Blow up of the solutions to a linear elliptic system involving Schr{\"o}dinger operators

TL;DR

This paper investigates the blow-up behavior of solutions to a 2x2 linear elliptic system involving Schr{"o}dinger operators as a parameter approaches a critical eigenvalue, considering potentials with superquadratic growth.

Contribution

It provides a detailed analysis of solution blow-up in a linear elliptic system with Schr{"o}dinger operators, including cases with double eigenvalues and superquadratic potentials.

Findings

Solutions blow up as the parameter approaches the principal eigenvalue.

The analysis covers systems with constant coefficient matrices and superquadratic potentials.

The behavior is characterized near the critical eigenvalue.

Abstract

We show how the solutions to a $2\times2$ linear system involving Schr{\"o}dinger operators blow up as the parameter $\mu$ tends to some critical value which is the principal eigenvalue of the system; here the potential is continuous positive with superquadratic growth and the square matrix of the system is with constant coefficients and may have a double eigenvalue.

Equations

$$LU:=(-\Delta +q(x))U=AU+\mu U+F(x)in{\mathbb{R}}^N,$$

$$U(x{)}_{|x|\to \infty }\to 0$$

$$Lu:=(-\Delta +q(x))u=\sigma u+f(x)\in {\mathbb{R}}^N,$$

$$\underset{|x|\Rightarrow +\infty }{\lim }u(x)=0.$$

$$f\ge 0,\not\equiv 0\Rightarrow u>0.$$

$$\mathcal{P}:=\{Q\in \mathcal{C}({\mathbb{R}}_{+},(0,\infty ))/\exists R_0>0,Q^{\prime }>0a.e.on[R_0,\infty ),{\int }_{R_0}^{\infty }Q(r{)}^{-1/2}<\infty \}.$$

$$Q_1(|x|)\le q(x)\le Q_2(|x|)\le C_0{Q}_1(|x|),\forall x\in {\mathbb{R}}^N,$$

$$\int_{R_{0}}^{\infty}(Q_{2}(s)-Q_{1}(s))\int_{R_{0}}^{s}exp\big{(}-\int_{r}^{s}[Q_{1}(t)^{1/2}+Q_{2}(t)^{1/2}]dt\big{)}drds<\infty.$$

$$u>c\varphi .$$

$$u\le \frac{C}{\Lambda -\sigma }\varphi .$$

$$\mathcal{X}:=\{h\in L^2({\mathbb{R}}^N):h/\varphi \in L^{\infty }({\mathbb{R}}^N)\},$$

$$u\le -c^{\prime }\varphi .$$

$$0<\frac{k^{\prime }}{\Lambda -\sigma }\varphi <u<\frac{K^{\prime }}{\Lambda -\sigma }\varphi .$$

$$-\frac{k"}{\Lambda -\sigma }\varphi <-u<-\frac{K"}{\Lambda -\sigma }\varphi <0.$$

$$u=u^1\varphi +u^{\perp };f=f^1\varphi +f^{\perp }.$$

$$L{u}^{\perp }=\sigma u^{\perp }+f^{\perp }$$

$$L{u}^1\varphi =\Lambda u^1\varphi =\sigma u^1\varphi +f^1\varphi .$$

$$u^1=\frac{f^1}{(\Lambda -\sigma )}\to \pm \infty as(\Lambda -\sigma )\to 0.$$

$$\frac{K^{\prime }}{\Lambda -\sigma }\varphi <u;u<\frac{K"}{\Lambda -\sigma }\varphi .$$

$$LU=AU+\mu U+F(x)in{\mathbb{R}}^N.$$

$$\left\{\begin{array}[]{lcl}Lu_{1}&=&au_{1}+bu_{2}+\mu u_{1}+f_{1}(x)\\ Lu_{2}&=&cu_{1}+du_{2}+\mu u_{2}+f_{2}(x)\end{array}\right.\,\;in\;{\mathbb{R}}^{N},.$$

$$u_1(x),u_2(x{)}_{|x|\to \infty }\to 0.$$

$$A=\left(\begin{array}[]{cccc}a&b\\ c&d\end{array}\right)\,\mbox{ with }\,b>0\,{\rm and}\,D:=(a-d)^{2}+4bc\geq 0.$$

