Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

Abstract

In this paper we deal with the nonlinear Schrödinger system

$$\begin{aligned} -\Delta u_i =\mu _i u_i^3 + \beta u_i \sum _{j\ne i} u_j^2 + \lambda _i u_i, \qquad u_1,\ldots , u_m\in H^1_0(\Omega ) \end{aligned}$$

in dimension 4, a problem with critical Sobolev exponent. In the competitive case (\(\beta <0\) fixed or \(\beta \rightarrow -\infty \)) or in the weakly cooperative case (\(\beta \ge 0\) small), we construct, under suitable assumptions on the Robin function associated to the domain \(\Omega \), families of positive solutions which blowup and concentrate at different points as \(\lambda _1,\ldots , \lambda _m\rightarrow 0\). This problem can be seen as a generalization for systems of a Brezis–Nirenberg type problem.

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Acknowledgments

Angela Pistoia was supported by GNAMPA and Sapienza Fondi di Ricerca. Hugo Tavares was partially supported by Fundação para a Ciência e Tecnologia through the program Investigador FCT and the project PEst-OE/EEI/-LA0009/2013, as well as by the ERC Advanced Grant 2013 N.339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”.

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Correspondence to Hugo Tavares.

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Dedicated to Prof. Paul Rabinowitz, with profound esteem and admiration.

Appendix A: Auxiliary estimates

Appendix A: Auxiliary estimates

We start by recalling here the notations

$$\begin{aligned}&U_{\delta ,\xi }(x)=c_4\frac{\delta }{\delta ^2+|x-\xi |^2},\qquad \psi ^0_{\delta ,\xi }=\delta \frac{\partial U_{\delta ,\xi }}{\partial \delta } \quad \text {and} \qquad \\&\quad \psi ^j_{\delta ,\xi }=\delta \frac{\partial U_{\delta ,\xi }}{\partial \xi _j},\ j=1,\ldots , 4. \end{aligned}$$

Lemma A.1

Given any \(K\Subset \Omega \subset \mathbb {R}^4\), we have

$$\begin{aligned} PU_{\delta ,\xi }=U_{\delta ,\xi }-\delta A H(\cdot ,\xi )+R_{\delta ,\xi }, \end{aligned}$$
(A.1)

with

$$\begin{aligned} \Vert R_{\delta ,\xi }\Vert _\infty =\text {o}(\delta ),\quad \left\| \frac{\partial R_{\delta ,\xi }}{\partial \delta } \right\| _\infty =\text {o}(1),\quad \left\| \frac{\partial R_{\delta ,\xi }}{\partial \xi _j}\right\| _\infty =\text {o}(\delta ) \end{aligned}$$

as \(\delta \rightarrow 0\), uniformly for \(\xi \in K\), with

$$\begin{aligned} A:=\int _{\mathbb {R}^4} U_{1,0}^3=c_4^3\int _{\mathbb {R}^4}\frac{1}{(1+|y|^2)^3}=\frac{c_4}{\alpha _4}. \end{aligned}$$

In particular, we have

$$\begin{aligned} PU_{\delta ,\xi }=U_{\delta ,\xi }-\delta A H(\cdot ,\xi )+\text {o}(\delta ),\quad P\psi ^0_{\delta ,\xi }=\psi _{\delta ,\xi }^0-\delta A H(\cdot ,\xi )+\text {o}(\delta )\nonumber \\ \end{aligned}$$
(A.2)

and

$$\begin{aligned} P\psi _{\delta ,\xi }^j=\psi _{\delta ,\xi }^j-\delta ^2 A\frac{\partial H}{\partial \xi }(\cdot ,\xi )+\text {o}(\delta ^2), \quad j=1,\ldots , 4, \end{aligned}$$
(A.3)

as \(\delta \rightarrow 0\), uniformly for \(\xi \in K\).

Proof

See Proposition 1 in [27]. \(\square \)

Lemma A.2

Take \(K\Subset \Omega \subset \mathbb {R}^4\). Then for every \(0<p< 2\),

$$\begin{aligned} \int _\Omega U_{\delta ,\xi }^p=\text {O}(\delta ^p) \quad \text { as } \;\delta \rightarrow 0, \end{aligned}$$

while

$$\begin{aligned} \int _\Omega U_{\delta ,\xi }^2= c_4^2\delta ^2 \omega _3 |\ln \delta |+\text {O}(\delta ^2) \quad \text { as } \;\delta \rightarrow 0 \end{aligned}$$

and, for \(2<p<4\),

$$\begin{aligned} \int _\Omega U_{\delta ,\xi }^p= \text {O}(\delta ^{4-p}) \quad \text { as } \;\delta \rightarrow 0, \end{aligned}$$

uniformly for \(\xi \in K\), where \(\omega _3\) denotes the measure of the unit sphere \(S^3\subset \mathbb {R}^4\).

