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Volume 8, Issue 1

# Multiple solutions for an elliptic system with indefinite Robin boundary conditions

Pablo Amster
• Corresponding author
• Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires; and IMAS – CONICET, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
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Published Online: 2017-07-08 | DOI: https://doi.org/10.1515/anona-2017-0034

## Abstract

Multiplicity of solutions is proved for an elliptic system with an indefinite Robin boundary condition under an assumption that links the linearisation at 0 and the eigenvalues of the associated linear scalar operator. Our result is based on a precise calculation of the topological degree of a suitable fixed point operator over large and small balls.

MSC 2010: 35J57; 35J60

## 1 Introduction

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

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

with the boundary condition

$\frac{\partial u}{\partial \nu }\left(x\right)=\gamma \left(x\right)u\left(x\right)\mathit{ }x\in \partial \mathrm{\Omega }.$(1.2)

Here L denotes the vector Laplacian given by $Lu:=\left(\mathrm{\Delta }{u}_{1},\mathrm{\dots },\mathrm{\Delta }{u}_{N}\right)$, $\mathrm{\Omega }\subset {ℝ}^{n}$ is a smooth bounded domain and ${g}_{0}:\overline{\mathrm{\Omega }}×{ℝ}^{N}\to {ℝ}^{N}$ is a smooth function. Without loss of generality, it shall be assumed throughout the paper that ${g}_{0}\left(x,0\right)=0$. We shall also assume that $p\in C\left(\overline{\mathrm{\Omega }},{ℝ}^{N}\right)$ and $\gamma \in {C}^{1}\left(\overline{\mathrm{\Omega }}\right)$. 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 $\gamma \left(x\right)>0$ or $\gamma \left(x\right)<0$, respectively. Our direct motivation for the present work can be found in [4], where a Painlevé II (ordinary) scalar equation was considered, namely

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

under the radiation boundary conditions ${u}^{\prime }\left(0\right)={a}_{0}u\left(0\right)$, ${u}^{\prime }\left(1\right)={a}_{1}u\left(1\right)$ 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 ${u}^{\prime \prime }\left(x\right)={g}_{0}\left(x,u\left(x\right)\right)+p\left(x\right)$, where ${g}_{0}$ is a general superlinear function, that is,

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

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

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

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

$-{\lambda }_{k}⪈\frac{\partial {g}_{0}}{\partial u}\left(\cdot ,0\right)⪈-{\lambda }_{k+1}$(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,

$\underset{|u|\to \mathrm{\infty }}{lim}\frac{〈{g}_{0}\left(x,u\right),u〉}{{|u|}^{2}}=+\mathrm{\infty }$

uniformly on x, where $〈\cdot ,\cdot 〉$ and $|\cdot |$ denote the usual inner product and norm of ${ℝ}^{N}$, 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

$\underset{|u|\to \mathrm{\infty }}{lim inf}\frac{〈{g}_{0}\left(x,u\right),u〉}{{|u|}^{2}}>\mu$(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 ${D}_{u}{g}_{0}\left(x,0\right)$ 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 ${G}_{0}:\overline{\mathrm{\Omega }}×{ℝ}^{N}\to ℝ$, then our main hypothesis is that the Hessian matrix at $u=0$ satisfies

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

where $A,B\in {ℝ}^{N×N}$ 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 ${\nu }_{k}\in {ℕ}_{0}$ 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, $〈Yu,u〉\le 〈Zu,u〉$ for all $u\in {ℝ}^{N}$.

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 ${D}_{u}{g}_{0}\left(x,0\right)$, 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.

## 2 Preliminaries

For convenience, throughout the paper we shall employ the notation $g\left(x,u\right):={g}_{0}\left(x,u\right)+p\left(x\right)$. It is clear that if ${g}_{0}$ satisfies (1.5), then so does g for arbitrary $p\in C\left(\overline{\mathrm{\Omega }}\right)$. Sometimes we shall express the condition in the following equivalent form: there exist ${C}_{\mu }\ge 0$ and $\epsilon >0$ such that

$〈g\left(x,u\right),u〉>\left(\mu +\epsilon \right){|u|}^{2}-{C}_{\mu }\mathit{ }x\in \overline{\mathrm{\Omega }},u\in {ℝ}^{N}.$(2.1)

Before establishing our main results, we need to recall some basic facts concerning the spectrum of the associated linear operator. Let us observe, in the first place, that the scalar operator $-\mathrm{\Delta }$ is symmetric for the Robin boundary condition. Thus, by standard arguments (see e.g. [5]) one can deduce the existence of a sequence of eigenvalues ${\lambda }_{1}\le {\lambda }_{2}\le {\lambda }_{3}\le \mathrm{\cdots }\to +\mathrm{\infty }$ and an associated orthonormal basis of ${L}^{2}\left(\mathrm{\Omega }\right)$ consisting of eigenfunctions ${\left\{{\phi }_{j}\right\}}_{j\in ℕ}$. For convenience, we shall denote ${\lambda }_{0}:=-\mathrm{\infty }$.

It is verified (see [1]) that ${\lambda }_{1}$ is simple and the associated eigenfunctions do not vanish on $\overline{\mathrm{\Omega }}$. From now on, we shall always assume that ${\phi }_{1}$ is such that ${\phi }_{1}>0$. Moreover, ${\phi }_{1}$ can be obtained as a global minimiser of the functional

$\mathcal{ℐ}\left(\phi \right):={\int }_{\mathrm{\Omega }}\frac{{|\nabla \phi \left(x\right)|}^{2}}{2}𝑑x-{\int }_{\partial \mathrm{\Omega }}\frac{\gamma \left(x\right)\phi {\left(x\right)}^{2}}{2}𝑑S$

subject to the restriction ${\parallel \phi \parallel }_{{L}^{2}}=1$. It follows that

${\lambda }_{1}{\parallel \phi \parallel }_{{L}^{2}}^{2}\le -{\int }_{\mathrm{\Omega }}\mathrm{\Delta }\phi \left(x\right)\phi \left(x\right)𝑑x$(2.2)

for any smooth function φ satisfying the Robin condition. In particular, adding $\left(\eta -{\lambda }_{1}\right){\parallel \phi \parallel }_{{L}^{2}}^{2}$ to both sides for some $\eta >0$, the Cauchy–Schwarz inequality yields the following estimate:

${\parallel \phi \parallel }_{{L}^{2}}\le \frac{1}{\eta }{\parallel \mathrm{\Delta }\phi +\left({\lambda }_{1}-\eta \right)\phi \parallel }_{{L}^{2}}.$(2.3)

#### Remark 2.1.

Using the standard norm of ${L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)$ still denoted by ${\parallel u\parallel }_{{L}^{2}}$, namely

${\parallel u\parallel }_{{L}^{2}}:={\left(\sum _{i=1}^{N}{\parallel {u}_{i}\parallel }_{{L}^{2}}^{2}\right)}^{1/2},$

we observe that (2.2) and (2.3) can be extended in an obvious way for vector functions, replacing Δ by L. In particular, the latter inequality implies, as an immediate consequence, the uniqueness of solutions, provided that $g\left(x,u\right)+{\lambda }_{1}u$ is (strictly) monotone increasing, that is,

$〈g\left(x,u\right)-g\left(x,v\right)+{\lambda }_{1}\left(u-v\right),u-v〉>0$

for all x and $u\ne v$. Clearly, this extends condition (1.3) for $N>1$.

