# Existence results for a class of nonlinear singular transport equations in bounded spatial domains

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal

► To cite this version:

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for a class of nonlinear singular transport equations in bounded spatial domains. Mathematical Methods in the Applied Sciences, 2019, 43 (4), pp.1685-1700. 10.1002/mma.5995 . hal-04903322

HAL Id: hal-04903322

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Submitted on 21 Jan 2025

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# Existence results for some nonlinear singular transport equations multidimensional case

Khalid Latrach<sup>a</sup> Hssaine Oummi<sup>b</sup> and Ahmed Zeghal<sup>b, *</sup>

<sup>a</sup>Université Clermont Auvergne

CNRS, LMBP

F-63000 Clermont-Ferrand, France

E-mail: khalid.latrach@uca.fr

<sup>b</sup>Laboratoire de Mathématiques et Applications

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques

B.P. 523, 23000 Béni-Mellal, Morocco

E-mail: a.zeghal@fstbm.ac.ma

January 21, 2025

## Abstract

The aim of this paper is to discuss the existence results for the nonlinear boundary value problem (1)-(2) on $L^p$-space, $p \ge 1$. This problem was already considered in [21, 28, 6] under the hypothesis that the collision frequency $\sigma(\cdot)$ is bounded. In this work we show that this hypothesis is not necessary, it suffices to assume that $\sigma(\cdot)$ is locally bounded. Our analysis uses the concept of Dunford-Pettis operators and a new version of Schauder's fixed point theorem.

**Keywords:** nonlinear transport equation; fixed point theorems; Dunford-Pettis operators; existence result; weak compactness.

**AMS subject classifications:**

## 1 Introduction

In this paper we are interested with existence of solution to the following boundary value problem

$$v.\nabla_x\psi(x, v) + \sigma(v)\psi(x, v) + \lambda\psi(x, v) = \int_V \kappa(v, v')f(x, v', \psi(x, v'))dv', \quad (1)$$

where $(x, v) \in D \times V$. Here $D$ is a smooth open subset of $\mathbb{R}^n$ and $V$ is a closed bounded subset of $\mathbb{R}^n$. This equation describes the transport of particles (neutrons, photons, molecules of gas, etc.) in the domain $D$. The function $\psi(x, v)$ represents the density number (or probability) of gas particles having the position $x$ and the velocity $v$. $\sigma(\cdot)$ and $\kappa(\cdot, \cdot)$ are measurable functions called,

<footnote>*Corresponding author</footnote>
respectively, the collision frequency and the scattering kernel. The map $f(\cdot, \cdot, \cdot)$ is a caratheodory function on $D \times V \times \mathbb{C}$ (see Definition ??) which defines a continuous superposition (or Nemytskii) operator which maps bounded sets into bounded sets. It is a nonlinear function of number density of gas particles $\psi$. We denote by $\lambda$ a complex number.

In our framework, we considered the absorbent boundary conditions

$$ \psi|_{\Gamma_{-}} = 0, \qquad (2) $$

where $\psi|_{\Gamma_{-}}$ is the restriction of $\psi$ to $\Gamma_{-}$ with $\Gamma_{-}$ is the incoming part of the phase boundary.

The goal of this paper is to complete the analysis started in the paper [6, 21, 28, 25, 26] where several existence results for the boundary value problem (??)-(??) are obtained in $L^p$-spaces, $1 \le p < +\infty$ in multidimensional as well as slab geometry. In the work [28], existence results were obtained in $L^1$-space under the hypothesis that the velocity space has a finite measure and the collision frequencies $\sigma(\cdot)$ is bounded. In [6] the authors has generalized the result stated in [28]. They consider a velocity space with an infinite measure and $\sigma(\cdot)$ is bounded. In the works [25, 26] the authors discussed the existence results of solutions of a mono-energetic singular transport equation on $L^p$-spaces, $1 \le p < \infty$ is slab geometry. This type of equation was investigated by several authors. Their main objective was to study the existence results the well-posedness and to discuss also the time asymptotic behaviour of the solution in $L^p$-spaces, with $1 \le p < \infty$. For more information in this direction, see for example [9, 10] and the references therein. In this work, we study the existence result of the solution for the multidimensional transport equation involving unbounded collision frequencies $\sigma(\cdot)$ and unbounded collision operator $K$ in $L^p$ space

$$ K : \varphi \rightarrow \int_V \kappa(v, v')\varphi(x, v')dv'. $$

More precisely, $\sigma(\cdot)$ and $K$ are assumed to satisfies the following conditions

* there exists a closed subset $S$ of $V$ with zero Lebesgue measure and a constant $\sigma_0$ such that

$$ \sigma(\cdot) \in L_{\text{loc}}^{\infty}(V \setminus S), \quad \sigma(v) > \sigma_0 \quad \text{a.e. on } V $$

* The collision operator $K$ is assumed to be bounded from the weighted space $L^p(D \times V, \sigma(v)dxdv)$ into $L^p(D \times V, dxdv)$ and viewed as an operator from $L^p(V, \sigma(v)dv)$ into $L^p(V, dv)$ is compact and positive.

