \frac{1} {\varepsilon _{n}^{2}} v_{\varepsilon _{n},\xi }\right \}_{n}$ converges to some $\gamma \in D^{1,2}(\mathbb{R}^{3})$ weakly in $D^{1,2}(\mathbb{R}_{+}^{n})$ and in $L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$. Finally, for any compact set $K \subset \mathbb{R}_{+}^{n}$ eventually $v_{\varepsilon _{n},\xi } \equiv \tilde{ v}_{\varepsilon _{n},\xi }$ on K. So it is easy to see that $\tilde{\gamma }=\gamma$.

We recall that $\vert g_{\xi }(\varepsilon z)\vert ^{1/2} = 1 + O(\varepsilon \vert z\vert )$ and $g_{\xi }^{ij}(\varepsilon z) =\delta _{ij} + O(\varepsilon \vert z\vert )$ so, by the weak convergence of $\left \{ \frac{1} {\varepsilon _{n}^{2}} v_{\varepsilon _{n},\xi }\right \}_{n}$ in $D^{1,2}(\mathbb{R}_{+}^{n})$, for any $\varphi \in C_{0}^{\infty }(\mathbb{R}_{+}^{n})$ we get

$$\displaystyle\begin{array}{rcl} & & \int \limits _{\text{supp }\varphi }\sum _{ij}\vert g_{\xi }(\varepsilon _{n}z)\vert ^{1/2}g_{\xi }^{ij}(\varepsilon _{ n}z)\partial _{i}\frac{\tilde{v}_{\varepsilon _{n},\xi }(z)} {\varepsilon _{n}^{2}} \partial _{j}\varphi (z)dz \\ & & \qquad \qquad \qquad \ \ =\int \limits _{\text{supp }\varphi }\sum _{ij}\vert g_{\xi }(\varepsilon _{n}z)\vert ^{1/2}g_{\xi }^{ij}(\varepsilon _{ n}z)\partial _{i}\frac{v_{\varepsilon _{n},\xi }(z)} {\varepsilon _{n}^{2}} \partial _{j}\varphi (z)dz \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \rightarrow \int \limits _{\mathbb{R}^{3}}\sum _{i}\partial _{i}\gamma (z)\partial _{i}\varphi (z)dz\text{ as }n \rightarrow \infty. {}\end{array}$$

(43)

Thus by (42) and by (43) and because $\left \{ \frac{1} {\varepsilon _{n}^{2}} \tilde{v}_{\varepsilon _{n},\xi }\right \}_{n}$ converges to γ weakly in $L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$ we get

$$\displaystyle{\int \limits _{\mathbb{R}_{+}^{n}}\sum _{i}\partial _{i}\gamma (z)\partial _{i}\varphi (z)dz = q\int \limits _{\mathbb{R}_{+}^{n}}U^{2}(z)\varphi (z)dz\text{ for all }\varphi \in C_{ 0}^{\infty }(\mathbb{R}_{ +}^{n}).}$$

So, finally, up to subsequences, $\left \{ \frac{1} {\varepsilon _{n}^{2}}\tilde{v}_{\varepsilon _{n},\xi }\right \}_{n}$ converges to γ, weakly in $L^{2^{{\ast}} }(\mathbb{R}_{+}^{n})$ and the function $\gamma \in D^{1,2}(\mathbb{R}_{+}^{n})$ is a weak solution of −Δ γ = qU ² in $\mathbb{R}_{+}^{n}$. □ 

Remark 5.3.

We remark that γ is positive and decays exponentially at infinity with its first derivative because it solves $-\varDelta \gamma = qU^{2}$ in $\mathbb{R}_{+}^{n}$. Moreover it is symmetric with respect to the first n − 1 variables.

Definition 5.4.

Let $\xi _{0} \in \partial M$. We introduce the functions $\mathcal{E}$ and $\tilde{\mathcal{E}}$ as follows.

$$\displaystyle{\mathcal{E}(y,x) = \left (\exp _{\xi (y)}^{\partial }\right )^{-1}(x) = \left (\exp _{\exp _{\xi _{ 0}}^{\partial }y}^{\partial }\right )^{-1}(\exp _{\xi _{ 0}}^{\partial }\bar{\eta }) =\tilde{ \mathcal{E}}(y,\bar{\eta })}$$

where $x,\xi (y) \in \partial M$, $y,\bar{\eta }\in B(0,R) \subset \mathbb{R}^{n-1}$ and $\xi (y) =\exp _{ \xi _{0}}^{\partial }y$, $x =\exp _{ \xi _{0}}^{\partial }\bar{\eta }$. Using Fermi coordinates, in a similar way we define

$$\displaystyle{\mathcal{H}(y,x) = \left (\psi _{\xi (y)}^{\partial }\right )^{-1}(x) = \left (\psi _{\exp _{\xi _{ 0}}^{\partial }y}^{\partial }\right )^{-1}\left (\psi _{\xi _{ 0}}^{\partial }(\bar{\eta },\eta _{ n})\right ) =\tilde{ \mathcal{H}}(y,\bar{\eta },\eta _{n}) = (\tilde{\mathcal{E}}(y,\bar{\eta }),\eta _{n})}$$

where x ∈ M, $\eta = (\bar{\eta },\eta _{n})$, with $\bar{\eta }\in B(0,R) \subset \mathbb{R}^{n-1}$ and 0 ≤ η _n  < R, $\xi (y) =\exp _{ \xi _{0}}^{\partial }y \in \partial M$ and $x =\psi _{ \xi _{0}}^{\partial }(\eta )$.