In order to understand the dependence of ${\phi }_{1}$ with respect to γ, observe that if $\stackrel{~}{\mathcal{ℐ}}$ is the functional associated to some $\stackrel{~}{\gamma }$ and if ${\stackrel{~}{\lambda }}_{1},{\stackrel{~}{\phi }}_{1}$ are the respective eigenvalue and eigenfunction, then, by virtue of the trace embedding ${H}^{1}\left(\mathrm{\Omega }\right)↪{L}^{2}\left(\partial \mathrm{\Omega }\right)$, we deduce that

$|\mathcal{ℐ}\left(\phi \right)-\stackrel{~}{\mathcal{ℐ}}\left(\phi \right)|\le c{\parallel \gamma -\stackrel{~}{\gamma }\parallel }_{\mathrm{\infty }}{\parallel \phi \parallel }_{{H}^{1}}^{2}.$

In particular,

${\stackrel{~}{\lambda }}_{1}-{\lambda }_{1}\le \stackrel{~}{\mathcal{ℐ}}\left({\phi }_{1}\right)-{\lambda }_{1}\le |\mathcal{ℐ}\left({\phi }_{1}\right)-\stackrel{~}{\mathcal{ℐ}}\left({\phi }_{1}\right)|\le c{\parallel \gamma -\stackrel{~}{\gamma }\parallel }_{\mathrm{\infty }}$

and

${\lambda }_{1}-{\stackrel{~}{\lambda }}_{1}\le \mathcal{ℐ}\left({\stackrel{~}{\phi }}_{1}\right)-{\stackrel{~}{\lambda }}_{1}\le |\mathcal{ℐ}\left({\stackrel{~}{\phi }}_{1}\right)-\stackrel{~}{\mathcal{ℐ}}\left({\stackrel{~}{\phi }}_{1}\right)|\le c{\parallel \gamma -\stackrel{~}{\gamma }\parallel }_{\mathrm{\infty }}.$

We conclude that if $\stackrel{~}{\gamma }\to \gamma$ uniformly, then ${\stackrel{~}{\lambda }}_{1}\to {\lambda }_{1}$. Furthermore, since $-\mathrm{\Delta }{\stackrel{~}{\phi }}_{1}={\stackrel{~}{\lambda }}_{1}{\stackrel{~}{\phi }}_{1}$ we deduce that the family $\left\{{\stackrel{~}{\phi }}_{1}\right\}$ is uniformly bounded in ${H}^{2}\left(\mathrm{\Omega }\right)$, and by a bootstrapping argument, we conclude that it is uniformly bounded in ${C}^{2}\left(\overline{\mathrm{\Omega }}\right)$. We claim that ${\stackrel{~}{\phi }}_{1}\to {\phi }_{1}$ for the ${C}^{1}$ norm. Indeed, otherwise we may suppose that ${\stackrel{~}{\phi }}_{1}$ converges for the ${C}^{1}$ norm to some $\phi \ne {\phi }_{1}$. Observe that ${\parallel \phi \parallel }_{{L}^{2}}=1$ and $\phi \ge 0$; moreover, since $\mathrm{\Delta }{\stackrel{~}{\phi }}_{1}\to -{\lambda }_{1}{\phi }_{1}$, it is readily seen that $\mathrm{\Delta }\phi =-{\lambda }_{1}\phi$ and hence $\phi ={\phi }_{1}$, a contradiction.

## 3 A space-dependent Hartman condition

Our general existence result is based on the fact that g satisfies the following Hartman type condition (the original condition was formulated in [6]):

• (H)

There exists a smooth function $R:\overline{\mathrm{\Omega }}\to \left(0,+\mathrm{\infty }\right)$ such that

$〈g\left(x,u\right),u〉\ge R\left(x\right)\mathrm{\Delta }R\left(x\right)+{|\nabla R\left(x\right)|}^{2}$(3.1)

for all $x\in \mathrm{\Omega }$ and all $u\in {ℝ}^{N}$ such that $|u|=R\left(x\right)$.

#### Lemma 3.1.

Assume that there exists $R\mathrm{:}\overline{\mathrm{\Omega }}\mathrm{\to }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{+}\mathrm{\infty }\mathrm{\right)}$ as before such that (3.1) holds and

$\frac{\partial R}{\partial \nu }\left(x\right)\ge R\left(x\right)\gamma \left(x\right)\mathit{ }x\in \partial \mathrm{\Omega }.$(3.2)

Then problem (1.1)–(1.2) has at least one solution u such that $\mathrm{|}u\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{|}\mathrm{\le }R\mathit{}\mathrm{\left(}x\mathrm{\right)}$ for all $x\mathrm{\in }\mathrm{\Omega }$.

#### Proof.

Let $P\left(x,u\right):=\mathrm{min}\left\{R\left(x\right),|u|\right\}\frac{u}{|u|}$. For fixed $\epsilon ,\theta >0$, a straightforward application of Schauder’s theorem shows that the truncated problem

$\left\{\begin{array}{cccc}\hfill Lu\left(x\right)-\epsilon u\left(x\right)& =g\left(x,P\left(x,u\left(x\right)\right)\right)-\epsilon P\left(x,u\left(x\right)\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill \frac{\partial u}{\partial \nu }\left(x\right)+\theta u\left(x\right)& =\left(\gamma \left(x\right)+\theta \right)P\left(x,u\left(x\right)\right),\hfill & & \hfill x\in \partial \mathrm{\Omega }\end{array}$

has at least one solution u. We claim that, in view of (3.1) and (3.2) it can be deduced that $|u\left(x\right)|\le R\left(x\right)$ for all x and, consequently, u is a solution of the original problem (1.1)–(1.2). Indeed, otherwise the function $\varphi \left(x\right):=|u\left(x\right)|-R\left(x\right)$ achieves a strictly positive maximum value at some point $\stackrel{^}{x}$. If $\stackrel{^}{x}\in \mathrm{\Omega }$, then