$${\xi }_1=\frac{a+d+\sqrt{D}}{2}\ge {\xi }_2=\frac{a+d-\sqrt{D}}{2}.$$

$$X_{k}=\left(\begin{array}[]{cccc}b\\ \xi_{k}-a\end{array}\right),$$

$$L(X\varphi )-AX\varphi =(\Lambda -{\xi }_1)X\varphi .$$

$$\nu =\Lambda -{\xi }_1$$

$$({\xi }_2-a)f_1^1<b{f}_2^1,$$

$$u_1,u_2\ge \frac{\gamma }{\nu -\mu }\varphi >0.$$

$$u_1,u_2\le -\frac{\gamma }{\nu -\mu }\varphi <0.$$

Full text

Blow up of the solutions to a linear elliptic

system involving Schrödinger operators

B.Alziary

J.Fleckinger

Institut de Mathématique, UMR5219

Ceremath - Université Toulouse 1

Abstract   We show how the solutions to a $2\times 2$ linear system involving Schrödinger operators blow up as the parameter $\mu$ tends to some critical value which is the principal eigenvalue of the system; here the potential is continuous positive with superquadratic growth and the square matrix of the system is with constant coefficients and may have a double eigenvalue.

1 Introduction

We study here the behavior of the solutions to a $2\times 2$ system (considered in its variational formulation):

[TABLE]

[TABLE]

where $q$ is a continuous positive potential tending to $+\infty$ at infinity with superquadratic growth; $U$ is a column vector with components $u_{1}$ and $u_{2}$ and $A$ is a $2\times 2$ square matrix with constant coefficients. $F$ is a column vector with components $f_{1}$ and $f_{2}$.

Such systems have been intensively studied mainly for $\mu=0$ and for $A$ with 2 distinct eigenvalues; here we consider also the case of a double eigenvalue. In both cases, we show the blow up of solutions as $\mu$ tends to some critical value $\nu$ which is the principal eigenvalue of System $(S)$. This extends to systems involving Schrödinger operators defined on ${\mathbb{R}}^{N}$ earlier results valid for systems involving the classical Laplacian defined on smooth bounded domains with Dirichlet boundary conditions.

This paper is organized as follows: In Section 2 we recall known results for one equation. In Section 3 we consider first the case where $A$ has two different eigenvalues and then we study the case of a double eigenvalue.

2 The equation

We shortly recall the case of one equation

[TABLE]

[TABLE]

$\sigma$ is a real parameter. Hypotheses

$(H_{q})$ $q$ is a positive continuous potential tending to $+\infty$ at infinity.

$(H_{f})$   $f\in L^{2}({\mathbb{R}}^{N})$, $f\geq 0$ and $f>0$ on some subset with positive Lebesgue measure.

It is well knwon that if $(H_{q})$ is satisfied, $L$ possesses an infinity of eigenvalues tending to $+\infty$: $\,0<\lambda_{1}<\lambda_{2}\leq\ldots.$ Notation: $(\Lambda,\phi)$   Denote by $\Lambda$ the smallest eigenvalue of $L$; it is positive and simple and denote by $\phi$ the associated eigenfunction, positive and with $L^{2}$-norm $\|\phi\|=1$. It is classical ([15], [19]) that if $f>0$ and $\sigma<\Lambda$ the positivity is improved, or in other words, the maximum principle (MP) is satisfied:

[TABLE]

Lately, for potentials growing fast enough (faster than the harmonic oscillator), another notion has been introduced ([4], [5], [9], [10]) which improves the maximum (or antimaximum principle): the ”groundstate positivity” (GSP) (resp. ” negativity” (GSN)) which means that there exists $k>0$ such that

$u>k\phi$ (GSP) (resp. $u<-k\phi$ (GSN))

.

We also say shortly ”fundamenal positivity” or” negativity”, or also ”$\phi$-positivity” or ”negativity”.

