Non-variational weakly coupled elliptic systems

Abstract

We establish the existence of a nonnegative fully nontrivial solution to a non-variational weakly coupled competitive elliptic system. We show that this kind of solutions belong to a topological manifold of Nehari-type, and apply a degree-theoretical argument on this manifold to derive existence.

1 Introduction and statement of results

In this paper we consider the existence of solutions to the elliptic system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_i = \mu _i u_i^p + \sum \limits _{j\ne i}\lambda _{ij}u_i^{\alpha _{ij}}u_j^{\beta _{ij}}, \\ u_i\ge 0,\ u_i\not \equiv 0 \text { in } \Omega , \\ u_i\in H^1_0(\Omega ), \quad i,j=1,\ldots ,\ell , \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$, $N\ge 2$, $1<p<\frac{N+2}{N-2}$ if $N\ge 3$, $1<p<\infty$ if $N=2$, $\mu _i>0$, $\lambda _{ij}<0$, $\alpha _{ij},\beta _{ij}>0$ and $\alpha _{ij}+\beta _{ij}<p$ for $i,j=1,\ldots ,\ell$, $j\ne i$. This system arises as a model for the steady state distribution of ell competing species coexisting in $\Omega$. Here $u_i$ represents the density of the i-th population, $\mu _i$ corresponds to the attraction between the species of the same kind, or more generally, $\mu _iu_i^p$ can be replaced by $f_i(u_i)$ and represent internal forces. The parameters $\lambda _{ij}$, $\lambda _{ji}$ (which may not be equal) correspond to the interaction (repulsion) between different species. In particular, if $\alpha _{ij}=\beta _{ij}=1$, then the interaction is of the Lotka-Volterra type while $\alpha _{ij}=1$, $\beta _{ij}=2$ corresponds to the interaction which appears in the Bose-Einstein condensates. In the latter case one also has $\lambda _{ij}=\lambda _{ji}$ and the system is variational.

In what follows we do not assume $\lambda _{ij}=\lambda _{ji}$ or $\beta _{ij}=\alpha _{ji}$. The system (1.1) is non-variational except for some very special choices of $\lambda _{ij}$, $\alpha _{ij}$ and $\beta _{ij}$. While there is an extensive literature concerning the existence (and multiplicity) of solutions for variational systems like (1.1), there are not so many results in the non-variational case. Here we could mention [1, 6,7,8,9] where, however, the right-hand sides are quite different from ours. In particular, in [6,7,8,9] the interaction term is of the Lotka-Volterra type (or is a variant of it) while the terms $f_i(u_i)$ are different from $\mu _iu_i^p$. For these $f_i$ one obtains uniform bounds on the solutions when $\lambda _{ij}\rightarrow -\infty$. Existence of such bounds allows to study the limiting behaviour of solutions. To be more precise, if $\lambda _{ij,n}\rightarrow -\infty$ and $(u_{1,n},\ldots ,u_{l,n})$ is a corresponding solution with uniform bound on each component, then one expects that $u_{i,n}\rightarrow u_i$ (in an appropriate space) and $u_i(x)\cdot u_j(x)=0$ a.e. in $\Omega$ for all $i\ne j$, i.e. different components separate spatially. This has been studied in the above mentioned papers. In [1, 6] the emphasis is in fact on the properties of limiting configurations, including regularity of free boundaries between the components.

The main result of this paper is the following

Theorem 1.1

The system (1.1) has a solution.

Existence proofs in the above-mentioned papers do not seem to be applicable here. Our problem can be reformulated as an operator equation in the space $\mathcal {H}:= H^1_0(\Omega )^\ell$ and one can use degree theory to obtain a nontrivial solution. However, this could give a semitrivial solution (i.e. $u_i=0$ for some but not all i). To rule out such solutions we introduce a Nehari-type manifold on which all u are fully nontrivial in the sense that no $u_i$ is identically zero, and then we apply a degree-theoretical argument on this manifold.

We do not know if there always exist solutions for (1.1) which are uniformly bounded, see Problem 5.5. Moreover, as we shall see in Sect. 5, under a suitable choice of exponents and parameters and for $\ell =2$ there exists a sequence of solutions which are synchronized in the sense that $u_{i,n}=t_{i,n}v_n$ ($i=1,2$) and such that $\Vert u_{i,n}\Vert \rightarrow \infty$ as $\lambda _{12,n},\lambda _{21,n}\rightarrow -\infty$. So the components neither separate spatially nor are bounded.

Let $u_i^+ := \max \{u_i,0\}$, $u_i^- := \min \{u_i,0\}$, and consider the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_i = \mu _i (u_i^+)^p + \sum \limits _{j\ne i}\lambda _{ij}(u_i^+)^{\alpha _{ij}}(u_j^+)^{\beta _{ij}}, \\ u_i\in H^1_0(\Omega ), \quad i,j=1,\ldots ,\ell . \end{array}\right. } \end{aligned}$$

(1.2)

In Proposition 3.4(v) we shall show that any fully nontrivial solution to this system also solves (1.1).

In what follows we shall work with (1.2) and we shall also need the parametrized system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_i = \mu _i (u_i^+)^p + t\sum \limits _{j\ne i}\lambda _{ij}(u_i^+)^{\alpha _{ij}}(u_j^+)^{\beta _{ij}}, \\ u_i\in H^1_0(\Omega ), \quad i,j=1,\ldots ,\ell , \quad 0\le t\le 1. \end{array}\right. } \end{aligned}$$

(1.3)

Note that (1.3) homotopies (1.2) to an uncoupled system. Since

$$\begin{aligned} t\sum \limits _{j\ne i}|\lambda _{ij}|(u_i^+)^{\alpha _{ij}}(u_j^+)^{\beta _{ij}} \le C(1+(u_1^+)^q+\cdots + (u_\ell ^+)^q), \end{aligned}$$

where $\alpha _{ij}+\beta _{ij}\le q<p$ for all i, j, the following statement holds true.

Lemma 1.2

All solutions $u =(u_1,\ldots ,u_\ell )$ of (1.3) are uniformly bounded in $L^\infty (\Omega )$ and hence in $H^1_0(\Omega )$. This bound is independent of $t\in [0,1]$.

This has been shown, in a much more general setting, in [13] for a single equation and in [11] for two equations. It is easy to see that the argument in [11] extends to an arbitrary number of equations. In both papers a blow-up procedure is used in order to reduce the problem to a Liouville-type result. For the reader’s convenience, in Appendix A we shall provide a simple proof of such reduction, adapted to our special case. The assumption $q<p$ is crucial for the validity of this lemma. Indeed, in [10] it has been shown that the conclusion may fail if $q=p$.

The paper is organized as follows. In Sect. 2 we state and prove a lemma for functions in ${\mathbb {R}}^\ell$. In Sect. 3 we define a Nehari-type manifold $\mathcal {N}$ similar to the one introduced in [5]. We also show that solutions to (1.2) correspond to solutions for an operator equation in an open subset of the product of the unit spheres $\mathscr {S}_i\subset H^1_0(\Omega )$, $1\le i\le \ell$. The idea comes from [4]. To our knowledge, this is the first time a Nehari-type manifold appears in a non-variational setting. Theorem 1.1 is proved in Sect. 4 and synchronized solutions are discussed in Sect. 5. As we have already mentioned, Lemma 1.2 is proved in Appendix A.

In the proof of Theorem 1.1 we shall employ a topological degree argument. Since our operator is not admissible for common infinite-dimensional degree theories, we introduce a sequence of finite-dimensional (“Galerkin-like”) approximations and use the Brouwer degree, see (4.6) and (4.9–4.11) below.

2 A lemma on functions in ${\mathbb {R}}^\ell$

Let $a_i,\alpha _{ij},\beta _{ij}>0$, $b_i,d_{ij}\ge 0$, $\alpha _{ij}+\beta _{ij}<p$ for all $i,j=1,\ldots ,\ell$, $j\ne i$. Define $M:(0,\infty )^\ell \rightarrow {\mathbb {R}}^\ell$ as

$$\begin{aligned}M(s) := (M_1(s),\ldots ,M_\ell (s)), \end{aligned}$$

where

$$\begin{aligned} M_i(s) := a_is_i - b_is_i^p + \sum \limits _{j\ne i}d_{ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}}, \qquad i,j= 1,\ldots ,\ell . \end{aligned}$$

Lemma 2.1

• (i) If $b_i=0$ for some i, then $M(s)\ne 0$ for any $s\in (0,\infty )^\ell$.