$0={\partial }_{j}\varphi \left(\stackrel{^}{x}\right)=〈\frac{u\left(\stackrel{^}{x}\right)}{|u\left(\stackrel{^}{x}\right)|},{\partial }_{j}u\left(\stackrel{^}{x}\right)〉-{\partial }_{j}R\left(\stackrel{^}{x}\right)$

for all j and $\mathrm{\Delta }\varphi \left(\stackrel{^}{x}\right)\le 0$, that is,

$0\ge \frac{-1}{{|u\left(\stackrel{^}{x}\right)|}^{3}}\sum _{j}{〈u\left(\stackrel{^}{x}\right),{\partial }_{j}u\left(\stackrel{^}{x}\right)〉}^{2}+\frac{1}{|u\left(\stackrel{^}{x}\right)|}\left({|\nabla u\left(\stackrel{^}{x}\right)|}^{2}+〈u\left(\stackrel{^}{x}\right),Lu\left(\stackrel{^}{x}\right)〉\right)-\mathrm{\Delta }R\left(\stackrel{^}{x}\right),$

where we employed the notation ${|\nabla u|}^{2}:={\sum }_{i=1}^{N}{|\nabla {u}_{i}|}^{2}$. Then

$〈\frac{u\left(\stackrel{^}{x}\right)}{|u\left(\stackrel{^}{x}\right)|},Lu\left(\stackrel{^}{x}\right)〉\le \mathrm{\Delta }R\left(\stackrel{^}{x}\right)+\frac{{|\nabla R\left(\stackrel{^}{x}\right)|}^{2}}{|u\left(\stackrel{^}{x}\right)|}\le \mathrm{\Delta }R\left(\stackrel{^}{x}\right)+\frac{{|\nabla R\left(\stackrel{^}{x}\right)|}^{2}}{R\left(\stackrel{^}{x}\right)}.$

Moreover, observe that

$〈Lu\left(\stackrel{^}{x}\right),u\left(\stackrel{^}{x}\right)〉=〈g\left(\stackrel{^}{x},P\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right)\right),u\left(\stackrel{^}{x}\right)〉+\epsilon 〈u\left(\stackrel{^}{x}\right)-P\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right)\right),u\left(\stackrel{^}{x}\right)〉>〈g\left(\stackrel{^}{x},P\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right)\right),u\left(\stackrel{^}{x}\right)〉$

and consequently, using (3.1), we obtain

$\mathrm{\Delta }R\left(\stackrel{^}{x}\right)+\frac{{|\nabla R\left(\stackrel{^}{x}\right)|}^{2}}{R\left(\stackrel{^}{x}\right)}>〈\frac{u\left(\stackrel{^}{x}\right)}{|u\left(\stackrel{^}{x}\right)|},g\left(\stackrel{^}{x},P\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right)\right)〉=\frac{1}{R\left(\stackrel{^}{x}\right)}〈P\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right),g\left(\stackrel{^}{x},P\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right)\right)〉\ge \mathrm{\Delta }R\left(\stackrel{^}{x}\right)+\frac{{|\nabla R\left(\stackrel{^}{x}\right)|}^{2}}{R\left(\stackrel{^}{x}\right)},$

Now suppose $\stackrel{^}{x}\in \partial \mathrm{\Omega }$, then

$0\le \frac{\partial \varphi }{\partial \nu }\left(\stackrel{^}{x}\right)=〈\frac{u\left(\stackrel{^}{x}\right)}{|u\left(\stackrel{^}{x}\right)|},\frac{\partial u}{\partial \nu }\left(\stackrel{^}{x}\right)〉-\frac{\partial R}{\partial \nu }\left(\stackrel{^}{x}\right).$

This implies that

$\frac{\partial R}{\partial \nu }\left(\stackrel{^}{x}\right)<\gamma \left(\stackrel{^}{x}\right)R\left(\stackrel{^}{x}\right),$

#### Remark 3.2.

Observe that if (3.2) is strict, then the previous proof is still valid for $\theta =0$. Furthermore, if (3.1) is strict, then $|u\left(x\right)| for all x.

In order to establish the existence of solutions, let us observe that, without loss of generality, we may assume that $\mathrm{\Omega }={F}^{-1}\left(-\mathrm{\infty },0\right)$ for some ${C}^{2}$ mapping $F:\overline{\mathrm{\Omega }}\to ℝ$ such that 0 is a regular value of F. Thus, the outer normal at $x\in \partial \mathrm{\Omega }$ is simply computed as $\nu \left(x\right)=\frac{\nabla F\left(x\right)}{|\nabla F\left(x\right)|}$. For convenience, let us fix the following constants:

$a>\underset{x\in \partial \mathrm{\Omega }}{\mathrm{max}}\frac{\gamma \left(x\right)}{|\nabla F\left(x\right)|},C:={\parallel a\mathrm{\Delta }F+2{a}^{2}{|\nabla F|}^{2}\parallel }_{\mathrm{\infty }}.$(3.3)

#### Theorem 3.3.

Let C be defined by (3.3) and assume that (1.5) holds with $\mu \mathrm{\ge }C$. Then problem (1.1)–(1.2) admits at least one solution for arbitrary p.

#### Proof.

Define $R\left(x\right):={e}^{aF\left(x\right)+b}$ with a as before and some $b>0$ to be specified. Direct computation shows that

$\nabla R\left(x\right)=aR\left(x\right)\nabla F\left(x\right)$

and

$\frac{\partial R}{\partial \nu }\left(x\right)=aR\left(x\right)\frac{\partial F}{\partial \nu }\left(x\right)\ge aR\left(x\right)>\gamma \left(x\right)R\left(x\right)$

for all $x\in \partial \mathrm{\Omega }$. On the other hand,

$\mathrm{\Delta }R\left(x\right)=a\mathrm{\Delta }F\left(x\right)R\left(x\right)+{a}^{2}{|\nabla F\left(x\right)|}^{2}R\left(x\right),$

and hence

$R\left(x\right)\mathrm{\Delta }R\left(x\right)+{|\nabla R\left(x\right)|}^{2}\le CR{\left(x\right)}^{2}$

with C given by (3.3). Then we may fix ${R}_{0}$ such that

$\frac{〈g\left(x,u\right),u〉}{{|u|}^{2}}>C$

for $|u|\ge {R}_{0}$ and $b\gg 0$ such that $R\left(x\right)\ge {R}_{0}$ for all x. Hence, for $|u|=R\left(x\right)$ it is seen that

$〈g\left(x,u\right),u〉>CR{\left(x\right)}^{2}\ge R\left(x\right)\mathrm{\Delta }R\left(x\right)+{|\nabla R\left(x\right)|}^{2}$

and the previous lemma applies. ∎

## 4.1 ${H}^{1}$ bounds

This section is devoted to obtain, when (1.5) is satisfied, suitable a priori ${H}^{1}$ bounds for the solutions of (1.1)–(1.2) depending only on μ and the constant ${C}_{\mu }$ in condition (2.1).