The first steps in this direction use a radial potential. Here we consider a small perturbation of a radial one as in [9]. The potential $q$   We define first a class ${\cal P}$ of radial potentials:

[TABLE]

The last inequality holds if $Q$ is growing sufficiently fast ($>r^{2}$). Now we give results of GSP or GSN for a potential $q$ which is a small perturbation of $Q$; we assume:

$(H^{\prime}_{q})$   $q$ satisfies $(H_{q})$ and there exists two functions $Q_{1}$ and $Q_{2}$ in ${\cal P}$, and two positive constants $R_{0}$ and $C_{0}$ such that

[TABLE]

[TABLE]

Denoting by $\Phi_{1}$ (resp. $\Phi_{2}$) the groundstate of $L_{1}:=-\Delta+Q_{1}$ (resp. $L_{2}=-\Delta+Q_{2}$), Corollary 3.3 in [9] says that all these groundstates are ”comparable” that is there exists constants $0<k_{1}\leq k_{2}\leq\infty$ such that $k_{1}\phi\leq\Phi_{1},\Phi_{2}\leq k_{2}\phi$. Finally

Theorem 1

(GSP) ([9])   If $(H^{\prime}_{q})$ and $(H_{f})$ are satisfied, then, for $\sigma<\Lambda$, there is a unique solution $u$ to $(E)$ which is positive, and there exists a constant $c>0$, such that

[TABLE]

Moreover, if also $f\leq C\phi$ with some constant $C>0$, then

[TABLE]

The space ${\cal X}\,$ It is convenient for several results to introduce the space of ”groundstate bounded functions”:

[TABLE]

equipped with the norm $\|h\|_{\cal X}=ess\sup_{{\mathbb{R}}^{n}}(|h|/\phi)$.

For a potential satisfying $(H^{\prime}_{q})$ and a function $f\in{\cal X}$, there is also a result of ”groundstate negativity” (GSN) for $(E)$; it is is an extension of the antimaximum principle, introduced by Clément and Peletier in 1978 ([13]) for the Laplacian when the parameter $\sigma$ crosses $\Lambda$.

Theorem 2

(GSN) ([9] )   Assume $(H^{\prime}_{q})$ and $(H_{f})$ are satisfied and $f\in{\cal X}$; then there exists $\delta(f)>0$ and a positive constant $c^{\prime}>0$ such that for all $\sigma\in(\Lambda,\Lambda+\delta)$,

[TABLE]

Remark 1

This holds also if we only assume $f^{1}:=\int f\phi>0.$

**Hypothesis $(H^{\prime}_{f})$ ** We consider now functions $f$ which are such that

$(H^{\prime}_{f})$:   $f\in{\cal X}$ and $f^{1}:=\int f\phi>0$.

Theorem 3

• Assume $(H^{\prime}_{q})$ and $(H^{\prime}_{f})$ are satisfied. Then there exists $\delta>0$ such that for $\Lambda-\delta<\sigma<\Lambda$ there exists positive constants $k^{\prime}$ and $K^{\prime}$, depending on $f$ and $\delta$ such that*

[TABLE]

If $\Lambda<\sigma<\Lambda+\delta$, there exists positive constants $k"$ and $K"$, depending on $f$ and $\delta$ such that

[TABLE]

This result extends earlier one in [17] and a a close result is Theorem 2.03 in [11]. It shows in particular that $u\in{\cal X}$ and $|u|\rightarrow\infty$ as $|\nu-\mu|\rightarrow 0$.

Proof:   Decompose $u$ and $f$ on $\phi$ and its orthogonal:

[TABLE]

We derive from $(E)$ $Lu=\sigma u+f$ :

[TABLE]

[TABLE]

We notice that since $q$ is smooth; so is $u$. Also, since $f\in{\cal X}$, $f^{\perp}$, $u$ and $u^{\perp}$ are also in ${\cal X}$ and hence are bounded. Choose $\sigma<\Lambda$ and assume $(H^{\prime}_{f})$. We derive from Equation (11) (by [6]Thm 3.2) that : $||u^{\perp}||_{{\cal X}}<K_{1}$. Therefore $|u^{\perp}|$ is bounded by some $cste.\phi>0$.

From Equation (12) we derive

[TABLE]

Choose $\Lambda-\delta$ small enough and $\sigma\in(\Lambda-\delta,\Lambda)$. Hence

[TABLE]

For $\sigma>\Lambda$. we do exactly the same, except that the signs are changed for $u^{1}$ in $(\ref{u1})$.

3 A $2\times 2$ Linear system

Consider now a linear system with constant coefficients.