• (ii) If $b_i>0$ for all i, then there exists $s\in (0,\infty )^\ell$ such that $M(s)=0$.

  Moreover, if $0<a\le a_i\le \overline{a}$, $0<b\le b_i\le \overline{b}$ and $d_{ij}\le \overline{d}$ for all i, j, then there exist $0<r<R$, depending only on $a,\overline{a},b,\overline{b},\overline{d}$, such that $s\in (r,R)^\ell$.

• (iii) The solution s in (ii) is unique.

• (iv) The solution s in (ii) depends continuously on $a_i,b_i>0,\ d_{ij}\ge 0$.

Proof

(i) :  If $b_i=0$ then

$$\begin{aligned} M_i(s) = a_is_i+\sum \limits _{j\ne i}d_{ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}}>0 \qquad \text {for all } s\in (0,\infty )^\ell . \end{aligned}$$

(ii) :  Let $0<r<R$ be such that, for every $i,j=1,\ldots ,\ell$,

$$\begin{aligned} a_it-b_it^p >0&\qquad \text {if \ }t\in (0,r], \\ a_it-b_it^p+\sum \limits _{j\ne i}d_{ij}t^{\alpha _{ij}+\beta _{ij}}<0&\qquad \text {if \ }t\in [R,\infty ) \end{aligned}$$

(such R exists because $\alpha _{ij}+\beta _{ij}<p$). If $s=(s_1,\ldots ,s_\ell )\in (0,\infty )^\ell$ and $s_i\ge s_j$ for all j, then

$$\begin{aligned} M_i(s) = a_is_i-b_is_i^p+\sum \limits _{j\ne i}d_{ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}} \le a_is_i-b_is_i^p+\sum \limits _{j\ne i}d_{ij}s_i^{\alpha _{ij}+\beta _{ij}}. \end{aligned}$$

Therefore, $M_i(s)<0$ whenever $s_i=\max \{s_1,\ldots ,s_\ell \}\ge R$, and $M_i(s)>0$ if $0<s_i\le r$. If $a\le a_i\le \overline{a}$, $b\le b_i\le \overline{b}$, $d_{ij}\le \overline{d}$, then

$$\begin{aligned} a_it-b_it^p\ge at-\overline{b}t^p, \qquad a_it-b_it^p+\sum \limits _{j\ne i}d_{ij}t^{\alpha _{ij}+\beta _{ij}} \le \overline{a}t-bt^p + \sum \limits _{j\ne i}\overline{d}t^{\alpha _{ij}+\beta _{ij}}, \end{aligned}$$

so r, R may be chosen as claimed.

Let

$$\begin{aligned} G(s) := \rho -s \qquad \text {where}\quad \rho :=\tfrac{r+R}{2}(1,\ldots ,1). \end{aligned}$$

Then $H(s,\tau ) := \tau M(s)+(1-\tau )G(s) \ne 0$ on the boundary of $[r,R]^\ell$ for every $\tau \in [0,1]$. Hence this is an admissible homotopy for the Brouwer degree (see e.g. [18, Appendix D] for the definition and properties of this degree). So

$$\begin{aligned} \deg (M, (r,R)^\ell , \rho ) = \deg (G,(r,R)^\ell , \rho ) = (-1)^\ell \end{aligned}$$

and $M(s)=0$ must have a solution.

(iii) :  If $M(s_1^0,\ldots ,s_\ell ^0)=0$, then $\widetilde{M}(1,\ldots ,1)=0$ where

$$\begin{aligned} \widetilde{M}_i(s) = \widetilde{a}_is_i-\widetilde{b}_1s_i^p+\sum \limits _{j\ne i}\widetilde{d}_{ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}} \end{aligned}$$

with $\widetilde{a}_i := a_is_i^0$, $\widetilde{b}_i := b_i(s_i^0)^p$, $\widetilde{d}_{ij} := d_{ij}(s_i^0)^{\alpha _{ij}}(s_j^0)^{\beta _{ij}}$. So we may assume without loss of generality that $M(1,\ldots ,1)=0$. Then,

$$\begin{aligned} a_i-b_i+\sum \limits _{j\ne i}d_{ij}=0. \end{aligned}$$

Suppose there is another solution $s=(s_1,\ldots ,s_\ell )$. Then, using the previous identity, we get

$$\begin{aligned} 0=a_is_i-b_is_i^p+\sum \limits _{j\ne i}d_{ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}} = a_is_i - \Big (a_i+\sum \limits _{j\ne i}d_{ij}\Big )s_i^p+\sum \limits _{j\ne i}d_{ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}}, \end{aligned}$$

and after rearranging the terms,

$$\begin{aligned} a_i(s_i-s_i^p) = \sum \limits _{j\ne i}d_{ij}(s_i^p-s_i^{\alpha _{ij}}s_j^{\beta _{ij}}). \end{aligned}$$

There are two possible cases: If $s_i>1$ for some i, we may assume without loss of generality that $s_i\ge s_j$ for all j. Then the left-hand side above is negative while the right-hand side is $\ge 0$, a contradiction. If, on the other hand, $0<s_i<1$ for some i, we may assume $s_i\le s_j$ for all j. Now the left-hand side is positive and the right-hand side is $\le 0$, a contradiction again.

(iv) :  If $a_{n,i},a_i,b_{n,i},b_i>0$, $d_{n,i},d_i\ge 0$, $a_{n,i}\rightarrow a_i$, $b_{n,i}\rightarrow b_i$, $d_{n,ij}\rightarrow d_{ij}$ then, as in (ii), there exist $0<r<R$ such that the unique solution $s_n$ to

$$\begin{aligned} M_{n,i}(s) := a_{n,i}s_i - b_{n,i}s_i^p + \sum \limits _{j\ne i}d_{n,ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}}=0, \qquad i,j= 1,\ldots ,\ell , \end{aligned}$$

belongs to $[r,R]^\ell$ for every n. Passing to a subsequence, we have that $s_n\rightarrow s\in [r,R]^\ell$ and $M(s)=0$. $\square$

3 A Nehari-type manifold

Let $\mathcal {H}:= H^1_0(\Omega )^\ell$, $u=(u_1,\ldots ,u_\ell )\in \mathcal {H}$. As convenient norms in $H^1_0(\Omega )$ and $\mathcal {H}$ we choose

$$\begin{aligned} \Vert u_i\Vert :=\left( \int _{\Omega }|\nabla u_i|^2\right) ^{\frac{1}{2}} \quad \text {and} \quad \Vert u\Vert := (\Vert u_1\Vert ^2+\cdots +\Vert u_\ell \Vert ^2)^{\frac{1}{2}}, \end{aligned}$$

and we denote by $\langle \,\cdot \,,\,\cdot \,\rangle$ the inner product in $H^1_0(\Omega )$. Let

$$\begin{aligned} I(u) := (I_1(u),\ldots ,I_\ell (u)) \end{aligned}$$

where $I_i: H^1_0(\Omega )\rightarrow H^1_0(\Omega )$ are given by

$$\begin{aligned} I_i(u):=u_i-K_i(u) \end{aligned}$$

(3.1)

and

$$\begin{aligned} \langle K_i(u),v\rangle := \int _{\Omega }\mu _i(u_i^+)^pv + \sum \limits _{j\ne i}\lambda _{ij}\int _{\Omega }(u_i^+)^{\alpha _{ij}}(u_j^+)^{\beta _{ij}}v\quad \forall v\in H^1_0(\Omega ). \end{aligned}$$

(3.2)

Lemma 3.1

If $u_n\rightharpoonup u$ weakly in $\mathcal {H}$, then $K_i(u_n)\rightarrow K_i(u)$ strongly in $H^1_0(\Omega )$ for each $i=1,\ldots ,\ell$.