To this end, let us firstly observe that necessarily $\mu \ge -{\lambda }_{1}$: indeed, for $\mu <-{\lambda }_{1}$ we may consider

$g\left(x,u\right):=-{\lambda }_{1}u+p\left(x\right),$

for which the problem has an unbounded set of solutions if all the coordinates of p are orthogonal (in the ${L}^{2}$ sense) to the eigenfunction associated to ${\lambda }_{1}$, and no solutions otherwise.

#### Remark 4.1.

In view of Theorem 3.3, the same choice of g leads to the conclusion that if C is as in (3.3) then $C\ge -{\lambda }_{1}$.

We claim that the condition $\mu \ge -{\lambda }_{1}$ is also sufficient for getting appropriate ${H}^{1}$ bounds. Indeed, using the extension of (2.2) for vector functions (see Remark 2.1), we can multiply the equation by u and integrate to obtain

${\int }_{\mathrm{\Omega }}〈g\left(x,u\left(x\right)\right),u\left(x\right)〉𝑑x={\int }_{\mathrm{\Omega }}〈Lu\left(x\right),u\left(x\right)〉𝑑x\le -{\lambda }_{1}{\parallel u\parallel }_{{L}^{2}}^{2}.$

From inequality (2.1) and using the fact that $\mu \ge -{\lambda }_{1}$ it is verified that $〈g\left(x,u\right),u\right)〉\ge \left(\epsilon -{\lambda }_{1}\right)|u|{}^{2}-C{}_{\mu }$, which in turn yields

${\lambda }_{1}{\parallel u\parallel }_{{L}^{2}}^{2}\le \left({\lambda }_{1}-\epsilon \right){\parallel u\parallel }_{{L}^{2}}^{2}+{C}_{\mu }|\mathrm{\Omega }|.$

Hence ${\parallel u\parallel }_{{L}^{2}}\le \frac{{C}_{\mu }}{\epsilon }$ and, moreover,

${\int }_{\mathrm{\Omega }}{|\nabla u\left(x\right)|}^{2}𝑑x\le \left({\lambda }_{1}-\epsilon \right){\parallel u\parallel }_{{L}^{2}}^{2}+{C}_{\mu }|\mathrm{\Omega }|+{\int }_{\partial \mathrm{\Omega }}\gamma \left(x\right){|u\left(x\right)|}^{2}𝑑S.$

Next, extend the outer normal to a smooth vector field $\nu :\overline{\mathrm{\Omega }}\to {ℝ}^{N}$ and define $\mathrm{\Phi }\left(x\right):=\gamma \left(x\right){|u\left(x\right)|}^{2}\nu \left(x\right)$. Then

${\int }_{\partial \mathrm{\Omega }}\gamma \left(x\right){|u\left(x\right)|}^{2}𝑑S={\int }_{\partial \mathrm{\Omega }}〈\mathrm{\Phi }\left(x\right),\nu \left(x\right)〉𝑑S={\int }_{\mathrm{\Omega }}\mathrm{div}\mathrm{\Phi }\left(x\right)𝑑x.$

Now let $D:={\mathrm{max}}_{j=1,\mathrm{\dots },n}{\parallel \gamma {\nu }_{j}\parallel }_{\mathrm{\infty }}$ and assume, without loss of generality, that $\epsilon <\frac{1}{D}$. Since

$\mathrm{div}\mathrm{\Phi }\left(x\right)=\sum _{j}\left[{\partial }_{j}\left(\gamma {\nu }_{j}\right)\left(x\right)|u\left(x\right){|}^{2}+2\gamma \left(x\right){\nu }_{j}\left(x\right)〈u\left(x\right),{\partial }_{j}u\left(x\right)〉\right]$$\le \sum _{j}\left[\left({\partial }_{j}\left(\gamma {\nu }_{j}\right)\left(x\right)+\frac{\gamma \left(x\right){\nu }_{j}\left(x\right)}{\epsilon }\right){|u\left(x\right)|}^{2}+\gamma \left(x\right){\nu }_{j}\left(x\right)\epsilon {|{\partial }_{j}u\left(x\right)|}^{2}\right],$

it follows that

${\int }_{\mathrm{\Omega }}{|\nabla u\left(x\right)|}^{2}𝑑x\le C\left(\epsilon \right){\parallel u\parallel }_{{L}^{2}}^{2}+D\epsilon {\parallel \nabla u\parallel }_{{L}^{2}}^{2},$

where $C\left(\epsilon \right):={\parallel {\sum }_{j}\left({\partial }_{j}\left(\gamma {\nu }_{j}\right)+\frac{\gamma {\nu }_{j}}{\epsilon }\right)\parallel }_{\mathrm{\infty }}+\left[\left(\frac{{\lambda }_{1}}{\epsilon }-1\right)+|\mathrm{\Omega }|\right]{C}_{\mu }$ and the desired bound is obtained.

Thus we have proved the following lemma.

#### Lemma 4.2.

Fix $\epsilon \mathrm{>}\mathrm{0}$ small enough, $\mu \mathrm{\ge }\mathrm{-}{\lambda }_{\mathrm{1}}$ and ${C}_{\mu }\mathrm{\ge }\mathrm{0}$. Then there exists an ${H}^{\mathrm{1}}$ bound for the solutions of (1.1)–(1.2) for any g satisfying (2.1).

#### Remark 4.3.

The preceding argument shows that, if $g\left(x,u\right)={\nabla }_{u}G\left(x,u\right)$, then the associated functional is coercive. However, the functional is not defined over ${H}^{1}\left(\mathrm{\Omega }\right)$ unless a growth condition is assumed for G. As we shall see, by enlarging μ in an appropriate way, we can still apply the variational method because g may be replaced by a suitable truncation.

## 4.2 ${L}^{\mathrm{\infty }}$ bounds

In this section we shall prove that if (1.5) holds with $\mu \ge C$, where C is defined as in (3.3), then the solutions of (1.1)–(1.2) also admit a priori bounds for the ${L}^{\mathrm{\infty }}$ norm.

