This work is devoted to the study of existence and multiplicity of vector solutions $u(x)=({u}_{1}(x),\mathrm{\dots},{u}_{N}(x))$ of the problem

$Lu(x)={g}_{0}(x,u(x))+p(x)\mathit{\hspace{1em}}x\in \mathrm{\Omega},$(1.1)

with the boundary condition

$\frac{\partial u}{\partial \nu}(x)=\gamma (x)u(x)\mathit{\hspace{1em}}x\in \partial \mathrm{\Omega}.$(1.2)

Here *L* denotes the vector Laplacian given by $Lu:=(\mathrm{\Delta}{u}_{1},\mathrm{\dots},\mathrm{\Delta}{u}_{N})$, $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$ is a smooth bounded domain
and ${g}_{0}:\overline{\mathrm{\Omega}}\times {\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$ is a smooth function.
Without loss of generality,
it shall be assumed throughout the paper
that ${g}_{0}(x,0)=0$.
We shall also assume
that $p\in C(\overline{\mathrm{\Omega}},{\mathbb{R}}^{N})$ and $\gamma \in {C}^{1}(\overline{\mathrm{\Omega}})$.
It is remarked that, unlike in the most standard Robin condition, we do not assume that $\gamma \le 0$.

The problem arises in many different applications; for instance (see [9] and the references therein) for some particular ${g}_{0}$, it can be seen as the
steady state of the population density of some species,
governed by a parabolic problem.
The boundary
condition corresponds to the law of the population flux,
given on the boundary by γ.
Thus, the inflow/outflow of population
to the region occurs when $\gamma (x)>0$ or $\gamma (x)<0$, respectively.
Our direct motivation for the present work can be found in
[4], where a
Painlevé II (ordinary) scalar equation
was considered,
namely

${u}^{\prime \prime}(x)=\frac{u{(x)}^{3}}{2}+\delta (x)u(x)+p$

under the radiation boundary conditions ${u}^{\prime}(0)={a}_{0}u(0)$, ${u}^{\prime}(1)={a}_{1}u(1)$ with ${a}_{0},{a}_{1}>0$.
It was shown that ${a}_{1}>0$
is compensated by the effect of the superlinear term $\frac{{u}^{3}}{2}$ in such a way that the associated functional is still coercive, giving rise to a unique solution when ${a}_{0}\ge {a}_{1}$ and exactly three solutions when ${a}_{0}<{a}_{1}$, provided that the positive parameter *p* is small. It is worth noticing that, in this model, δ is a linear function depending on ${a}_{0}$ and ${a}_{1}$; thus, the uniqueness/multiplicity of solutions can be understood as a consequence of the interaction between the nonlinear term of the equation and the Robin coefficients.
The previous results have been extended in [3] and [2] for the equation
${u}^{\prime \prime}(x)={g}_{0}(x,u(x))+p(x)$,
where ${g}_{0}$ is a general superlinear function, that is,

$\underset{|u|\to \mathrm{\infty}}{lim}\frac{{g}_{0}(x,u)}{u}=+\mathrm{\infty}$

uniformly on *x*. Indeed, it was shown that the interaction between $g,{a}_{0}$ and ${a}_{1}$ can be expressed in a very
precise way in terms of the spectrum of the associated linear operator $-{u}^{\prime \prime}$ under the Robin conditions: if

$\frac{\partial {g}_{0}}{\partial u}(x,u)>-{\lambda}_{1}$(1.3)

for all *x* and all *u*, then the problem has a unique solution,
and if

$-{\lambda}_{k}\u2a88\frac{\partial {g}_{0}}{\partial u}(\cdot ,0)\u2a88-{\lambda}_{k+1}$(1.4)

for some odd *k*, then the problem has at least three solutions when ${\parallel p\parallel}_{\mathrm{\infty}}$ is small.
Furthermore,
if (1.4) holds with *k* even, then the problem has at least five solutions for small *p*, provided that ${\lambda}_{1}<0$.

The main purpose of this paper consists in obtaining suitable extensions of the results mentioned above in two directions: in the first place, we shall consider a system instead of a scalar equation and in the second place, we shall not be longer dealing with ODEs but with a PDE system.
We shall study system (1.1) in terms of the eigenvalues
${\lambda}_{1}<{\lambda}_{2}\le {\lambda}_{3}\le \mathrm{\cdots}\to +\mathrm{\infty}$
of the scalar problem $-\mathrm{\Delta}\phi =\lambda \phi $
associated to the Robin
condition ${\frac{\partial \phi}{\partial \nu}|}_{\partial \mathrm{\Omega}}=\gamma \phi $
and ${\lambda}_{0}:=-\mathrm{\infty}$.
At first sight, it seems reasonable to assume that ${g}_{0}$ is superlinear, that is,