Proof

Since $p,\,\alpha _{ij}+\beta _{ij}<\frac{N+2}{N-2}$ for $N\ge 3$, after passing to a subsequence $u_{n,i}^+\rightarrow u_i^+$ strongly in $L^{p+1}(\Omega )$ and in $L^{\alpha _{ij}+\beta _{ij}+1}(\Omega )$ for every $j\ne i$. Using Hölder’s and Sobolev’s inequalities we obtain

$$\begin{aligned}&|\langle K_i(u_n)-K_i(u),v\rangle | \le C\Big (\big |(u_{n,i}^+)^p-(u_i^+)^p\big |_{\frac{p+1}{p}}\\&\quad + \sum _{j\ne i}\big |(u_{n,i}^+)^{\alpha _{ij}}-(u_i^+)^{\alpha _{ij}}\big |_{\frac{\alpha _{ij}+\beta _{ij}+1}{\alpha _{ij}}}+ \sum _{j\ne i}\big |(u_{n,j}^+)^{\beta _{ij}}-(u_j^+)^{\beta _{ij}}\big |_{\frac{\alpha _{ij}+\beta _{ij}+1}{\beta _{ij}}}\Big )\Vert v\Vert , \end{aligned}$$

where $|\,\cdot \,|_r$ denotes the norm in $L^r(\Omega )$. From [18, Theorem A.2] we derive

$$\begin{aligned} \sup _{v\ne 0}\frac{|\langle K_i(u_n)-K_i(u),v\rangle |}{\Vert v\Vert }\longrightarrow 0. \end{aligned}$$

Hence, $K_i(u_n)\rightarrow K_i(u)$ strongly in $H^1_0(\Omega )$, as claimed. $\square$

We define a Nehari-type set $\mathcal {N}$ by putting

$$\begin{aligned} \mathcal {N}:= \{u\in \mathcal {H}: u_i\ne 0 \text { \ and \ } \langle I_i(u),u_i\rangle = 0 \text { \ for all \ } i=1,\ldots ,\ell \}. \end{aligned}$$

Lemma 3.2

$\mathcal {N}$ is closed in $\mathcal {H}$.

Proof

Since $\lambda _{ij}<0$, it follows from the Sobolev inequality that

$$\begin{aligned} \Vert u_i\Vert ^2 \le \mu _i\int _{\Omega }(u_i^+)^{p+1} \le C_i\Vert u_i\Vert ^{p+1} \end{aligned}$$

for some $C_i>0$. Hence there exists $d_0>0$ such that, if $(u_1,\ldots ,u_\ell )\in \mathcal {N}$, then $\Vert u_i\Vert \ge d_0$ for all i. This shows that $\mathcal {N}$ is closed in $\mathcal {H}$. $\square$

For $u :=(u_1,\ldots ,u_\ell )\in \mathcal {H}$, $s:=(s_1,\ldots ,s_\ell )\in (0,\infty )^\ell$ and $su := (s_1u_1,\ldots ,s_\ell u_\ell )$, we define

$$\begin{aligned} M_u(s) := (M_{u,1}(s),\ldots ,M_{u,\ell }(s)), \end{aligned}$$

where

$$\begin{aligned} M_{u,i}(s) := \langle I_i(su),u_i\rangle = a_{u,i}s_i-b_{u,i}s_i^p + \sum \limits _{j\ne i}d_{u,ij}s_i^{\alpha _{ij}}s_j^{\beta _{ij}} \end{aligned}$$

and

$$\begin{aligned} a_{u,i} := \Vert u_i\Vert ^2, \quad b_{u,i} := \int _{\Omega }\mu _i(u_i^+)^{p+1}, \quad d_{u,ij} := \int _{\Omega }(-\lambda _{ij})(u_i^+)^{\alpha _{ij}+1}(u_j^+)^{\beta _{ij}}. \end{aligned}$$

Lemma 3.3

• (i) If $a_{u,i}\ne 0$ and $b_{u,i}=0$ for some i, then $M_{u}(s)\ne 0$ for any $s\in (0,\infty )^\ell$.

• (ii) If $a_{u,i},\,b_{u,i}>0$ for all i, then there exists a unique $s_u\in (0,\infty )^\ell$ such that $M_u(s_u)= 0$. Moreover, if $0<a\le a_{u,i}\le \overline{a}$, $0<b\le b_{u,i}\le \overline{b}$ and $d_{u,ij}\le \overline{d}$ for all i, j, then there exist $0<r<R$, depending only on $a,\overline{a},b,\overline{b},\overline{d}$, such that $s_u\in (r,R)^\ell$.

Proof

This is an immediate consequence of Lemma 2.1. $\square$

Let

$$\begin{aligned} \mathscr {S}:= \{v\in H^1_0(\Omega ): \Vert v\Vert = 1\}, \quad \mathcal {T}:= \mathscr {S}^\ell , \end{aligned}$$

and

$$\begin{aligned} \mathcal {U}:&= \{u\in \mathcal {T}: s_u\in (0,\infty )^\ell \text { exists with }M_u(s_u)=0\}\nonumber \\&= \{u\in \mathcal {T}: u_i^+\ne 0\text { for all } i=1,\ldots ,\ell \}. \end{aligned}$$

(3.3)

The tangent space of $\mathcal {T}$ at u is

$$\begin{aligned} T_u(\mathcal {T}):=\{(v_1,\ldots ,v_\ell )\in \mathcal {H}:\langle u_i,v_i\rangle =0\text { for all }i=1,\ldots ,\ell \}. \end{aligned}$$

(3.4)

Proposition 3.4

• (i) $\mathcal {U}$ is a nonempty open subset of $\mathcal {T}$ and $\mathcal {U}\ne \mathcal {T}$.

• (ii) The mapping $m: \mathcal {U}\rightarrow \mathcal {N}$ given by $m(u) := s_uu$ is a homeomorphism. In particular, $\mathcal {N}$ is a topological manifold.

• (iii) If $(u_n)$ is a sequence in $\mathcal {U}$ such that $u_n\rightarrow u\in \partial \mathcal {U}$, then $s_{u_n}\rightarrow \infty$ (and hence $\Vert m(u_n)\Vert \rightarrow \infty$).

• (iv) Let $S:\mathcal {U}\rightarrow \mathcal {H}$ be given by

  $$\begin{aligned} S(u) := I(s_uu) = s_uu-K(s_uu). \end{aligned}$$

  Then $S(u)\in T_u(\mathcal {U})$ for every $u\in \mathcal {U}$.

• (v) $S(u) = 0$ if and only if $m(u)=s_uu$ is a solution for (1.1).

Proof

(i) :  That $\mathcal {U}$ is neither empty nor the whole $\mathcal {T}$ is obvious and, since $u\mapsto u_i^+$ is continuous [2, Lemma 2.3], it is easily seen from the second line of (3.3) that $\mathcal {U}$ is open in $\mathcal {T}$.

(ii) :  If $u\in \mathcal {U}$, then $s_uu\in \mathcal {N}$ because $\langle I_i(s_uu),s_{u,i}u_i\rangle = s_{u,i}M_{u,i}(s_u)= 0$ for all i. So m is well defined. If $(u_n)$ is a sequence in $\mathcal {U}$ and $u_n\rightarrow u\in \mathcal {U}$, then $a_{u_n,i}\rightarrow a_{u,i}$, $b_{u_n,i}\rightarrow b_{u,i}$ and $d_{u_n,ij}\rightarrow d_{u,ij}$ for all i, j. By Lemma 2.1(iv), $s_{u_n}\rightarrow s_u$. Hence, m is continuous.

If $u\in \mathcal {N}$, then $u_i^+\ne 0$ for all i. Otherwise, $0=\langle I_i(u),u_i\rangle =\Vert u_i\Vert ^2$, a contradiction. Hence, the inverse of m satisfies

$$\begin{aligned} m^{-1}(u) := \left( \frac{u_1}{\Vert u_1\Vert }, \ldots , \frac{u_\ell }{\Vert u_\ell \Vert }\right) \in \mathcal {U}, \end{aligned}$$

and it is obviously continuous.

(iii) :  Let $(u_n)$ be a sequence in $\mathcal {U}$ such that $u_n\rightarrow u\in \partial \mathcal {U}$. If $(s_{u_n})$ is bounded, then, after passing to a subsequence, $s_{u_n}\rightarrow s_*$. Since $\mathcal {N}$ is closed, $s_*u\in \mathcal {N}$ and hence $u\in \mathcal {U}$. This is impossible because $\mathcal {U}$ is open.

(iv) :  Since $\langle I_i(s_uu),u_i\rangle = M_{u,i}(s_u)= 0$ for all i, we have that $S(u)\in T_u(\mathcal {T})$ according to (3.4).