Lemma 5.5.

It holds

$$\displaystyle{\frac{\partial \tilde{\mathcal{E}}_{k}} {\partial y_{j}} (0,0) = -\delta _{jk}\text{ for }j,k = 1,\ldots,n - 1}$$

Proof.

We recall that $\tilde{\mathcal{E}}(y,\bar{\eta }) = \left (\exp _{\xi (y)}^{\partial }\right )^{-1}(\exp _{\xi _{0}}^{\partial }\bar{\eta })$. Let us introduce, for $y,\bar{\eta }\in B(0,R) \subset \mathbb{R}^{n-1}$

$$\displaystyle\begin{array}{rcl} F(y,\bar{\eta })& =& \left (\exp _{\xi _{0}}^{\partial }\right )^{-1}\left (\exp _{\xi (y)}^{\partial }(\bar{\eta })\right ) {}\\ \varGamma (y,\bar{\eta })& =& \left (y,F(y,\bar{\eta })\right ). {}\\ \end{array}$$

We notice that $\varGamma ^{-1} = (y,\tilde{\mathcal{E}}(y,\bar{\eta }))$. We can easily compute the derivative of Γ. We have

$$\displaystyle{\varGamma '(\hat{y},\hat{\eta })[\tilde{y},\tilde{\eta }] = \left (\begin{array}{cc} \text{Id}_{\mathbb{R}^{n-1}} & 0 \\ F_{y}'(\hat{y},\hat{\eta })&F_{\eta }'(\hat{y},\hat{\eta }) \end{array} \right )\left (\begin{array}{c} \tilde{y}\\ \tilde{\eta } \end{array} \right ),}$$

thus

$$\displaystyle{\left (\varGamma ^{-1}\right )^{{\prime}}(\hat{y},\hat{\eta })[\tilde{y},\tilde{\eta }] = \left (\begin{array}{cc} \text{Id}_{\mathbb{R}^{n-1}} & 0 \\ -\left (F_{\eta }'(\hat{y},\hat{\eta })\right )^{-1}F_{y}'(\hat{y},\hat{\eta })&\left (F_{\eta }'(\hat{y},\hat{\eta })\right )^{-1} \end{array} \right )\left (\begin{array}{c} \tilde{y}\\ \tilde{\eta } \end{array} \right )}$$

Now, by direct computation we have that

$$\displaystyle{F_{\eta }'(0,\hat{\eta }) = \text{Id}_{\mathbb{R}^{n-1}}\text{ and }F_{y}'(\hat{y},0) = \text{Id}_{\mathbb{R}^{n-1}},}$$

so $\frac{\partial \tilde{\mathcal{E}}_{k}} {\partial y_{j}} (0,0) = \left (-\left (F_{\eta }'(0,0)\right )^{-1}F_{y}'(0,0)\right )_{jk} = -\delta _{jk}$. □ 

Lemma 5.6.

We have that

$$\displaystyle\begin{array}{rcl} \tilde{\mathcal{H}}(0,\bar{\eta },\eta _{n})& =& (\bar{\eta },\eta _{n})\text{ for }\bar{\eta } \in \mathbb{R}^{n-1},\eta _{ n} \in \mathbb{R}_{+} {}\\ \frac{\partial \tilde{\mathcal{H}}_{k}} {\partial y_{j}} (0,0,\eta _{n})& =& -\delta _{jk}\text{ for }j,k = 1,\ldots,n - 1,\eta _{n} \in \mathbb{R}_{+} {}\\ \frac{\partial \tilde{\mathcal{H}}_{n}} {\partial y_{j}} (y,\bar{\eta },\eta _{n})& =& 0\text{ for }j = 1,\ldots,n - 1,y,\bar{\eta }\in \mathbb{R}^{n-1},\eta _{ n} \in \mathbb{R}_{+} {}\\ \end{array}$$

Proof.

The first two claims follow immediately by Definition 5.4 and Lemma 5.5. For the last claim, observe that $\tilde{\mathcal{H}}_{k}(y,\bar{\eta },\eta _{n}) =\tilde{ \mathcal{E}}_{k}(y,\bar{\eta })$ which does not depend on η _n as well as its derivatives. □