[TABLE]

As above, $L:=-\Delta+q$ where the potential $q$ satisfies $(H^{\prime}_{q})$, and where $\mu$ is a real parameter. $L$ can be detailed as 2 equations:

[TABLE]

[TABLE]

Assume

[TABLE]

Note that $b>0$ does not play any role since we can always change the order of the equations.

The eigenvalues of $A$ are

[TABLE]

As far as we know, all the previous studies suppose that the largest eigenvalue $\xi_{1}$ is simple (i.e. $D=(a-d)^{2}+4bc>0$). Here we also study, in the second subsection, the case of a double eigenvalue $\xi_{1}=\xi_{2}$, that is $D=0$; this implies necessarily $bc<0$ and necessarily the matrix is not cooperative.

3.1 Case $\xi_{1}>\xi_{2}$

This is the classical case where $\xi_{1}$ is simple. Set $\xi_{1}>\xi_{2}$. The eigenvectors are

[TABLE]

Set $X:=X_{1}$.

As above, denote by $(\Lambda,\phi)$, $\phi>0$, the principal eigenpair of the operator $L=(-\Delta+q(x))$.

It is easy to see that

[TABLE]

Hence

[TABLE]

is the principal eigenvalue of $(S)$ with associated eigenvector $X\phi$. Note that the components of $X\phi$ do not change sign, but, in the case of a non cooperative matrix they are not necessarily both positive. We prove:

Theorem 4

• Assume $(H^{\prime}_{q})$and $(H_{A})$; $f_{1}$ and $f_{2}$ satisfy $(H^{\prime}_{f})$; assume also $D>0$ and $d-a>0$. If*

[TABLE]

there exists $\delta>0$, independant of $\mu$, such that if $\nu-\delta<\mu<\nu$, there exists a positive constant $\gamma$ depending only on $F$ such that

[TABLE]

If $\nu<\mu<\nu+\delta$, the sign are reversed:

[TABLE]

Remark 2

If $(H^{\prime}_{q})$ and $(H_{A})$ are satisfied; if $f_{1}$ and $f_{2}$ satisfy $(H^{\prime}_{f})$ as in Theorem 4, but if $d-a<0$ we have, if $(a-\xi_{2})f_{1}^{1}+bf_{2}^{1}>0$ and $\nu-\delta<\mu<\nu$

[TABLE]

Remark 3

It is noticeable that for all these cases, $|u_{1}|$, $|u_{2}|$ $\rightarrow+\infty$ as $|\nu-\mu|\rightarrow 0$.

These results extend Theorem 4.2 in [4].

**Proof:  ** As in [3], we use $J$ the associated Jordan matrix (which in this case is diagonal) and $P$ the change of basis matrix which are such that

[TABLE]

Here

[TABLE]

[TABLE]

Denoting $\tilde{U}=P^{-1}U$ and $\tilde{F}=P^{-1}F$, we derive from System $(S)$ (after multiplication by $\ P^{-1}U$ to the left):

[TABLE]

Since $J$ is diagonal we have two independant equations:

[TABLE]

The projection on $\phi$ and on its orthogonal for $k=1$ and $2$ gives

[TABLE]

hence

[TABLE]

[TABLE]

If both $f_{k}$ verify $(H^{\prime}_{f})$ , they are are in ${\cal X}$ and bounded and hence both ${\tilde{f}}_{k}^{\perp}$ are bounded; therefore, by (20) both ${\tilde{u}}_{k}^{\perp}$ are also bounded.

We derive from (19) that

[TABLE]

Consider again Equation (19) for $k=2$; obviously, $(\tilde{u}_{2})^{1}$ stays bounded as $\mu\rightarrow\nu=\Lambda-\xi_{1}$ and therefore $\tilde{u}_{2}$ stays bounded. .