(v) :  If $u\in \mathcal {U}$ satisfies $S(u)=0$, then ${\bar{u}}:=s_uu\in \mathcal {N}$ and ${\bar{u}}$ is a weak solution to the system (1.2) (see (3.1) and (3.2)). Multiplying the i-th equation in (1.2) by $u_i^-:=\min \{{\bar{u}}_i,0\}$ and integrating gives $\int _{\Omega }|\nabla u_i^-|^2 = 0$. Hence $u_i^- = 0$, i.e., ${\bar{u}}_i\ge 0$ for all i. As ${\bar{u}}\in \mathcal {N}$, we have that ${\bar{u}}_i\ne 0$. This proves that ${\bar{u}}$ solves (1.1). The converse is obvious. $\square$

Remark 3.5

If $\alpha _{ij}\ge 1$ for all i and all $j\ne i$, then, as ${\bar{u}}_i$ above satisfies the i-th equation in (1.1), we have

$$\begin{aligned} -\Delta {\bar{u}}_i +c(x){\bar{u}}_i\ge 0 \quad \text {where } c(x) := -\sum \limits _{j\ne i}\lambda _{ij}{\bar{u}}_i^{\alpha _{ij}-1}\bar{u}_j^{\beta _{ij}}. \end{aligned}$$

Since all $u_i$ are continuous in $\overline{\Omega }$ and $c\ge 0$, it follows from the strong maximum principle (see e.g. [14, Theorem 3.5]) that our solution is strictly positive in $\Omega$ in this case.

4 Proof of Theorem 1.1

In this section the sub- or superscript t will be used in order to emphasize that we are concerned with the system (1.3). So, e.g.,

$$\begin{aligned} I_{t}(u):=u-K_{t}(u),\qquad S_t(u):=I_{t}(s^t_uu)=s^t_uu-K_t(s^t_uu), \end{aligned}$$

(4.1)

with

$$\begin{aligned} \langle K_{t,i}(u),v\rangle := \int _{\Omega }\mu _i(u_i^+)^pv + t\sum \limits _{j\ne i}\lambda _{ij}\int _{\Omega }(u_i^+)^{\alpha _{ij}}(u_j^+)^{\beta _{ij}}v, \end{aligned}$$

and

$$\begin{aligned} \mathcal {N}_t := \{u\in \mathcal {H}: u_i\ne 0,\ \langle I_{t,i}(u),u_i\rangle = 0 \text { for all } i=1,\ldots ,\ell \}. \end{aligned}$$

According to this notation, $\mathcal {N}_1 =\mathcal {N}$. When $t=1$, we shall sometimes omit the sub- or superscript t.

Consider first the system (1.3) with $t=0$. In this case the equations are uncoupled, the set

$$\begin{aligned} \mathcal {N}_0=\{u\in \mathcal {H}:u_i\ne 0,\ \Vert u_i\Vert ^2=\int _{\Omega }\mu _i(u_i^+)^{p+1}\text { \ for all \ }i=1,\ldots ,\ell \} \end{aligned}$$

is the product of the usual Nehari manifolds associated to these equations, and the components of $s_u^0=(s_{u,1}^0\ldots ,s_{u,\ell }^0)$ are

$$\begin{aligned} s_{u,i}^0=\left( \int _{\Omega }\mu _i(u_i^+)^{p+1}\right) ^{-\frac{1}{p-1}}, \qquad u\in \mathcal {U}. \end{aligned}$$

The function $I_0$ (cf. (4.1)) is the gradient vector field of the functional $\mathcal {J}:\mathcal {H}\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \mathcal {J}(u):= \frac{1}{2}\sum _{i=1}^\ell \Vert u_i\Vert ^2 - \frac{1}{p+1}\sum _{i=1}^\ell \int _{\Omega }\mu _i(u_i^+)^{p+1}. \end{aligned}$$

Note that

$$\begin{aligned} \mathcal {J}(u)=(\tfrac{1}{2}-\tfrac{1}{p+1})\Vert u\Vert ^2\quad \text {if \ }u\in \mathcal {N}_0. \end{aligned}$$

$\mathcal {J}$ has a minimizer $\widetilde{u}_0=(\widetilde{u}_{0,1},\ldots ,\widetilde{u}_{0,\ell })$ on $\mathcal {N}_0$ with $\widetilde{u}_{0,i}>0$ and $\widetilde{u}_0$ is a solution to the system (1.3) with $t=0$. Each component $\widetilde{u}_{0,i}$ is a positive least energy solution to the i-th equation of this system. Let $\Psi :\mathcal {U}\rightarrow {\mathbb {R}}$ be given by

$$\begin{aligned} \Psi (u) :&=\mathcal {J}(s_u^0u) =\left( \tfrac{1}{2}-\tfrac{1}{p+1}\right) |s_u^0|^2\nonumber \\&=\left( \tfrac{1}{2}-\tfrac{1}{p+1}\right) \sum _{i=1}^\ell \left( \int _{\Omega }\mu _i(u_i^+)^{p+1}\right) ^{-\frac{2}{p-1}}. \end{aligned}$$

(4.2)

By [18, Proposition 1.12] one has that $\Psi \in \mathcal {C}^2(\mathcal {U},{\mathbb {R}})$. It is easily seen that

$$\begin{aligned} \Psi '(u)v=\mathcal {J}'(s^0_uu)[s^0_uv]=\langle S_0(u),s^0_uv\rangle \quad \text {for all \ }u\in \mathcal {U}, \ v\in T_u(\mathcal {U}), \end{aligned}$$

(4.3)

and that u is a critical point of $\Psi$ if and only if $u\in \mathcal {U}$ and $m_0(u)=s^0_uu$ is a critical point of $\mathcal {J}$, see [4, Theorem 3.3]. Let $u_0:=m_0^{-1}(\widetilde{u}_0)$. Then $u_0$ is a minimizer for $\Psi$.

Invoking Lemma 1.2 we may choose $R>0$ such that all solutions to the systems (1.3) are contained in the open ball $B_R(0)\subset \mathcal {H}$, where R is independent of $t\in [0,1]$. Then, by Proposition 3.4,

$$\begin{aligned} \{u\in \mathcal {U}:S_t(u)=0\}\subset \mathcal {V}_t:=m_t^{-1}(B_R(0)\cap \mathcal {N}_t). \end{aligned}$$

(4.4)

For $a\le d$ let

$$\begin{aligned} \Psi ^d:=\{u\in \mathcal {U}:\Psi (u)\le d\}, \quad \Psi _a^d := \{u\in \mathcal {U}: a\le \Psi (u)\le d\}. \end{aligned}$$

It follows from Proposition 3.4(iii) that the set $\Psi ^d$ is closed in $\mathcal {T}$ for any $d\in {\mathbb {R}}$. Note that $\lambda _{ij}<0$ implies $s_{u,i}^t\ge s_{u,i}^0$ for every $u\in \mathcal {U}$, $t\in [0,1]$, $i=1,\ldots \ell$. So if $|s_u^t|<R$, then $|s_u^0|<R$; hence $\mathcal {V}_t\subset \mathcal {V}_0$ and, setting $c:=(\frac{1}{2}-\frac{1}{p+1})R^2$, we have that the closure of $\mathcal {V}_t$ in $\mathcal {T}$ satisfies

$$\begin{aligned} \overline{\mathcal {V}}_t\subset \overline{\mathcal {V}}_0\subset \Psi ^c\qquad \forall t\in [0,1]. \end{aligned}$$

(4.5)

For each $i=1,\ldots ,\ell$ and $k\ge 2$ we choose an ascending sequence $(E_{k,i})$ of linear subspaces of $H^1_0(\Omega )$ such that $\dim E_{k,i} = k$, $u_{0,i}\in E_{2,i}$ ($u_0$ is the minimizer chosen above) and $\overline{\bigcup _{k\ge 1} E_{k,i}} = H^1_0(\Omega )$. We define

$$\begin{aligned} E_k:=E_{k,1}\times \cdots \times E_{k,\ell }\subset \mathcal {H}\end{aligned}$$

(4.6)

and denote by $P_k$ the orthogonal projector of $\mathcal {H}$ onto $E_k$.