$\underset{|u|\to \mathrm{\infty}}{lim}\frac{\u3008{g}_{0}(x,u),u\u3009}{{|u|}^{2}}=+\mathrm{\infty}$

uniformly on *x*, where $\u3008\cdot ,\cdot \u3009$ and $|\cdot |$ denote the usual inner product and norm of ${\mathbb{R}}^{N}$, respectively.
However, we shall employ a weaker condition, namely, that there exists a constant $\mu \ge -{\lambda}_{1}$
depending only on Ω and γ such that, if

$\underset{|u|\to \mathrm{\infty}}{lim\; inf}\frac{\u3008{g}_{0}(x,u),u\u3009}{{|u|}^{2}}>\mu $(1.5)

uniformly on *x*,
then the problem has a solution (see Theorem 3.3). This constant can be estimated: for example, it will be shown that it always suffices to take

$\mu :=-{\lambda}_{1}+{\parallel \frac{\nabla {\phi}_{1}}{{\phi}_{1}}\parallel}_{\mathrm{\infty}}^{2},$

where ${\phi}_{1}>0$ is the (unique) principal eigenfunction such that ${\parallel {\phi}_{1}\parallel}_{{L}^{2}}=1$.
For the ODE case $n=1$ or
when *g* grows like ${|u|}^{q}$ for some $q\le \frac{n+2}{n-2}$, it can be seen that the condition $\mu =-{\lambda}_{1}$ is optimal for our purposes.

Extra solutions will be obtained when *p* is
small, under an appropriate
condition on the interaction between
${D}_{u}{g}_{0}(x,0)$ and
the spectrum of the linear associated operator.
In more precise terms, if for simplicity we assume
that ${g}_{0}={\nabla}_{u}{G}_{0}$ for some smooth function
${G}_{0}:\overline{\mathrm{\Omega}}\times {\mathbb{R}}^{N}\to \mathbb{R}$, then our main hypothesis is that
the Hessian matrix at $u=0$ satisfies

$A\le -{H}_{u}{G}_{0}(x,0)\le B,$

where $A,B\in {\mathbb{R}}^{N\times N}$ are symmetric matrices with eigenvalues ${a}_{1}\le \mathrm{\cdots}\le {a}_{N}$ and
${b}_{1}\le \mathrm{\cdots}\le {b}_{N}$ such that

${\lambda}_{{\nu}_{k}}<{a}_{k}\le {b}_{k}<{\lambda}_{{\nu}_{k}+1}$(1.6)

for some ${\nu}_{k}\in {\mathbb{N}}_{0}$ and all $k=1,\mathrm{\dots},N$. Here, the ordering $\le $ over the set of symmetric functions is understood in the usual sense that
$Y\le Z$ if and only if $Z-Y$ is positive semi-definite or equivalently, $\u3008Yu,u\u3009\le \u3008Zu,u\u3009$ for all $u\in {\mathbb{R}}^{N}$.

It shall be seen that if
${\nu}_{1}+\mathrm{\cdots}+{\nu}_{N}$ is
an odd number, then the problem
has at least two solutions (and generically three), provided that ${\parallel p\parallel}_{{L}^{2}}$ is small enough.
In fact, the result holds for a general function ${g}_{0}$, with the role of the Hessian matrix now played by the Jacobian matrix
${D}_{u}{g}_{0}(x,0)$, provided it is symmetric. In contrast with the
case ${g}_{0}={\nabla}_{u}{G}_{0}$, the general situation does not have variational structure and is presented in our main result through the Leray–Schauder degree method (see Theorem 5.2 below).
To our knowledge, there are no previous similar results in the literature concerning the multiplicity of solutions for an elliptic system with indefinite Robin conditions.

The paper is organised as follows. In the next section, we recall some basic facts concerning the spectrum of the associated
linear operator.
In Section 3, we prove the existence of at least one solution by means of an appropriate space-dependent Hartman condition.
In Section 4, we obtain ${H}^{1}$ bounds for all solutions, provided that $\mu \ge -{\lambda}_{1}$. Furthermore, enlarging μ if necessary, we
also obtain ${L}^{\mathrm{\infty}}$ bounds, which shall be the key for proving our multiplicity result. This task is accomplished in Section 5, where we firstly prove a lemma that helps to compute the degree of
certain linear operators and then apply the lemma for the computation of the degree of $I-K$, where *K* is an appropriate
compact fixed point operator.
Roughly speaking, for $p=0$ we shall prove that the degree is equal to 1 over a large ball and equal to -1 over a small neighbourhood of the origin. This guarantees, when *p* is small, the existence of a
solution close to the origin and typically two ‘large’ solutions.

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