Spikes of the two-component elliptic system in ${\mathbb {R}}^4$ with the critical Sobolev exponent

Abstract

Consider the following elliptic system:

$$\begin{aligned} \left\{ \begin{aligned}&-\varepsilon ^2\Delta u_1+\lambda _1u_1=\mu _1u_1^3+\alpha _1u_1^{p-1}+\beta u_2^2u_1 \quad \text {in }\quad \Omega ,\\&-\varepsilon ^2\Delta u_2+\lambda _2u_2=\mu _2u_2^3+\alpha _2u_2^{p-1}+\beta u_1^2u_2 \quad \text {in }\quad \Omega ,\\&u_1,u_2>0\quad \text {in }\quad \Omega ,\quad u_1=u_2=0\quad \text {on }\quad \partial \Omega , \end{aligned}\right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^4$ is a bounded domain, $\lambda _i,\mu _i,\alpha _i>0$ $(i=1,2)$ and $\beta \not =0$ are constants, $\varepsilon >0$ is a small parameter and $2<p<2^*=4$. By using variational methods, we study the existence of ground state solutions to this system for sufficiently small $\varepsilon >0$. The concentration behaviors of least-energy solutions as $\varepsilon \rightarrow 0^+$ are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as $\varepsilon \rightarrow 0^+$.

Appendix

In this section, we will list known results that will be used frequently in this paper. Let $\mathcal {H}_{i,{\mathbb {R}}^4}$ be the Hilbert space $H^1({\mathbb {R}}^4)$ that is equipped with the inner product

$$\begin{aligned} \langle u,v\rangle _{i,{\mathbb {R}}^4}=\int _{{\mathbb {R}}^4}\nabla u\nabla v+\lambda _i uv dx. \end{aligned}$$

The corresponding norm is specified by $\Vert u\Vert _{i,{\mathbb {R}}^4}=\langle u,u\rangle _{i,{\mathbb {R}}^4}^{\frac{1}{2}}$. Define

$$\begin{aligned} \mathcal {E}_{i,{\mathbb {R}}^4}(u)=\frac{1}{2}\Vert u\Vert _{i,{\mathbb {R}}^4}^2-\frac{\alpha _i}{p}\Vert u\Vert _{L^p({\mathbb {R}}^4)}^p-\frac{\mu _i}{4}\Vert u\Vert _{L^4({\mathbb {R}}^4)}^4. \end{aligned}$$

(6.1)

Then it is well known that $\mathcal {E}_{i,{\mathbb {R}}^4}(u)$ is of class $C^2$ in $\mathcal {H}_{i,{\mathbb {R}}^4}$. Set

$$\begin{aligned} \mathcal {M}_{i,{\mathbb {R}}^4}=\left\{ u\in \mathcal {H}_{i,{\mathbb {R}}^4}\backslash \{0\}\mid \mathcal {E}_{i,{\mathbb {R}}^4}'(u)u=0\right\} . \end{aligned}$$

(6.2)

and define

$$\begin{aligned} d_{i,{\mathbb {R}}^4}=\inf _{\mathcal {M}_{i,{\mathbb {R}}^4}}\mathcal {E}_{i,{\mathbb {R}}^4}(u). \end{aligned}$$

(6.3)

Proposition 6.1

$0< d_{i,{\mathbb {R}}^4}<\frac{1}{4\mu _i}\mathcal {S}^2$ holds for both $i=1,2$, where $\mathcal {S}$ is the optimal embedding constant from $H^1({\mathbb {R}}^4)\rightarrow L^4({\mathbb {R}}^4)$, which is defined by

$$\begin{aligned} \mathcal {S}=\inf \left\{ \Vert \nabla u\Vert _{L^2({\mathbb {R}}^4)}^2\mid u\in H^1({\mathbb {R}}^4), \Vert u\Vert _{L^4({\mathbb {R}}^4)}^2=1\right\} . \end{aligned}$$

Moreover, $d_{i,{\mathbb {R}}^4}$ is attained by some $U_{i,{\mathbb {R}}^4}\in \mathcal {M}_{i,{\mathbb {R}}^4}$, which is also a solution of the following equation