With this aim, assume that (2.1) holds with $\mu \ge C$ and set $R\left(x\right)={e}^{aF\left(x\right)+b}$ as before such that (3.2) holds strictly. Fix ${R}_{0}$ as in the proof of Theorem 3.3 and set b large enough such that $R\left(x\right)>{R}_{0}$ for all x. From the previous computations it is deduced that if u is a solution and $|u\left(x\right)|\ge R\left(x\right)$ for some x, then the absolute maximum of the function $\varphi \left(x\right)=|u\left(x\right)|-R\left(x\right)$ is achieved at the boundary. Indeed, with the previous notation observe that if the (nonnegative) maximum value of ϕ is achieved at some $\stackrel{^}{x}\in \mathrm{\Omega }$, then it is verified as before that

$〈u\left(\stackrel{^}{x}\right),Lu\left(\stackrel{^}{x}\right)〉\le C{|u\left(\stackrel{^}{x}\right)|}^{2}<〈g\left(\stackrel{^}{x},u\left(\stackrel{^}{x}\right)\right),u\left(\stackrel{^}{x}\right)〉,$

a contradiction. Let u be a solution and suppose that $|u\left(x\right)|\ge R\left(x\right)$ for some x. The function $\psi \left(b\right):=\mathrm{max}\varphi \left(x\right)$ is continuous with respect to b, so (since $\psi \left(b\right)\to -\mathrm{\infty }$ as $b\to +\mathrm{\infty }$) enlarging b, if necessary we may assume that $\psi \left(b\right)=0$, that is, $|u\left(x\right)| for $x\in \mathrm{\Omega }$ and $|u\left(\stackrel{^}{x}\right)|=R\left(\stackrel{^}{x}\right)$ for some $\stackrel{^}{x}\in \partial \mathrm{\Omega }$. This implies that

$\gamma \left(\stackrel{^}{x}\right)R\left(\stackrel{^}{x}\right)<\frac{\partial R}{\partial \nu }\left(\stackrel{^}{x}\right)\le \gamma \left(\stackrel{^}{x}\right)|u\left(\stackrel{^}{x}\right)|=\gamma \left(\stackrel{^}{x}\right)R\left(\stackrel{^}{x}\right),$

In summary, there exists a constant b such that if u is a solution of the problem, then $|u\left(x\right)|. Furthermore, observe that the choice of R depends only on the fact that $〈g\left(u\right),u〉>C{|u|}^{2}$ for $|u|>{R}_{0}$ and the constant C itself is independent of g; thus, replacing g by the function

we see that the resulting problem has the same solutions. Hence, we may assume that g grows linearly with respect to u. In particular, if $g={\nabla }_{u}G$, then the existence of at least one solution is also deduced by variational methods (see Remark 4.3).

#### Remark 4.4.

An alternative choice of R in the Hartman condition is

$R\left(x\right)=:c{\phi }_{1}\left(x\right)$

for some c sufficiently large, provided that

$〈g\left(x,u\right),u〉\ge R\left(x\right)\mathrm{\Delta }R\left(x\right)+{|\nabla R\left(x\right)|}^{2}={c}^{2}\left(-{\lambda }_{1}{\phi }_{1}{\left(x\right)}^{2}+{|\nabla {\phi }_{1}\left(x\right)|}^{2}\right)$

for $|u|=c|{\phi }_{1}\left(x\right)|$, that is,

$〈g\left(x,u\right),u〉\ge {|u|}^{2}\left(-{\lambda }_{1}+\frac{{|\nabla {\phi }_{1}\left(x\right)|}^{2}}{{\phi }_{1}{\left(x\right)}^{2}}\right).$

Thus, it suffices to consider

$\mu :=-{\lambda }_{1}+{\parallel \frac{\nabla {\phi }_{1}}{{\phi }_{1}}\parallel }_{\mathrm{\infty }}^{2}.$(4.1)

Obviously, this R does not satisfy (3.2) strictly, so the previous argument for the ${L}^{\mathrm{\infty }}$ bound fails. However, we may overcome this difficulty by taking, instead, $\stackrel{~}{\gamma }:=\gamma +\eta$ for some η sufficiently small such that $\stackrel{~}{\mu }<\mu +\epsilon$ with ε as in (2.1) and $\stackrel{~}{R}:=c{\stackrel{~}{\phi }}_{1}$, which satisfies the Hartman condition and

$\frac{\partial \stackrel{~}{R}}{\partial \nu }=\stackrel{~}{\gamma }\stackrel{~}{R}>\gamma \stackrel{~}{R}.$

## 5.1 A useful lemma

The key for a proof of our multiplicity result shall be the computation of the degree of an associated linear operator. For $\eta >0$ and a fixed continuous matrix function $M\in C\left(\overline{\mathrm{\Omega }},{ℝ}^{N×N}\right)$ let us define the compact linear operator ${K}_{M}:{L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)\to {L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)$ given by ${K}_{M}v:=u$, the unique solution of the problem

$\left\{\begin{array}{cccc}\hfill Lu\left(x\right)+\left({\lambda }_{1}-\eta \right)u\left(x\right)& =M\left(x\right)v\left(x\right)+\left({\lambda }_{1}-\eta \right)v\left(x\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill \frac{\partial u}{\partial \nu }\left(x\right)& =\gamma \left(x\right)u\left(x\right),\hfill & & \hfill x\in \partial \mathrm{\Omega }.\end{array}$

Then we have the following lemma.

#### Lemma 5.1.

Let $M\mathrm{\in }C\mathit{}\mathrm{\left(}\overline{\mathrm{\Omega }}\mathrm{,}{\mathrm{R}}^{N\mathrm{×}N}\mathrm{\right)}$ be a symmetric matrix function such that $A\mathrm{\le }\mathrm{-}M\mathrm{\le }B$, where the symmetric matrices $A\mathrm{,}B\mathrm{\in }{\mathrm{R}}^{N\mathrm{×}N}$ satisfy (1.6). Then $\mathrm{ker}\mathit{}\mathrm{\left(}I\mathrm{-}{K}_{M}\mathrm{\right)}\mathrm{=}\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}$. Furthermore, if $V\mathrm{\subset }{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{,}{\mathrm{R}}^{N}\mathrm{\right)}$ is a bounded open neighbourhood of 0, then

$\mathrm{deg}\left(I-{K}_{M},V,0\right)={\left(-1\right)}^{{\nu }_{1}+\mathrm{\cdots }+{\nu }_{N}}.$

#### Proof.

Following the ideas in [7], we can verify that the problem

$Lu\left(x\right)=M\left(x\right)u\left(x\right)$

has no nontrivial solutions satisfying the boundary condition (1.2). This implies that $I-{K}_{M}$ vanishes only at $u=0$ and, consequently, $\mathrm{deg}\left(I-{K}_{M},V,0\right)$ is well defined.