For $k=1$, $(\tilde{u}_{1})^{1}=\frac{(\tilde{f}_{1})^{1}}{\nu-\mu}\rightarrow\infty$ as $\mu\rightarrow\nu=\Lambda-\xi_{1}$, since $(\tilde{f}_{1})^{1}=\frac{1}{\xi_{1}-\xi_{2}}((a-\xi_{2})f_{1}^{1}+bf_{2}^{1})>0$,; this is the condition which appears in Theorem 4. Then, we simply apply Theorem 3 to (18) for $k=1$ and deduce that there existes $\delta>0$, such that, for $|\Lambda-\xi_{1}-\mu|=|\nu-\mu|<\delta$, there exists a positive constant $C>0$ such that $\tilde{u}_{1}\phi\geq\frac{C}{\nu-\mu}\phi$. Now, it follows from $U=P\tilde{U}$, that

[TABLE]

As $\nu-\mu\rightarrow 0$, since $\tilde{u}_{2}$ stays bounded, $u_{1}$ behaves as $b(\tilde{u}_{1})^{1}\phi$, $u_{2}$ as $(\xi_{1}-a)(\tilde{u}_{1})^{1}\phi$. More precisely, if $|\mu-\nu|$ small enough

[TABLE]

where $K$ is a positive constant depending only on $F$.

Remark 4

Indeed, we always assume that $b>0$, hence $u_{1}>0$ for $\nu-\mu>0$ small enough. For the sign of $u_{2}$ we remark that $(\xi_{1}-a)$ and $(d-a)$ have the same sign.

3.2 Case $\xi_{1}=\xi_{2}$

Consider now the case where the coefficients of the matrix $A$ satisfy $(H_{A})$ and

[TABLE]

Of course this implies $bc<0$ and since $b>0$ , then $c<0$: we have a non cooperative system. Now $\xi_{1}=\xi_{2}=\xi=\frac{a+d}{2}$. We prove here

Theorem 5

Assume $(H^{\prime}_{q})$ and $(H_{A})$ with $(a-d)^{2}+4bc=0$; assume also that $f_{1},f_{2}$ satisfy $(H^{\prime}_{f})$ and :

[TABLE]

If $\mu<\nu=\Lambda-\xi$, $\nu-\mu<\delta$, small enough, there exists a positive constant $\gamma$ such that

[TABLE]

If $\nu=\Lambda-\xi<\mu<\nu+\delta$, ($\delta$ small enough), there exists a positive constant $\gamma^{\prime}$ such that

[TABLE]

Remark 5

Note that the condition $\frac{(a-d)}{2}f_{1}^{1}+bf_{2}^{1}>0$ in the theorem above is the same than in theorem 4 $(\xi_{2}-a)f_{1}^{1}<bf_{2}^{1}$, since in theorem 5 $\xi_{2}=\xi=\frac{a-d}{2}$.

Prrof   The eigenvector associated to eigenvalue $\xi$ is

[TABLE]

The vector $X\phi$ is thus an eigenvector for $L-A$,

[TABLE]

We use again $J$ the associated Jordan matrix and $P$ the change of basis matrix; we have

[TABLE]

Here

[TABLE]

[TABLE]

As above, setting $\tilde{U}=P^{-1}U$ and $\tilde{F}=P^{-1}F$, we derive from System $(S)$

[TABLE]

We do not have anymore a decoupled system but

[TABLE]

If $\xi+\mu<\Lambda$ (that is $\mu<\nu$) and if $\tilde{f}_{2}=\dfrac{(a-d)}{2b}f_{1}+f_{2}$ and $\tilde{f}_{1}=\frac{-2}{a-d}f_{2}$ satisfies $(H^{\prime}_{f})$, hence are in ${\cal X}$ and $\frac{(a-d)}{2b}f_{1}^{1}+f_{2}^{1}>0$. By Theorem 3 applied to the second equation, there exists a constant $K>0$, such that $\tilde{u}_{2}>\frac{K}{\nu-\mu}\phi$. Hence, for $\nu-\mu$ small enough fo any $\tilde{f}_{1}\in{\cal X}$, $\tilde{u}_{2}+\tilde{f}_{1}>0$ and is in $X$; then again Theorem 3 for the first equation implies that there exists a constant $K^{\prime}>0$, such that $\tilde{u}_{1}>\frac{K^{\prime}}{\nu-\mu}\phi$.

Since here $a>d$., there exists a constant $\gamma>0$,

[TABLE]

Again $u_{1}\rightarrow+\infty$ as $\nu-\mu\rightarrow 0$ and $u_{2}$ $\rightarrow-\infty$ as $\nu-\mu\rightarrow 0$.

If $\mu>\Lambda$ ( and $\mu-\nu>0$ small enough we have analogous calculation with signs reversed.

Remark 6

The results in theorem 5 coincide with those of theroem 4 in the case $D=0$.