Proof

See the results in [50]. $\square$

Let $\Omega \subset {\mathbb {R}}^4$ be a bounded domain, $\lambda _i,\mu _i,\alpha _i>0(i=1,2)$ constants, $\varepsilon >0$ a small parameter and $2<p<2^*=4$. Let

$$\begin{aligned} \mathcal {E}_{i,\varepsilon ,\Omega }(u)=\frac{1}{2}\Vert u\Vert _{i,\varepsilon ,\Omega }^2-\frac{\alpha _i}{p}\Vert u\Vert _{L^p(\Omega )}^p-\frac{\mu _i}{4}\Vert u\Vert _{L^4(\Omega )}^4. \end{aligned}$$

(6.4)

Then it is well known that $\mathcal {E}_{i,\varepsilon ,\Omega }(u)$ is of class $C^2$ in $\mathcal {H}_{i,\varepsilon ,\Omega }$. Set

$$\begin{aligned} \mathcal {M}_{i,\varepsilon ,\Omega }=\left\{ u\in \mathcal {H}_{i,\varepsilon ,\Omega }\backslash \{0\}\mid \mathcal {E}_{i,\varepsilon ,\Omega }'(u)u=0\right\} \end{aligned}$$

(6.5)

and define

$$\begin{aligned} d_{i,\varepsilon ,\Omega }=\inf _{\mathcal {M}_{i,\varepsilon ,\Omega }}\mathcal {E}_{i,\varepsilon ,\Omega }(u),\quad i=1, 2. \end{aligned}$$

(6.6)

Proposition 6.2

Let $\varepsilon >0$ be sufficiently small. Then, $\varepsilon ^4C'\le d_{i,\varepsilon ,\Omega }\le \frac{\varepsilon ^4}{4\mu _i}\mathcal {S}^2-\varepsilon ^4C$ for both $i=1,2$. Moreover, there exists ${\widetilde{U}}_{i,\varepsilon }\in \mathcal {M}_{i,\varepsilon ,\Omega }$ such that $\mathcal {E}_{i,\varepsilon ,\Omega }({\widetilde{U}}_{i,\varepsilon })=d_{i,\varepsilon ,\Omega }$, which is also a solution of the following equation:

$i=1,2$, Moreover, ${\widetilde{U}}_{i,\varepsilon }(\varepsilon y+q_i^\varepsilon )\rightarrow v_i^0$ strongly in $\mathcal {H}_{i,{\mathbb {R}}^4}$ as $\varepsilon \rightarrow 0^+$ and $\{{\widetilde{U}}_{i,\varepsilon }\}$ is uniformly bounded in $L^\infty (\Omega )$, where $q_i^\varepsilon$ is the maximum point of ${\widetilde{U}}_{i,\varepsilon }$ and $v_i^0$ is a least-energy solution of $(\mathcal {P}_{i})$.

Proof

By the results in [8], $\varepsilon ^4 C'\le d_{i,\varepsilon ,\Omega }\le \frac{\varepsilon ^4}{4\mu _i}\mathcal {S}^2-\varepsilon ^4C$ for both $i=1,2$. Regarding the remaining results, we believe that they exist but we can not find the references; thus, we will sketch their proofs here. We only present the proof for $(\mathcal {P}_{1,\varepsilon })$ since that of $(\mathcal {P}_{2,\varepsilon })$ is similar. Once again, by the result in [8], $d_{1,\varepsilon ,\Omega }<\frac{\varepsilon ^4}{4\mu _1}\mathcal {S}^2-\varepsilon ^4C$ for sufficiently small $\varepsilon >0$. Since $p>2$, $\mathcal {M}_{i,\varepsilon ,\Omega }$ is a natural constraint. Hence, by applying the concentration-compactness principle in a standard way, we can show that $(\mathcal {P}_{i,\varepsilon })$ has a least-energy solution ${\widetilde{U}}_{i,\varepsilon }$ that satisfies ${\widetilde{U}}_{i,\varepsilon }\in \mathcal {M}_{i,\varepsilon ,\Omega }$ and $\mathcal {E}_{i,\varepsilon ,\Omega }({\widetilde{U}}_{i,\varepsilon })=d_{i,\varepsilon ,\Omega }$. Now, let us consider the functions ${\widetilde{U}}_{i,\varepsilon }(\varepsilon y+q_i^\varepsilon )$. By a similar argument to that used for Proposition 3.2, we can show that the only solution of the following equation is $u\equiv 0$:

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\lambda _1u=\mu _1u^3+\alpha _1u^{p-1}\quad \text {in }\quad {\mathbb {R}}^4_+,\\&u\ge 0\quad \text {in }\quad {\mathbb {R}}^4_+,\quad u=0\quad \text {on }\quad \partial {\mathbb {R}}^4_+.\end{aligned}\right. \end{aligned}$$

Thus, if $\Omega ^*_{i,\varepsilon }\rightarrow {\mathbb {R}}^4_+$ as $\varepsilon \rightarrow 0^+$, then by a similar argument to that used for Lemma 4.2, we can obtain a contradiction since $d_{1,\varepsilon ,\Omega }<\frac{\varepsilon ^4}{4\mu _1}\mathcal {S}^2-\varepsilon ^4C$ for $\varepsilon >0$ sufficiently small. Hence, $\Omega ^*_{i,\varepsilon }\rightarrow {\mathbb {R}}^4$ as $\varepsilon \rightarrow 0^+$. Now, by the result in [8] and a similar argument to that used in Case. 1 of the proof to Proposition 4.1, we can show that ${\widetilde{U}}_{i,\varepsilon }(\varepsilon y+q_i^\varepsilon )\rightarrow v_i^0$ strongly in $\mathcal {H}_{i,{\mathbb {R}}^4}$ as $\varepsilon \rightarrow 0^+$, where $v_i^0$ is a least-energy solution of $(\mathcal {P}_{i})$. The uniform boundedness of $\{{\widetilde{U}}_{i,\varepsilon }\}$ in $L^\infty (\Omega )$ can be obtained via standard elliptic estimates (cf. [8, 10]). $\square$

Let

$$\begin{aligned} \mathcal {I}_{\varepsilon }(\overrightarrow{{\mathbf {u}}})=\sum _{i=1}^2\left( \frac{\varepsilon ^2}{2}\Vert \nabla u_i\Vert _{L^2({\mathbb {R}}^4)}^2-\frac{\mu _i}{4}\Vert u_i\Vert _{L^4({\mathbb {R}}^4)}^4\right) -\frac{\beta }{2}\Vert u_1u_2\Vert _{L^2({\mathbb {R}}^4)}^2. \end{aligned}$$

(6.7)

Then $\mathcal {I}_{\varepsilon }$ is of class $C^2$ in $\mathcal {D}=D^{1,2}({\mathbb {R}}^4)\times D^{1,2}({\mathbb {R}}^4)$. Set $A_\varepsilon =\inf _{\mathcal {V}_{\varepsilon }}\mathcal {I}_{\varepsilon }(\overrightarrow{{\mathbf {u}}})$, where

$$\begin{aligned} \mathcal {V}_{\varepsilon }=\left\{ \overrightarrow{{\mathbf {u}}}\in \widetilde{\mathcal {D}}\mid \mathcal {I}'_{\varepsilon }(\overrightarrow{{\mathbf {u}}})\overrightarrow{{\mathbf {u}}}_1=\mathcal {I}'_{\varepsilon }(\overrightarrow{{\mathbf {u}}})\overrightarrow{{\mathbf {u}}}_2=0\right\} \end{aligned}$$

(6.8)

with $\widetilde{\mathcal {D}}=(D^{1,2}({\mathbb {R}}^4)\backslash \{0\})\times (D^{1,2}({\mathbb {R}}^4)\backslash \{0\})$.

Proposition 6.3

$A_\varepsilon =\varepsilon ^4A_1$ holds. Moreover, $A_1=\frac{1}{4\mu _1}\mathcal {S}^2+\frac{1}{4\mu _2}\mathcal {S}^2$ for $\beta <0$ and $A_1=\frac{k_1+k_2}{4}\mathcal {S}^2$ for $0<\beta <\min \{\mu _1,\mu _2\}$ or $\beta >\max \{\mu _1,\mu _2\}$ with $k_1,k_2$ satisfying

$$\begin{aligned} \left\{ \begin{aligned} \mu _1k_1+\beta k_2&=1,\\ \mu _2k_2+\beta k_1&=1.\end{aligned}\right. \end{aligned}$$

(6.9)

Proof

See the results in [10]. $\square$