Next, observe that ${K}_{M}v$ can be computed as

${K}_{M}v\left(x\right)=-{\int }_{\mathrm{\Omega }}{G}_{\eta }\left(x,y\right){M}^{\eta }\left(y\right)v\left(y\right)𝑑y,$

where ${G}_{\eta }$ is the Green function associated to the scalar operator $-\mathrm{\Delta }-\left({\lambda }_{1}-\eta \right)I$ under the Robin boundary conditions and

${M}^{\eta }:=M+\left({\lambda }_{1}-\eta \right){I}_{N},$

with the identity matrix ${I}_{N}\in {ℝ}^{N×N}$. Recall that ${G}_{\eta }$ can be written as

${G}_{\eta }\left(x,y\right)=\sum _{j=1}^{\mathrm{\infty }}\frac{{\phi }_{j}^{\eta }\left(x\right){\phi }_{j}^{\eta }\left(y\right)}{{\lambda }_{j}^{\eta }},$

where $\left\{{\phi }_{j}^{\eta }\right\}$ is an orthonormal basis of eigenfunctions associated to the eigenvalues ${\lambda }_{j}^{\eta }$. It is clear that ${\lambda }_{j}^{\eta }={\lambda }_{j}-{\lambda }_{1}+\eta$ and we may take ${\phi }_{j}^{\eta }={\phi }_{j}$. Thus,

$\left(I-{K}_{M}\right)u=u+\sum _{j=1}^{\mathrm{\infty }}\frac{{\phi }_{j}}{{\lambda }_{j}^{\eta }}{\int }_{\mathrm{\Omega }}{\phi }_{j}\left(y\right){M}^{\eta }\left(y\right)u\left(y\right)𝑑y.$

Moreover, it is seen that the degree, regarded as a function of M, is continuous, that is, locally constant. Hence, using the fact that the subset of ${ℝ}^{N×N}$ defined by is connected (because it is convex), we may assume that M is a constant matrix, say $M=-A$.

Writing ${u}^{j}:={\int }_{\mathrm{\Omega }}{\phi }_{j}\left(x\right)u\left(x\right)𝑑x$, the coordinate of the vector function u in the basis $\left\{{\phi }_{j}\right\}$, we obtain

$\left(I-{K}_{M}\right)u=u+\sum _{j=1}^{\mathrm{\infty }}\frac{{\phi }_{j}}{{\lambda }_{j}^{\eta }}{M}^{\eta }{u}^{j}=\sum _{j=1}^{\mathrm{\infty }}{\phi }_{j}{M}_{j}^{\eta }{u}^{j},$

where

${M}_{j}^{\eta }:={I}_{N}+\frac{{M}^{\eta }}{{\lambda }_{j}^{\eta }}=\frac{{\lambda }_{j}{I}_{N}+M}{{\lambda }_{j}-{\lambda }_{1}+\eta }.$

Observe also that

${\parallel \sum _{j=q+1}^{\mathrm{\infty }}\frac{{\phi }_{j}}{{\lambda }_{j}^{\eta }}{u}^{j}\parallel }_{{L}^{2}}\le \frac{{\parallel u\parallel }_{{L}^{2}}}{|{\lambda }_{q+1}^{\eta }|}\to 0$

uniformly for ${\parallel u\parallel }_{{L}^{2}}\le \rho$. This implies that, for some large enough q, it suffices to compute the degree restricted to the subspace spanned by ${\left\{{\phi }_{j}\right\}}_{j\le q}$ or, equivalently, the degree of the mapping

$\left({u}^{1},\mathrm{\dots },{u}^{q}\right)↦\left({M}_{1}^{\eta }{u}^{1},\mathrm{\dots },{M}_{q}^{\eta }{u}^{q}\right).$

In other words, it suffices to compute the sign of the determinant of the block matrix

$\left(\begin{array}{c}\hfill {M}_{1}^{\eta }\hfill \\ \hfill \hfill & \hfill {M}_{2}^{\eta }\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \mathrm{\ddots }\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {M}_{q}^{\eta }\hfill \end{array}\right).$

Let ${w}^{k}$ be the eigenvector of the matrix $M=A$ associated to the eigenvalue ${a}_{k}$, then

$\left({\lambda }_{j}{I}_{N}+M\right){w}^{k}=\left({\lambda }_{j}+{a}_{k}\right){w}^{k}$

and, since ${\lambda }_{j}^{\eta }>0$, it follows that

$\mathrm{sgn}\left[\mathrm{det}\left({M}_{j}^{\eta }\right)\right]=\mathrm{sgn}\left(\prod _{k=1}^{N}\left({\lambda }_{j}+{a}_{k}\right)\right).$

Thus, if $q>{\nu }_{k}$ for all k, it is seen that

$\mathrm{deg}\left(I-{K}_{M},V,0\right)=\mathrm{sgn}\left(\prod _{k=1}^{N}\prod _{j=1}^{q}\left({\lambda }_{j}+{a}_{k}\right)\right)$

and the result is deduced from the fact that

${\lambda }_{{\nu }_{k}}+{a}_{k}<0<{\lambda }_{{\nu }_{k}+1}+{a}_{k}.\mathit{∎}$

## 5.2 Multiple solutions for p small

#### Theorem 5.2.

Assume that ${D}_{u}\mathit{}{g}_{\mathrm{0}}\mathit{}\mathrm{\left(}x\mathrm{,}\mathrm{0}\mathrm{\right)}$ is symmetric and that (1.5) holds with C as in (3.3). Furthermore, assume that there exist symmetric matrices A and B with respective eigenvalues ${a}_{\mathrm{1}}\mathrm{\le }\mathrm{\cdots }\mathrm{\le }{a}_{N}$ and ${b}_{\mathrm{1}}\mathrm{\le }\mathrm{\cdots }\mathrm{\le }{b}_{N}$ satisfying

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

for some ${\nu }_{k}\mathrm{\in }{\mathrm{N}}_{\mathrm{0}}$, and all $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}N$ such that

$A\le -{D}_{u}{g}_{0}\left(x,0\right)\le B$

for all x. If ${\nu }_{\mathrm{1}}\mathrm{+}\mathrm{\cdots }\mathrm{+}{\nu }_{N}$ is odd, then there exists $r\mathrm{>}\mathrm{0}$ such that the problem has at least two solutions for all p such that ${\mathrm{\parallel }p\mathrm{\parallel }}_{{L}^{\mathrm{2}}}\mathrm{<}r$.