Lemma 4.1

Given $d>0$ there exists $k_d\in \mathbb {N}$ such that

$$\begin{aligned} P_k(S_t(u))\ne 0\qquad \text {for all \ }u\in (\Psi ^d\smallsetminus \mathcal {V}_t)\cap E_k, \ k\ge k_d, \ t\in [0,1]. \end{aligned}$$

Proof

Arguing by contradiction, assume that there exist $k_n\rightarrow \infty$, $t_n\in [0,1]$ and $u_n\in (\Psi ^d\smallsetminus \mathcal {V}_{t_n})\cap E_{k_n}$ such that

$$\begin{aligned} P_{k_n}(S_{t_n}(u_n))=s_{u_n}^{t_n}u_n-P_{k_n}K_{t_n}(s_{u_n}^{t_n}u_n)=0\qquad \forall n\in \mathbb {N}. \end{aligned}$$

(4.7)

As $u_n\in \Psi ^d$, we derive from (4.2) that $\int _{\Omega }\mu _i(u_{n,i}^+)^{p+1}\ge b$ for some $b>0$ and all n, i. In the notation of Lemma 3.3, we have $a_{u_n,i}=1$ and, using the Hölder and the Sobolev inequalities, $b\le b_{u_n,i}\le \overline{b}$ and

$$\begin{aligned} d_{u_n,ij} = t_n\int _{\Omega }(-\lambda _{ij}) (u_{n,i}^+)^{\alpha _{ij}+1}(u_{n,j}^+)^{\beta _{ij}} \le \overline{d} \end{aligned}$$

for some $\overline{b},\overline{d}>0$. So Lemma 3.3 asserts that $(s^{t_n}_{u_n,i})$ is bounded and bounded away from 0 for each i. Therefore, after passing to a subsequence, $s^{t_n}_{u_n,i}\rightarrow s_i>0$, $t_n\rightarrow t$ and $u_n\rightharpoonup u$ weakly in $\mathcal {H}$. By Lemma 3.1, $K_{t_n}(s_{u_n}^{t_n}u_n) \rightarrow K_t(su)$ strongly in $\mathcal {H}$, and we easily deduce that $P_{k_n}K_{t_n}(s_{u_n}^{t_n}u_n) \rightarrow K_t(su)$ strongly in $\mathcal {H}$. Now we derive from (4.7) that $s_{u_n}^{t_n}u_n\rightarrow su$ strongly in $\mathcal {H}$ and $su-K_{t}(su)=0$. Therefore, $su\in \mathcal {N}_t$, $s=s^t_u$ and $S_t(u)=0$. On the other hand, as $u_n\notin \mathcal {V}_{t_n}$, we have that $\Vert s^{t_n}_{u_n}u_n\Vert \ge R$. Hence, $\Vert s^t_uu\Vert \ge R$. This is a contradiction. $\square$

Lemma 4.2

Let c be as in (4.5). Then $\Psi ^c\cap E_k$ is contractible in itself for each large enough k.

Proof

Let $\eta : [0,1]\times \mathcal {U}\rightarrow \mathcal {U}$ be given by

$$\begin{aligned} \eta (\tau ,u) := \left( \frac{(1-\tau )u_1+\tau u_{0,1}}{\Vert (1-\tau )u_1+\tau u_{0,1}\Vert },\ldots , \frac{(1-\tau )u_\ell +\tau u_{0,\ell }}{\Vert (1-\tau )u_\ell +\tau u_{0,\ell }\Vert }\right) , \end{aligned}$$

where $u_0$ is the previously chosen minimizer for $\Psi$ on $\mathcal {U}$. Note that $\eta$ is well defined and maps into $\mathcal {U}$ because $u_{0,i}>0$ in $\Omega$ and $u_i^+\ne 0$ for all i. Moreover, if $u\in E_k$, then $\eta (\tau ,u)\in E_k$ for each $k\ge 2$. So $\eta$ is a deformation of $\mathcal {U}\cap E_k$ into $u_0$ and, in particular, of $\Psi ^c\cap E_k$ into $u_0$ in $\mathcal {U}\cap E_k$.

We claim that there exists $\delta _0>0$ such that

$$\begin{aligned} \int _{\Omega }[((1-\tau )u_i+\tau u_{0,i})^+]^{p+1} \ge \delta _0\quad \text {for all \ }\tau \in [0,1], \ u\in \Psi ^c, \ i=1,\ldots ,\ell . \end{aligned}$$

Otherwise, there would exist $\tau _n\in [0,1]$ and $u_n\in \Psi ^c$ such that

$$\begin{aligned} (1-\tau _n)\int _{\Omega }(u_{n,i}^+)^{p+1}\le \int _{\Omega }[((1-\tau _n)u_{n,i}+\tau _n u_{0,i})^+]^{p+1}\rightarrow 0 \end{aligned}$$

(4.8)

(the inequality is satisfied because $u_{0,i}>0$). From (4.2) we see that there exists $\delta >0$ such that $\int _{\Omega }(u_i^+)^{p+1}\ge \delta$ for all $u\in \Psi ^c$ and all i. Hence, $\tau _n\rightarrow 1$. Since $(u_n)$ is bounded in $\mathcal {H}$, a subsequence of $(u_{n,i})$ converges in $L^{p+1}(\Omega )$. Therefore,

$$\begin{aligned} \int _{\Omega }[((1-\tau _n)u_{n,i}+\tau _n u_{0,i})^+]^{p+1}\rightarrow \int _{\Omega }u_{0,i}^{p+1}\ge \delta , \end{aligned}$$

a contradiction to (4.8).

So, for every $\tau \in [0,1], \ u\in \Psi ^c, \ i=1,\ldots ,\ell ,$ we have

$$\begin{aligned} \int _{\Omega }(\eta _i(\tau ,u)^+)^{p+1}&= \int _{\Omega }\frac{[((1-\tau )u_i+\tau u_{0,i})^+]^{p+1}}{\Vert (1-\tau )u_i+\tau u_{0,i}\Vert ^{p+1}} \\&\ge \int _{\Omega }[((1-\tau )u_i+\tau u_{0,i})^+]^{p+1} \ge \delta _0, \end{aligned}$$

and we deduce from (4.2) that there exists $d>c$ such that

$$\begin{aligned} \eta (\tau ,u)\in \Psi ^d\cap E_k\qquad \text {for all \ }\tau \in [0,1], \ u\in \Psi ^c\cap E_k, \ k\ge 2. \end{aligned}$$

Next we show that $\Psi |_{\,\mathcal {U}\cap E_k}$ does not have a critical value in [c, d] for any large enough k. Indeed, if $u_k\in \Psi _c^d$ is a critical point of $\Psi |_{\,\mathcal {U}\cap E_k}$, then, according to (4.3),

$$\begin{aligned} \langle S_0(u_k), s^0_{u_k}v\rangle = 0 \quad \text {for all } v\in T_{u_k}(\mathcal {U}\cap E_k), \end{aligned}$$

i.e., $P_kS_0(u_k)=0$. Since $u_k\in \Psi _c^d\subset \Psi ^d\smallsetminus \mathcal {V}_t$ (see (4.5)), $k<k_d$ according to Lemma 4.1.

Now Proposition 3.4(iii) allows us to use the negative gradient flow of $\Psi |_{\,\mathcal {U}\cap E_k}$ in the standard way to obtain a retraction $\varrho :\Psi ^d\cap E_k\rightarrow \Psi ^c\cap E_k$; see, e.g., [3, Theorem I.3.2]. Then, $\varrho \circ \eta : [0,1]\times (\Psi ^c\cap E_k)\rightarrow \Psi ^c\cap E_k$ is a deformation of $\Psi ^c\cap E_k$ into a point. $\square$

The following statement is an immediate consequence of Lemma 4.2 and basic properties of homology (see e.g. [12, Sections III.4 and III.5]).

Corollary 4.3

Denote the q-th singular homology with coefficients in a field $\mathbb {F}$ by $\mathrm {H}_q(\cdot )$. Then $\mathrm {H}_0(\Psi ^c\cap E_k) = \mathbb {F}$ and $\mathrm {H}_q(\Psi ^c\cap E_k) = 0$ for $q\ne 0$. In particular, the Euler characteristic

$$\begin{aligned} \chi (\Psi ^c\cap E_k) := \sum _{q\ge 0}(-1)^q\dim _\mathbb {F} \mathrm {H}_q(\Psi ^c\cap E_k) = 1 \end{aligned}$$

for every large enough k.