Proof

Take \(R={{\mathrm{dist}}}(\xi ,\partial \Omega )/2\). Then

$$\begin{aligned} \int _\Omega U_{\delta ,\xi }^p&=c_4^p \int _{B_{R}(\xi )} \frac{\delta ^p}{(\delta ^2+|x-\xi |^2)^p}+\text {O}(\delta ^p)\\&=c_4^p \delta ^{4-p} \omega _3 \int _0^{R/\delta } \frac{t^3}{(1+t^2)^p}\, \mathrm{d}t+\text {O}(\delta ^p). \end{aligned}$$

We have \(t^3/(1+t^2)^p\le t^{3-2p}\) for every \(t\ge 0\). Thus, if \(p\ne 3/2\),

$$\begin{aligned} \int _\Omega U_{\delta ,\xi }^p=\text {O}(\delta ^{4-p}) + c_4^p \delta ^{4-p} \omega _3 \left[ t^{4-2p}\right] _{t=1}^{t=R/\delta }+\text {O}(\delta ^p)=\text {O}(\delta ^{4-p})+\text {O}(\delta ^p). \end{aligned}$$

Thus \(\int _\Omega U_{\delta ,\xi }^p=\text {O}(\delta ^p)\) if \(0<p<2\), and \(\int _\Omega U_{\delta ,\xi }^p=\text {O}(\delta ^{4-p})\) if \(2<p<4\).

Likewise, we have that

$$\begin{aligned} \int _\Omega U_{\delta ,\xi }^2&= c_4^2 \delta ^2 \omega _3 \int _0^{R/\delta } \frac{t^3}{(1+t^2)^2}\, \mathrm{d}t + \text {O}(\delta ^2)\\&=\frac{c_4^2}{2}\delta ^2 \omega _3 \left[ \frac{1}{1+t^2}+\ln (1+t^2)\right] ^{t=R/\delta }_{t=0}+\text {O}(\delta ^2)\\&=-c_4^2 \delta ^2 \omega _3 \ln \delta +\text {O}(\delta ^2), \end{aligned}$$

as \(\delta \rightarrow 0\). \(\square \)

Next we will present some asymptotic estimates related to the competition term of the system under consideration. For that, we will need the following simple pointwise estimates.

Lemma A.3

Let \(F:\mathbb {R}\rightarrow \mathbb {R}\) be defined by \(F(s)=(s^+)^4/4\). Then there exists \(c>0\) such that, for every \(a,b\in \mathbb {R}\),

$$\begin{aligned} |F(a+b)-F(a)-F'(a)b|\le & {} c (a^2b^2 +b^4), \\ |F'(a+b)-F'(a)-F''(a)b|\le & {} c(|a| b^2 +|b|^3), \end{aligned}$$

and

$$\begin{aligned} |F''(a+b)-F''(a)|\le c (|a| |b| + b^2). \end{aligned}$$

Moreover, for every \(p>1\) there exists \(C>0\) such that

$$\begin{aligned} \left| |a+b|^p-|a|^p\right| \le C \left( |a|^{p-1}|b| + |b|^p \right) , \end{aligned}$$

for every \(a,b\in \mathbb {R}\).

Proof

The proof is very elementary, and follows simply from a Taylor’s expansion with Lagrange-type remainder. \(\square \)

Lemma A.4

Given \(p,q>0\) and \(\eta >0\) small, we have that

$$\begin{aligned} \int _\Omega U_{\delta _1,\xi _1}^p U_{\delta _2,\xi _2}^q \le \text {O}(\delta _2^q)\int _\Omega U_{\delta _1,\xi _1}^p + \text {O}(\delta _1^p) \int _\Omega U_{\delta _2,\xi _2}^q+\text {O}(\delta _1^p\delta _2^q). \end{aligned}$$

as \((\delta _1,\delta _2)\rightarrow (0,0)\), uniformly for all \(\xi _1,\xi _2\in \Omega \) such that \(|\xi _1-\xi _2|\ge 2\eta \), \({{\mathrm{dist}}}(\xi _1,\partial \Omega ),{{\mathrm{dist}}}(\xi _2,\partial \Omega )\ge 2\eta \).