#### Proof.

Assume firstly that $p=0$. According to Section 4.1, we may fix a constant $\mathcal{ℛ}$ depending only on μ and ${C}_{\mu }$ such that ${\parallel u\parallel }_{{L}^{2}}<\mathcal{ℛ}$ for any solution u. Moreover, as mentioned in Section 4.2, we may also assume that ${g}_{0}$ has linear growth; thus, for fixed $\eta >0$, the operator $K:{L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)\to {L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)$ given by $Kv:=u$, the unique solution of the problem

$\left\{\begin{array}{cccc}\hfill Lu\left(x\right)+\left({\lambda }_{1}-\eta \right)u\left(x\right)& ={g}_{0}\left(x,v\left(x\right)\right)+\left({\lambda }_{1}-\eta \right)v\left(x\right),\hfill & & \hfill x\in \mathrm{\Omega },\\ \hfill \frac{\partial u}{\partial \nu }\left(x\right)& =\gamma \left(x\right)u\left(x\right),\hfill & & \hfill x\in \partial \mathrm{\Omega }\end{array}$

is well defined and clearly compact. Let ${K}_{M}$ be defined as in Lemma 5.1, with $M=\left(\mu +\eta \right){I}_{N}$, then the operator $sK+\left(1-s\right){K}_{M}$ has no fixed points on $\partial {B}_{\mathcal{ℛ}}\left(0\right)$ for $0\le s\le 1$, because ${g}_{s}\left(x,u\right):=s{g}_{0}\left(x,u\right)+\left(1-s\right)Mu$ satisfies (2.1). Since $\mu \ge -{\lambda }_{1}$, the assumptions of Lemma 5.1 hold with $A=B=M$ and ${\nu }_{k}=0$ for all k, then

$\mathrm{deg}\left(I-K,{B}_{\mathcal{ℛ}}\left(0\right),0\right)=\mathrm{deg}\left(I-{K}_{M},{B}_{\mathcal{ℛ}}\left(0\right),0\right)=1.$

Next we shall prove that if $\rho >0$ is small enough then $Ku\ne u$ on $\partial {B}_{\rho }\left(0\right)$ and

$\mathrm{deg}\left(I-K,{B}_{\rho }\left(0\right),0\right)=-1.$

To this end, set $M\left(x\right):={D}_{u}{g}_{0}\left(x,0\right)$ and observe that if $v\in {B}_{\rho }\left(0\right)$, then by letting ${L}_{\eta }:=L+\left({\lambda }_{1}-\eta \right)I$, we deduce from estimate (2.3) that

${\parallel Kv-{K}_{M}v\parallel }_{{L}^{2}}\le \frac{1}{\eta }{\parallel {L}_{\eta }\left(Kv-{K}_{M}v\right)\parallel }_{{L}^{2}}=o\left(\rho \right).$

Due to the linearity and compactness of ${K}_{M}$, it is easy to verify that there exists a constant $\theta >0$ such that

${\parallel v-{K}_{M}v\parallel }_{{L}^{2}}\ge \theta \rho$

for all $v\in \partial {B}_{\rho }\left(0\right)$. It follows, for $\rho >0$ sufficiently small, that the operator defined by $sK+\left(1-s\right){K}_{M}$ has no fixed points on $\partial {B}_{\rho }\left(0\right)$ for $s\in \left[0,1\right]$, because

${\parallel v-sKv-\left(1-s\right){K}_{M}v\parallel }_{{L}^{2}}\ge {\parallel v-{K}_{M}v\parallel }_{{L}^{2}}-{\parallel {K}_{M}v-Kv\parallel }_{{L}^{2}}\ge \theta \rho -o\left(\rho \right)>0$

for $v\in \partial {B}_{\rho }\left(0\right)$. This implies, for small ρ, that the degree of $I-K$ is well defined and, according to Lemma 5.1,

$\mathrm{deg}\left(I-K,{B}_{\rho }\left(0\right),0\right)=\mathrm{deg}\left(I-{K}_{M},{B}_{\rho }\left(0\right),0\right)={\left(-1\right)}^{{\nu }_{1}+\mathrm{\cdots }+{\nu }_{N}}=-1,$

which proves the claim. Moreover, by the excision property of the degree,

$\mathrm{deg}\left(I-K,{B}_{\mathcal{ℛ}}\left(0\right)\setminus {B}_{\rho }\left(0\right),0\right)=2.$

Since the degree is locally constant with respect to the third coordinate, it is deduced that, for any $P\in {L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)$ sufficiently close to 0, the degrees $\mathrm{deg}\left(I-K,{B}_{\rho }\left(0\right),P\right)$ and $\mathrm{deg}\left(I-K,{B}_{\mathcal{ℛ}}\left(0\right)\setminus {B}_{\rho }\left(0\right),P\right)$ are well defined and equal to -1 and 2, respectively.

In order to complete the proof, for each $p\in {L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)$ we define $P:=\mathrm{\Theta }\left(p\right)$ as the unique solution of the linear problem

${L}_{\eta }P=p,\frac{\partial P}{\partial \nu }{|}_{\partial \mathrm{\Omega }}=\gamma P.$

The previous estimate (2.3) yields ${\parallel P\parallel }_{{L}^{2}}\le \frac{1}{\eta }{\parallel p\parallel }_{{L}^{2}}$; that is, if p is close to 0 then so is P and the problem $u-Ku=P$ has at least one solution in ${B}_{\rho }\left(0\right)$ and another one in ${B}_{\mathcal{ℛ}}\left(0\right)\setminus {B}_{\rho }\left(0\right)$. The result is thus deduced from the fact that if $u-Ku=P$, then

$Lu+\left({\lambda }_{1}-\eta \right)u=LKu+\left({\lambda }_{1}-\eta \right)Ku+p={g}_{0}\left(\cdot ,u\right)+\left({\lambda }_{1}-\eta \right)u+p,$

and

$\frac{\partial u}{\partial \nu }{|}_{\partial \mathrm{\Omega }}=\frac{\partial \left(Ku\right)}{\partial \nu }{|}_{\partial \mathrm{\Omega }}+\frac{\partial P}{\partial \nu }{|}_{\partial \mathrm{\Omega }}=\gamma \left(Ku+P\right)=\gamma u.$

In other words, u is a solution of (1.1)–(1.2), which completes the proof. ∎

#### Remark 5.3.

Using the Sard–Smale theorem (see [8]), we deduce that the problem has generically at least three solutions, namely, there exists a residual set $\mathrm{\Sigma }\subset {L}^{2}\left(\mathrm{\Omega },{ℝ}^{N}\right)$ such that if $p\in \mathrm{\Sigma }\cap {B}_{r}\left(0\right)$, then the problem has at least three solutions.