For $u_0$ as above, let

$$\begin{aligned} \sigma _i:\mathscr {S}\smallsetminus \{-u_{0,i}\}\rightarrow ({\mathbb {R}}u_{0,i})^\perp =:F_i \end{aligned}$$

be the stereographic projection. The product $\sigma =(\sigma _1,\ldots ,\sigma _\ell )$ of the stereographic projections is a diffeomorphism. So its derivative at u

$$\begin{aligned} \sigma '(u): T_u(\mathcal {U})\rightarrow F := F_1 \times \ldots \times F_\ell \end{aligned}$$

is an isomorphism for every $u\in \mathcal {U}$. Note that, as $u_{0,i}\in E_{2,i}$, we have that $\sigma _i((\mathscr {S}\cap E_k)\smallsetminus \{-u_{0,i}\})\subset F_i\cap E_k$ for all $k\ge 2$.

Proof of Theorem 1.1

Let $\mathcal {O}:=\sigma (\mathcal {V}_0)$ with $\mathcal {V}_0$ as in (4.4). As $u_0\in \mathcal {V}_0$ we have that $0\in \mathcal {O}$, and as $\overline{\mathcal {V}}_0\subset \mathcal {U}$ and $-u_0\notin \mathcal {U}$, $\mathcal {O}$ is bounded in F. Set $\mathcal {O}_k:=\mathcal {O}\cap E_k$ and $F_k:=F\cap E_k$. Then $\mathcal {O}_k$ is a bounded open neighborhood of 0 in $F_k$, and $\overline{\mathcal {O}}_k\subset \sigma (\Psi ^c\cap E_k)$ for c as in (4.5).

Fix $k_c\in \mathbb {N}$ as in Lemma 4.1. Recall that

$$\begin{aligned} S_t(u) =s^t_uu- K_t(s^t_uu)\in T_u(\mathcal {U})\qquad \forall u\in \mathcal {U}\end{aligned}$$

(see Proposition 3.4(iv)). Define $G_{t,k}:\sigma (\mathcal {U}\cap E_k)\rightarrow F_k$ by

$$\begin{aligned} G_{t,k}(w) := (\sigma '(\sigma ^{-1}(w))\circ P_k\circ S_t\circ \sigma ^{-1})(w). \end{aligned}$$

(4.9)

Note that

$$\begin{aligned} G_{t,k}(w)=0\Longleftrightarrow P_k(S_t(\sigma ^{-1}(w)))=0. \end{aligned}$$

So, if $k\ge k_c$, $w\in \sigma (\mathcal {U}\cap E_k)$ and $G_{t,k}(w)=0$, Lemma 4.1 asserts that $w\in \mathcal {O}_k$. In particular, $G_{t,k}(w)\ne 0$ for every $w\in \partial \mathcal {O}_k$. From the homotopy and the excision properties of the Brouwer degree we get that

$$\begin{aligned} \deg (G_{1,k},\mathcal {O}_k,0)=\deg (G_{0,k},\mathcal {O}_k,0)=\deg (G_{0,k},\sigma (\Psi ^c\cap E_k),0). \end{aligned}$$

(4.10)

On the other hand, using (4.3) and (4.9) we get

$$\begin{aligned} (\Psi \circ \sigma ^{-1})'(w)z&=\Psi '(\sigma ^{-1}(w))[(\sigma ^{-1})'(w)z]\\&=s_{\sigma ^{-1}(w)}^0\langle P_k(S_0(\sigma ^{-1}(w))),\,(\sigma ^{-1})'(w)z\rangle \\&=s_{\sigma ^{-1}(w)}^0\langle (\sigma ^{-1})'(w)(G_{0,k}(w)),\,(\sigma ^{-1})'(w)z\rangle \\&=\frac{4s_{\sigma ^{-1}(w)}^0}{(\Vert w\Vert ^2+1)^2}\langle G_{0,k}(w),z\rangle \qquad \forall w\in \sigma (\mathcal {U}\cap E_k), \ z\in F_k. \end{aligned}$$

The last identity is obtained by a simple calculation, see e.g. [15, Lemma 3.4]. Since $(\Psi \circ \sigma ^{-1})|_{\sigma (\mathcal {U}\cap E_k)}$ is of class $\mathcal {C}^2$ (see (4.2)) and $-(\Psi \circ \sigma ^{-1})'(w)$ points into $(\Psi \circ \sigma ^{-1})^c$ for all $w\in (\Psi \circ \sigma ^{-1})^c_c$, from [3, Theorem II.3.3] and Corollary 4.3 we obtain

$$\begin{aligned} \deg (G_{0,k},\sigma (\Psi ^c\cap E_k),0)&=\deg (((\Psi \circ \sigma ^{-1})|_{\sigma (\mathcal {U}\cap E_k)})',\sigma (\Psi ^c\cap E_k),0)\nonumber \\&= \chi (\sigma (\Psi ^c\cap E_k)) = \chi (\Psi ^c\cap E_k) = 1. \end{aligned}$$

(4.11)

Combining (4.10) and (4.11) gives

$$\begin{aligned} \deg (G_{1,k},\mathcal {O}_k,0)=1. \end{aligned}$$

Hence, for each $k\ge k_c$ there exists $w_k\in \mathcal {O}_k$ such that $G_{k,1}(w_k) = 0$. Then $u_k := \sigma ^{-1}(w_k)\in \mathcal {V}_0\cap E_k\subset \Psi ^c\cap E_k$ satisfies $P_k(S(u_k))=0$, i.e.,

$$\begin{aligned} s_{u_k}u_k = P_kK(s_{u_k}u_k). \end{aligned}$$

(4.12)

As in the proof of Lemma 4.1 (with $t_n$ replaced by 1 and $s_{u_n}^{t_n}u_{n}$ by $s_{u_k}u_k$) one shows that $(s_{u_k,i})$ is bounded and bounded away from 0 for each i. So passing to a subsequence, $s_{u_k}\rightarrow s$ and $u_k\rightharpoonup u$ weakly in $\mathcal {H}$. Taking limits in (4.12) and using Lemma 3.1, we obtain that $s_{u_k}u_k\rightarrow su$ strongly in $\mathcal {H}$ and $su=K(su)$. Hence, $su\in \mathcal {N}$, $s=s_u$ and $S(u) = s_uu-K(s_uu) = 0$. So, according to Proposition 3.4(v), $s_uu$ is a solution to (1.1). $\square$

5 Synchronized solutions

A solution $u=(u_1,\ldots ,u_\ell )$ to (1.1) is called synchronized if $u_i=t_iv$ and $u_j=t_jv$ for some $i\ne j$, $v\in H^1_0(\Omega )\smallsetminus \{0\}$ and $t_1,t_2>0$. In this section we consider a system of 2 equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1 = \mu _1 u_1^p + \lambda _{12}u_1^{\alpha _{12}}u_2^{\beta _{12}}, \\ -\Delta u_2 = \mu _2 u_2^p + \lambda _{21}u_2^{\alpha _{21}}u_1^{\beta _{21}}, \\ u_1,u_2\ge 0 \text { in } \Omega , \quad u_1,u_2\in H^1_0(\Omega )\smallsetminus \{0\}. \end{array}\right. } \end{aligned}$$

(5.1)

Recall that according to our assumptions $\alpha _{12}+\beta _{12}<p$ and $\alpha _{21}+\beta _{21}<p$.