Proof

This is a simple consequence of the fact that

$$\begin{aligned}&\int _\Omega U_{\delta _1,\xi _1}^p U_{\delta _2,\xi _2}^q = \int _{B_\eta (\xi _1)} U_{\delta _1,\xi _1}^p U_{\delta _2,\xi _2}^q+ \int _{B_\eta (\xi _2)} U_{\delta _1,\xi _1}^p U_{\delta _2,\xi _2}^q\\&\quad +\int _{\Omega {\setminus } (B_\eta (\xi _1)\cup B_{\eta }(\xi _2))}U_{\delta _1,\xi _1}^p U_{\delta _2,\xi _2}^q \end{aligned}$$

together with the fact that \(U_{\delta _i,\xi _i}\le C \delta _i\) on \(\Omega {\setminus } B_\eta (\xi _i)\), for some \(C>0\) independent of \(\delta _i\). \(\square \)

Lemma A.5

Given \(j=0,\ldots , 4\) and \(\eta >0\) small, we have

$$\begin{aligned}&\left\| (PU_{\delta _2,\xi _2})^2(P\psi _{\delta _1,\xi _1}^j) \right\| _{4/3}=\text {O}(\delta _{1}\delta _2) \quad \text { and }\quad \\&\quad \left\| (PU_{\delta _1,\xi _1})(PU_{\delta _2,\xi _2})(P\psi _{\delta _1,\xi _1}^j)\right\| _{4/3}=\text {O}(\delta _{1}\delta _2) \end{aligned}$$

as \((\delta _1,\delta _2)\rightarrow (0,0)\), uniformly for all \(\xi _1,\xi _2\in \Omega \) such that \(|\xi _1-\xi _2|\ge 2\eta \), \({{\mathrm{dist}}}(\xi _1,\partial \Omega ),{{\mathrm{dist}}}(\xi _2,\partial \Omega )\ge 2\eta \).

Proof

To simplify notations, define \(U_i:=U_{\delta _i,\xi _i}\) and \(\psi _i^j:=\psi ^j_{\delta _i,\xi _i}\), for \(i=1,2\), \(j=0,\ldots , 4\). We have (\(j=0\))

$$\begin{aligned} \left\| (PU_2)^2(P\psi _1^0) \right\| _{4/3}^{4/3}=\int _\Omega |U_{2}|^{8/3}|\psi ^0_1|^{4/3}+R_{\varvec{\delta },\varvec{\xi }}, \end{aligned}$$

with

$$\begin{aligned} R_{\varvec{\delta },\varvec{\xi }}= & {} \int _\Omega \left[ (PU_2)^{8/3}-U_2^{8/3}\right] \left[ |P\psi _1^0|^{4/3}-|\psi _1^0|^{4/3} \right] \\&+\int _\Omega U_2^{8/3}\left[ |P\psi _1^0|^{4/3}-|\psi _1^0|^{4/3} \right] +\;\int _\Omega |\psi _1^0|^{4/3}\left[ (PU_2)^{8/3}-U_2^{8/3}\right] , \end{aligned}$$

so that, by taking in consideration Lemmas A.1, A.3 and A.4, and the fact that \(|\psi _1^0|\le C U_1\) for some \(C>0\), we have

$$\begin{aligned} |R_{\varvec{\delta },\varvec{\xi }}|\le & {} \int _\Omega \left| U_2^{5/3}\text {O}(\delta _2) +\text {O}(\delta _2^{8/3}) \right| \left| |\psi _1^0|^{1/3}\text {O}(\delta _1)+\text {O}(\delta _1^{4/3})\right| \\&+\int _\Omega U_2^{8/3} \left| |\psi _1^0|^{1/3}\text {O}(\delta _1) + \text {O}(\delta _1^{4/3})\right| \\&+\;\int _\Omega |\psi _1^0|^{4/3}\left| U_2^{5/3}\text {O}(\delta _2)+\text {O}(\delta _2^{8/3})\right| =\text {O}(\delta _1^{4/3}\delta _2^{4/3}) \end{aligned}$$

Since also

$$\begin{aligned} \int _\Omega (U_2)^{8/3}|\psi _1^0|^{4/3}&\le C \int _\Omega (U_2)^{8/3}(U_1)^{4/3}\le \text {O}(\delta _1^{4/3})\int _\Omega (U_2)^{8/3}+\text {O}(\delta _2^{8/3})\\&\quad \times \int _\Omega (U_1)^{4/3}+\text {O}(\delta _1^{4/3}\delta _2^{4/3})=\text {O}(\delta _1^{4/3}\delta _2^{4/3}), \end{aligned}$$

we have

$$\begin{aligned} \left\| (PU_2)^2(P\psi _1^0)\right\| _{4/3}^{4/3}=\text {O}(\delta _1^{4/3}\delta _2^{4/3}). \end{aligned}$$