## 6 Further discussion and open problems

The existence of solutions for problem (1.1)–(1.2) was obtained under condition (1.5). Remarkably, the bounds of Section 4.1 imply by themselves that the degree of the operator $I-K$ defined in the proof of Theorem 5.2 over a large ball is equal to 1, thus proving the existence of at least one solution. Since the ${H}^{1}$ bounds require only that $\mu \ge -{\lambda }_{1}$, one might be tempted to believe that the latter condition is sufficient for all our purposes; however, the operator K is not well defined for arbitrary g and this is why we needed also the ${L}^{\mathrm{\infty }}$ bounds. As an immediate consequence, we conclude that the condition $\mu =-{\lambda }_{1}$ is optimal when $n=1$. This is still true for $n=2$, provided that g has polynomial growth and also when $n\ge 3$, provided that g behaves asymptotically as ${|u|}^{q}$ with $q\le \frac{n+2}{n-2}$. So it is an open problem to determine, for the general case, the optimal value of μ when $n\ge 2$. The value given by (4.1) is clearly larger than $-{\lambda }_{1}$ and also satisfies the following lower bound: let

$\kappa ={\parallel \frac{\nabla {\phi }_{1}}{{\phi }_{1}}\parallel }_{\mathrm{\infty }}^{2},$

then

${|\nabla {\phi }_{1}|}^{2}\le \kappa {|{\phi }_{1}|}^{2}\mathit{ }\text{and}\mathit{ }\int {|\nabla {\phi }_{1}|}^{2}\le \kappa .$

Since ${\lambda }_{1}=\int {|\nabla {\phi }_{1}|}^{2}-{\int }_{\partial \mathrm{\Omega }}\gamma {\phi }_{1}^{2}$, it is deduced that

${\parallel \frac{\nabla {\phi }_{1}}{{\phi }_{1}}\parallel }_{\mathrm{\infty }}^{2}-{\lambda }_{1}\ge {\int }_{\partial \mathrm{\Omega }}\gamma {\phi }_{1}^{2}.$

On the other hand, observe that the use of a variable R in the Hartman condition can be avoided if $\mu =0$ and $\gamma \le 0$. The fact that $\gamma \nleqq 0$ is compensated by enlarging the value of μ. It is worth mentioning that, as shown in [5], the problem can always be reformulated into an equivalent one with $\gamma \le 0$; however, under this transformation the relation between ${g}_{0}$ and the eigenvalues becomes less clear.

Also, it would be interesting to analyse the case in which $D{g}_{0}\left(x,0\right)$ is not necessarily symmetric. It is worth noticing that symmetry was used only in order to apply the lemma on bilinear symmetric forms used in [7], but, at first sight, more general results are possible. This is left as a second open problem.

Another matter concerns further multiplicity. In the same line of Theorem 5.2, one may try to see, when ${\nu }_{1}+\mathrm{\cdots }+{\nu }_{N}$ is even, if it is possible to obtain five solutions, as it happens for the ODE case presented in [2]. This is true for the scalar case $N=1$, for which the function $R\left(x\right)$ defined in Section 3 can be used as an upper solution and, if $\frac{\partial {g}_{0}}{\partial u}\left(x,0\right)<0$ in $\overline{\mathrm{\Omega }}$, then $\alpha :=r{\phi }_{1},$ with $r>0$ small enough serves as a lower solution. Taking

${U}_{\alpha ,\beta }:=\left\{u\in C\left(\overline{\mathrm{\Omega }},ℝ\right):\alpha

we have that

$\mathrm{deg}\left(I-K,{U}_{\alpha ,\beta },0\right)=1.$

In the same way, an ordered couple $\stackrel{~}{\alpha }<\stackrel{~}{\beta }<0$ of a lower and an upper solution is readily obtained. Furthermore, we may take $\rho >0$ small enough such that

${U}_{\alpha ,\beta }\cap {B}_{\rho }\left(0\right)={U}_{\stackrel{~}{\alpha },\stackrel{~}{\beta }}\cap {B}_{\rho }\left(0\right)=\mathrm{\varnothing };$

thus, if condition (1.4) is satisfied with $k>0$ even, then

$\mathrm{deg}\left({U}_{\alpha ,\beta }\setminus \left({B}_{\rho }\left(0\right)\cup {U}_{\alpha ,\beta }\cup {U}_{\stackrel{~}{\alpha },\stackrel{~}{\beta }}\right)=-2,$

so generically the problem has at least five solutions. Moreover, observe that the condition $\frac{\partial g}{\partial u}\left(x,0\right)<0$ is fulfilled automatically if k is sufficiently large: for instance, for the ODE studied in [2] with ${a}_{0},{a}_{1}>0$, it suffices to take $k\in 2ℕ$ because it is verified that ${\lambda }_{2}>0$.

Finally, we remark that condition (1.6) makes sense only when ${\lambda }_{{\nu }_{k}}\ne {\lambda }_{{\nu }_{k}+1}$. Unlike the case $n=1$, for which an elementary study of the Wronskian determinant shows that all the eigenvalues are simple, nothing is known about the multiplicity of higher order eigenvalues when $n>1$. This might reduce the number of possible different situations in Theorem 5.2: for example, for the periodic ODE one has

$0={\lambda }_{1}<{\lambda }_{2}={\lambda }_{3}<{\lambda }_{4}={\lambda }_{5}<\mathrm{\cdots }.$

In particular, all the eigenvalues, except the first one, have even multiplicity. This means that, if (1.6) holds, then ${\nu }_{k}=0$ or ${\nu }_{k}$ is odd. Thus, the assumption that ${\nu }_{1}+\mathrm{\cdots }+{\nu }_{N}$ is odd simply says, in this case, that $\mathrm{#}\left\{k:{\nu }_{k}\ne 0\right\}$ is odd.

## Acknowledgements

The author thanks Professor Mónica Clapp for her thoughtful comments and to the anonymous referee for the very careful reading of the manuscript and fruitful corrections and remarks.

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Revised: 2017-05-11

Accepted: 2017-05-17

Published Online: 2017-07-08

Funding Source: Secretaria de Ciencia y Tecnica, Universidad de Buenos Aires

Award identifier / Grant number: 20020120100029BA

Funding Source: Consejo Nacional de Investigaciones Científicas y Técnicas

Award identifier / Grant number: PIP11220130100006CO

This work was partially supported by projects UBACyT 20020120100029BA and CONICET PIP 11220130100006CO.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 603–614, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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