Theorem 5.1

The system (5.1) has a synchronized solution if and only if $\alpha _{12}+\beta _{12}=\alpha _{21}+\beta _{21}=:q$ and

$$\begin{aligned} \frac{\lambda _{12}}{\lambda _{21}} = \left( \frac{\mu _1}{\mu _2}\right) ^{(\alpha _{21}-\beta _{12}-1)/(p-1)}. \end{aligned}$$

(5.2)

Proof

Inserting $u_1=t_1v$, $u_2=t_2v$ into (5.1) we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} -t_1\Delta v = \mu _1 t_1^pv^p + \lambda _{12}t_1^{\alpha _{12}}t_2^{\beta _{12}}v^{\alpha _{12}+\beta _{12}} \\ -t_2\Delta v = \mu _2 t_2^pv^p + \lambda _{21}t_2^{\alpha _{21}}t_1^{\beta _{21}}v^{\alpha _{21}+\beta _{21}}. \end{array}\right. } \end{aligned}$$

Dividing the first equation by $t_1$, the second one by $t_2$ and subtracting gives

$$\begin{aligned} (\mu _1t_1^{p-1}-\mu _2t_2^{p-1})v^p + (\lambda _{12}t_1^{\alpha _{12}-1}t_2^{\beta _{12}}v^{\alpha _{12}+\beta _{12}} - \lambda _{21}t_2^{\alpha _{21}-1}t_1^{\beta _{21}}v^{\alpha _{21}+\beta _{21}}) = 0. \end{aligned}$$

So $\alpha _{12}+\beta _{12}=\alpha _{21}+\beta _{21}=q$,

$$\begin{aligned} \mu _1t_1^{p-1}-\mu _2t_2^{p-1}=0 \qquad \text {and} \qquad \lambda _{12}t_1^{\alpha _{12}-1}t_2^{\beta _{12}} = \lambda _{21}t_2^{\alpha _{21}-1}t_1^{\beta _{21}}. \end{aligned}$$

Inserting the solution

$$\begin{aligned} t_2 = \left( \frac{\mu _1}{\mu _2}\right) ^{1/(p-1)}t_1 \end{aligned}$$

of the first equation into the second one gives (5.2).

We have shown that the conditions in Theorem 5.1 are necessary. It remains to show that they are also sufficient. To this aim observe that, if $\alpha _{12}+\beta _{12}=\alpha _{21}+\beta _{21}=:q$, (5.2) holds true, and w satisfies

$$\begin{aligned} -\Delta w = \mu _1 w^p - aw^q,\qquad w\ge 0,\ w\in H_0^1(\Omega )\smallsetminus \{0\}, \end{aligned}$$

(5.3)

with

$$\begin{aligned} a := -\lambda _{12}\left( \frac{\mu _1}{\mu _2}\right) ^{\beta _{12}/(p-1)}, \end{aligned}$$

then $\left( w,\left( \frac{\mu _1}{\mu _2}\right) ^{1/(p-1)}w\right)$ solves the system (5.1). Consider the functional

$$\begin{aligned} \Phi (w) := \frac{1}{2}\int _{\Omega }|\nabla w|^2 +\frac{a}{q+1}\int _{\Omega }(w^+)^{q+1} - \frac{\mu _1}{p+1}\int _{\Omega }(w^+)^{p+1}. \end{aligned}$$

By standard arguments (see e.g. [17] or [18]), $\Phi$ is of class $\mathcal {C}^1$ and critical points of $\Phi$ are solutions to the equation

$$\begin{aligned} -\Delta w +a(w^+)^q = \mu _1 (w^+)^p. \end{aligned}$$

(5.4)

We shall complete the proof by showing that $\Phi$ has a nontrivial critical point $w\ge 0$. We use the mountain pass theorem (see e.g. [17] or [18]). By easy calculations (as e.g. in [18, Proof of Theorem 1.19]), $\Phi$ has the mountain pass geometry. Here it is important that $p>2$ and $p>q$. Next we show that $\Phi$ satisfies the Palais-Smale condition. Let $(w_n)$ be such that $\Phi (w_n)\rightarrow c$ and $\Phi '(w_n)\rightarrow 0$. Then

$$\begin{aligned} c+1+\Vert w_n\Vert&\ge \Phi (w_n)-\frac{1}{p+1}\Phi '(w_n)w_n \\&= \left( \frac{1}{2}-\frac{1}{p+1}\right) \int _{\Omega }|\nabla w_n|^2 +a\left( \frac{1}{q+1}-\frac{1}{p+1}\right) \int _{\Omega }(w_n^+)^{q+1} \end{aligned}$$

for all n large enough. Hence $(w_n)$ is bounded, so passing to a subsequence, $w_n\rightarrow w$ weakly in $H^1_0(\Omega )$, and strongly in $L^p(\Omega )$ and $L^q(\Omega )$. It follows by a standard argument (see e.g. [18, Proof of Lemma 1.20]) that $w_n\rightarrow w$ strongly also in $H^1_0(\Omega )$. Finally, multiplying (5.4) by $w^-$ gives $\int _{\Omega }|\nabla w^-|^2 = 0$, so $w^-=0$. The proof is complete. $\square$

Remark 5.2

It is easy to show that if $q=p$, then there are no synchronized solutions for $-\lambda _{ij}$ sufficiently large, as is well known in the variational case, see e.g. [4, Proposition 3.2].

Remark 5.3

Let $\lambda _{ij,n}<0$, $i\ne j$, and let $u_n=(u_{n,1},\ldots ,u_{n,\ell })$ be a solution to (1.1) with $\lambda _{ij}$ replaced by $\lambda _{ij,n}$. It is easy to see that, if the sequence $(u_n)$ is bounded in $\mathcal {H}$, the components $u_{n,i}$ separate spatially as $\lambda _{ij,n}\rightarrow -\infty$. More precisely, after passing to a subsequence, $u_{n,i}\rightarrow u_i\ne 0$ weakly in $H_0^1(\Omega )$ and strongly in $L^p(\Omega )$ for each i, and $u_{i}(x)\cdot u_{ j}(x)=0$ a.e. in $\Omega$ for $i\ne j$. There is an extensive literature on spatial separation of solutions and limiting profiles, under the assumption that the sequence $(u_n)$ is bounded and under different assumptions on the nonlinearities. See e.g. [6, 7, 16] and the references therein.

Obviously, synchronized solutions to (1.1) do not separate spatially. So we cannot expect the sequence $(w_n)$ given by (5.3) to be bounded. Indeed, we have the following

Proposition 5.4

Let $(w_n)$ be a sequence of solutions to (5.3) with $a=a_n$. If $a_n\rightarrow \infty$, then $(w_n)$ is unbounded in $H_0^1(\Omega )$.

Proof

Suppose $(w_n)$ is bounded. Then, passing to a subsequence, $w_n\rightarrow w$ weakly in $H^1_0(\Omega )$, strongly in $L^p(\Omega )$ and in $L^q(\Omega )$. Since

$$\begin{aligned} \int _{\Omega }|\nabla w_n|^2+a_n\int _{\Omega }w_n^q = \mu _1\int _{\Omega }w_n^p, \end{aligned}$$

we have that $w_n\rightarrow 0$ in $L^q(\Omega )$. So $w=0$ and therefore $w_n\rightarrow 0$ strongly in $H^1_0(\Omega )$. This is a contradiction because by the Sobolev inequality,

$$\begin{aligned} \int _{\Omega }|\nabla w_n|^2 \le \mu _1\int _{\Omega }w_n^p \le C\left( \int _{\Omega }|\nabla w_n|^2\right) ^{p/2} \end{aligned}$$

for some constant C, so $\Vert w_n\Vert$ is bounded away from 0. $\square$

It is well known that, when the system (1.1) is variational, least energy solutions are bounded in $\mathcal {H}$, independently of $\lambda _{ij}$. We close this section with the following open question.

Problem 5.5

Given $\lambda _{ij,n}\rightarrow -\infty$ for $i\ne j$, does the system (1.1) with $\lambda _{ij}$ replaced by $\lambda _{ij,n}$ have a solution $u_n$ such that the sequence $(u_{n,i})$ is bounded in $H_0^1(\Omega )$ for all i?

Appendix

In this appendix we prove Lemma 1.2. We employ some arguments which may be found in [11, 13]. First we note that by standard regularity results the solutions $u_i$ of (1.3) are in $\mathcal {C}^2(\Omega )\cap \mathcal {C}(\overline{\Omega })$.

Suppose there exists a sequence of solutions $(u_n)$ with $|u_n|_\infty \rightarrow \infty$. Passing to a subsequence, we may assume $|u_{n,i}|_\infty \rightarrow \infty$ and $|u_{n,i}|_\infty \ge |u_{n,j}|_\infty$ for some i and all j. There exists $x_n\in \Omega$ such that

$$\begin{aligned} \max _{x\in \Omega } u_{n,i}(x) = u_{n,i}(x_n). \end{aligned}$$

Let $\beta := \frac{2}{p-1}$ and choose $\varrho _n$ so that

$$\begin{aligned} \varrho _n^\beta \,|u_{n,i}|_\infty = 1. \end{aligned}$$

Then $\varrho _n\rightarrow 0$ and passing to a subsequence, $x_n\rightarrow x_0\in \overline{\Omega }$. Let

$$\begin{aligned} \Omega _n := \{y\in \mathbb {R}^N: \varrho _ny+x_n\in \Omega \} \end{aligned}$$

and

$$\begin{aligned} v_{n,j}(y) := \varrho _n^\beta u_{n,j}(\varrho _ny+x_n), \quad j=1,\ldots , \ell . \end{aligned}$$

(A.1)

Then,

$$\begin{aligned} 0\le v_{n,i}\le 1, \quad v_{n,i}(0)=1 \quad \text {and } v_{n,j}|_{\partial \Omega _n} = 0 \text { for all } j. \end{aligned}$$

(A.2)

Passing to a subsequence, there are two possible cases and we shall complete the proof by ruling out both of them. Denote the distance from x to a set A by d(x, A).