For \(j\ge 1\), we can reason in an analogous way: by using the fact that \(|\psi _1^j|\le C U_1^2\) for some \(C>0\), we obtain

$$\begin{aligned} \Vert (PU_2)^2(P\psi _1^j)\Vert _{4/3}^{4/3}=\int _\Omega (U_{2})^{8/3}|\psi ^j_1|^{4/3}+\text {O}(\delta _1^{4/3}\delta _2^{4/3})=\text {O}(\delta _1^{4/3}\delta _2^{4/3}). \end{aligned}$$

As for the second conclusion of the lemma, reasoning in the same line, we write (\(j=0\))

$$\begin{aligned} \Vert (PU_1)(PU_2)(P\psi _1^0)\Vert _{4/3}^{4/3}=\int _\Omega (U_{1})^{4/3} (U_2)^{4/3}|\psi ^0_1|^{4/3}+{\tilde{R}}_{\varvec{\delta },\varvec{\xi }}, \end{aligned}$$

with

$$\begin{aligned} {\tilde{R}}_{\varvec{\delta },\varvec{\xi }}= & {} \int _\Omega \left[ (PU_1)^{4/3}-U_1^{4/3}\right] \left[ (PU_2)^{4/3}-U_2^{4/3}\right] \left[ |P\psi _1^0|^{4/3}-|\psi _1^0|^{4/3}\right] \\&+\;\int _\Omega U_1^{4/3} U_2^{4/3} \left[ |P\psi _1^0|^{4/3}-|\psi _1^0|^{4/3}\right] \\&+ \int _\Omega U_1^{4/3} \left[ (PU_2)^{4/3}-U_2^{4/3}\right] |\psi _1^0|^{4/3}\\&+\;\int _\Omega \left[ (PU_1)^{4/3}-U_1^{4/3}\right] U_2^{4/3} |\psi _1^0|^{4/3}+\int _\Omega \left[ (PU_1)^{4/3}-U_1^{4/3}\right] \\&\times \left[ (PU_2)^{4/3}-U_2^{4/3}\right] |\psi _1^0|^{4/3}\\&+\;\int _\Omega \left[ (PU_1)^{4/3}-U_1^{4/3}\right] U_2^{4/3} \left[ |P\psi _1^0|^{4/3}-|\psi _1^0|^{4/3}\right] \\&+\;\int _\Omega U_1^{4/3} \left[ (PU_2)^{4/3}-U_2^{4/3}\right] \left[ |P\psi _1^0|^{4/3}-|\psi _1^0|^{4/3}\right] \end{aligned}$$

By using once again Lemmas A.1, A.3 and A.4, we can prove that \({\tilde{R}}_{\varvec{\delta },\varvec{\xi }}=\text {O}(\delta _1^{4/3}\delta _2^{4/3})\). Since moreover (by Lemma A.4)

$$\begin{aligned} \int _\Omega U_{1}^{4/3} U_2^{4/3}|\psi ^0_1|^{4/3}\le C\int _\Omega U_1^{8/3}U_2^{4/3}=\text {O}(\delta _1^{4/3}\delta _2^{4/3}), \end{aligned}$$

we conclude, as wanted, that

$$\begin{aligned} \Vert (PU_1)(PU_2)(P\psi _1^0)\Vert _{4/3}^{4/3}=\text {O}(\delta _1^{4/3}\delta _2^{4/3}). \end{aligned}$$

The fact that

$$\begin{aligned} \Vert (PU_1)(PU_2)(P\psi _1^j)\Vert _{4/3}^{4/3}=\text {O}(\delta _1^{4/3}\delta _2^{4/3}) \quad \text { for } j=1,\ldots ,4, \end{aligned}$$

follows in an analogous way, using this time the estimate: \(|\psi _1^j|\le C U_1^{2}\), for some \(C>0\). \(\square \)

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Pistoia, A., Tavares, H. Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions. J. Fixed Point Theory Appl. 19, 407–446 (2017). https://doi.org/10.1007/s11784-016-0360-6

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Keywords

  • Blowup and concentrating solutions
  • Brezis–Nirenberg type problems
  • Competitive and weakly cooperative systems
  • Critical Sobolev Exponent
  • Cubic Schrödinger systems
  • Lyapunov–Schmidt reduction

Mathematics Subject Classification

  • 35A15
  • 35J20
  • 35J47