Case 1. $\frac{d(x_n,\partial \Omega )}{\varrho _n}\rightarrow \infty$.

Since $\varrho _ny+x_n\in \Omega$ if $|y| < \frac{d(x_n,\partial \Omega )}{\varrho _n}$, for each $R>0$ there exists $n_0$ such that $B_R(0)\subset \Omega _n$ whenever $n\ge n_0$. For $y\in B_R(0)$ and $n\ge n_0$ we have

$$\begin{aligned} -\Delta _y v_{n,i}&= \varrho _n^{\beta +2}\Delta _x u_{n,i} = \varrho _n^{\beta +2}\Big (\mu _iu_{n,i}^p+\sum \limits _{j\ne i}\lambda _{ij}u_{n,i}^{\alpha _{ij}}u_{n,j}^{\beta _{ij}}\Big ) \\&= \varrho _n^{\beta +2-\beta p}\mu _i v_{n,i}^p + \sum \limits _{j\ne i} \varrho _n^{\beta +2-\beta (\alpha _{ij}+\beta _{ij})}\lambda _{ij}v_{n,i}^{\alpha _{ij}}v_{n,j}^{\beta _{ij}}. \end{aligned}$$

Since $\beta +2-\beta p = 0$ and $\gamma _{ij} := \beta +2-\beta (\alpha _{ij}+\beta _{ij}) > 0$, we can re-write this identity as

$$\begin{aligned} -\Delta v_{n,i} = \mu _i v_{n,i}^p + \sum \limits _{j\ne i} \varrho _n^{\gamma _{ij}}\lambda _{ij}v_{n,i}^{\alpha _{ij}}v_{n,j}^{\beta _{ij}}. \end{aligned}$$

By elliptic estimates, $(v_{n,i})$ is bounded in $W^{2,q}(B_R(0))$ for some $q>N$. So passing to a subsequence, $v_{n,i} \rightarrow v_i$ weakly in $W^{2,q}(B_R(0))$ and strongly in $\mathcal {C}^1(B_R(0))$. Since $\varrho _n^{\gamma _{ij}}\rightarrow 0$, $v_i$ is a nonnegative solution to the equation

$$\begin{aligned} -\Delta v = \mu _iv^p \end{aligned}$$

in $B_R(0)$. Let now $R_m\rightarrow \infty$. Then for each m we get a solution $v_{i,m}$ of the above equation in $B_{R_m}(0)$. Passing to subsequences and applying the diagonal procedure, we see that $v_{i,m}\rightarrow w$, weakly in $W^{2,q}_{loc}(\mathbb {R}^N)$ and strongly in $\mathcal {C}^1_{loc}(\mathbb {R}^N)$. So $-\Delta w = \mu _iw^p$ in $\mathbb {R}^N$, $w\ge 0$, $w(0)=1$ according to (A.2), and $w\in \mathcal {C}^2(\mathbb {R}^N)$ by Schauder estimates. Replacing w with cw for a suitable $c>0$ we may assume $\mu _i=1$. Hence it follows from [13, Theorem 1.2] that $w=0$ which rules out Case 1.

Case 2. $\frac{d(x_n,\partial \Omega )}{\varrho _n}\rightarrow d\in [0,\infty )$.

It is clear that $x_0\in \partial \Omega$ and we may assume without loss of generality that $x_0=0$ and $\nu = (0,\ldots ,0,1)$ is the unit outer normal to $\partial \Omega$ at $x_0$. Let

$$\begin{aligned} \mathbb {H}^N:= \{y\in \mathbb {R}^N: y_N<d\} \quad \text {where } y=(y_1,\ldots ,y_N). \end{aligned}$$

We shall need the following result.

Lemma A.1

• (i) Let $A\subset \mathbb {H}^N$ be compact. Then there exists $n_0$ such that $\varrho _ny+x_n\in \Omega$ for all $n\ge n_0$ and $y\in A$.

• (ii) Let $A\subset \mathbb {R}^N\smallsetminus \overline{\mathbb {H}^N}$ be compact. Then there exists $n_0$ such that $\varrho _ny+x_n\notin \Omega$ for all $n\ge n_0$ and $y\in A$.

Proof

(i) :  Since A is compact, there exists $\varepsilon >0$ such that $y_N<d-2\varepsilon$ for all $y\in A$. For each n there exists $\widehat{x}_n\in \partial \Omega$ which is closest to $x_n$, i.e., $d(x_n,\partial \Omega ) = |x_n-\widehat{x}_n|$. As $\partial \Omega$ is tangent to the hyperplane $x_N=0$ at 0,

$$\begin{aligned} \frac{|x_n-\widehat{x}_n|}{\varrho _n} = \frac{\widehat{x}_{n,N}-x_{n,N}}{\varrho _n} + o(1). \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{x_{n,N}-\widehat{x}_{n,N}}{\varrho _n}< -d+\varepsilon \qquad \text {and} \qquad y_{N} + \frac{x_{n,N}-\widehat{x}_{n,N}}{\varrho _n} < -\varepsilon \end{aligned}$$

for all $y\in A$ if n is large enough. There exists $C>0$ such that

$$\begin{aligned} \left| y+\frac{x_n-\widehat{x}_n}{\varrho _n}\right| \le C \quad \text {for all } y\in A. \end{aligned}$$

Using this, we see that there is $n_0$ such that, if $n\ge n_0$ and $y\in A$, then $\varrho _ny+x_n-\widehat{x}_n\in \Omega$ and, as $\widehat{x}_n\in \partial \Omega$ and $\partial \Omega$ is tangent to the hyperplane $x_N=0$ at 0, $\varrho _ny+x_n = \varrho _ny+(x_n-\widehat{x}_n) + \widehat{x}_n\in \Omega$.

(ii) :  This time $y_N>d+2\varepsilon$ for $y\in A$,

$$\begin{aligned} \frac{x_{n,N}-\widehat{x}_{n,N}}{\varrho _n}> -d-\varepsilon \quad \text {and} \quad y_{n,N} + \frac{x_{n,N}-\widehat{x}_{n,N}}{\varrho _n} > \varepsilon \end{aligned}$$

if n is sufficiently large, and the conclusion follows by a similar argument as above. $\square$

Now we can continue with Case 2. Let $\omega _R := B_R(0)\cap \{y\in \mathbb {R}^N: y_N<d-1/R\}$. Then $\overline{\omega }_R\subset \Omega _n$ for all $n\ge n_0$ by Lemma A.1(i). Let $v_{n,i}$ be given by (A.1) and using (A.2) extend it by 0 outside $\Omega _n$. According to Lemma A.1(ii), if $A\subset \mathbb {R}^N\smallsetminus \overline{\mathbb {H}^N}$ is compact, then $\varrho _ny+x_n\notin \Omega$ for all $y\in A$ and n large enough. So

$$\begin{aligned} \lim _{n\rightarrow \infty }v_{n,i}(x)=0 \quad \text {for all } x\notin \mathbb {H}^N. \end{aligned}$$

(A.3)

We can repeat the argument of Case 1 which now gives a nonegative solution to the equation $-\Delta w = \mu _iw^p$ in $\mathbb {H}^N$ such that $w(0)=1$. By (A.3), $w=0$ on $\partial \Omega$. As before, $w\in \mathcal {C}^2(\mathbb {H}^N)$, and since the extended functions $v_{n,i}$ are continuous in $\mathbb {R}^N$, $w\in \mathcal {C}^0(\overline{\mathbb {H}^N})$. So $w=0$ according to [13, Theorem 1.3], a contradiction. Hence also Case 2 is ruled out.