These two assumptions have been made first by Chabi and Mokhtar-kharroubi [8], [31, Chapter 9] and were motivated by free gas models (see [33, 34]) (see also ([9], [10],[29]). The first assumption means that the singularities of the collision frequency are contained in a set of zero Lebesgue measure. This hypothesis introduce some mathematical difficulties which we will circumvent by using the proprieties of weakly compact sets and the fixed point theory.

The outline of this work is as follows. In section 2, we introduce the functional setting of the problem and we fix the notations. Thus, we define the streaming operator and present some preliminary results required in the sequel. In section 3, we establish the compactness result. The main results of this section is Theorem ?? . Finally, we present the existence result of the problem (??)-(??) for wide classes of functions $f(\cdot, \cdot, \cdot)$ and scattering kernels $k(\cdot, \cdot, \cdot)$. We first transform the problem into a fixed point one involving an operator depending on the parameter $\lambda$, say, $\psi = G(\lambda)\psi$. The Proof is based on Schauder's fixed point theorem for weakly topology (a new version of Schauder's theorem).
## 2 Notations and preliminaries

In this section, we introduce the different notations and some preliminary results which we shall need in the sequel. To this end, let

$$X_p := L^p(D \times V; dxdv), \quad 1 \le p < \infty,$$

where $D$ is a smooth bounded open convex subset of $\mathbb{R}^n$. For $(x, v) \in \bar{D} \times \bar{V}$, we set

$$t^{-}(x, v) = \sup\{t > 0, x + sv \in D, 0 < s < t\}.$$

We denote by

$$\Gamma_{\pm} = \{(x, v) \in \partial D \times V, \pm v \cdot \nu_x \ge 0\},$$

where $\nu_x$ stands for the outer unit normal vector at $x \in \partial D$. Hence, for $(x, v) \in \Gamma_-$, one has $t^{-}(x, v) = 0$ and in all case $x - t^{-}(x, v)v \in \Gamma_-$. The number $t^{-}(x, v)$ is the time required by a particle having the position $x \in D$ and the velocity $v \in V$ to go out $D$.

We need also the following spaces

$$X_p^{\sigma} := L^p(D \times V; \sigma(v)dxdv), \quad L^p := L^p(V; dv) \quad \text{and} \quad L^{p, \sigma} := L^p(V; \sigma(v)dv).$$

Now we are in position to define the streaming operator $T$

$$\left\{ \begin{array}{l} T : D(T) \subseteq X_p^{\sigma} \to X_p \\ \psi \to T\psi(x, v) = -v \cdot \nabla_x \psi(x, v) - \sigma(v)\psi(x, v) \\ D(T) = \{ \psi \in X_p^{\sigma} \text{ such that } v \cdot \nabla_x \psi \in X_p \text{ and } \psi|_{\Gamma_-} = 0 \}. \end{array} \right.$$

Let us now introduce the following assumption

(A1) there exist a closed subset $S \subset V$ with zero Lebesgue measure and a constant $\sigma_0 > 0$ such that $\sigma(\cdot) \in L_{\text{loc}}^{\infty}(V \setminus S)$, and $\sigma(v) > \sigma_0$ for a.e. $v \in V$.

Let $\varphi \in X_p^{\sigma}$ and consider the problem

$$(\lambda - T)\psi = \varphi, \qquad (3)$$

where $\lambda$ is a complex number and the unknown $\psi$ must be sought in $D(T)$.

Set $G_{\sigma_0}$ the half plane of the complexe space defined by

$$G_{\sigma_0} := \{ \lambda \in \mathbb{C} \text{ such that } \operatorname{Re}\lambda > -\sigma_0 \}.$$

It is clear that the resolvent set of $T$ contains $G_{\sigma_0}$ and, for $\lambda \in G_{\sigma_0}$, the resolvent of $T$ is given by

$$(\lambda - T)^{-1}\varphi(x, v) = \int_{0}^{t^{-}(x, v)} e^{-t(\lambda + \sigma(v))} \varphi(x - tv, v)dt.$$

**Lemma 2.1.** Let $\lambda \in G_{\sigma_0}$ and assume that the condition (A1) holds true. Then $(\lambda - T)^{-1}$ belongs to $\mathcal{L}(X_p, X_p^{\sigma})$ and satisfies

$$\| (\lambda - T)^{-1} \|_{\mathcal{L}(X_p, X_p^{\sigma})} \le \sup_{v \in V} \frac{\sigma(v)^{\frac{1}{p}}}{\operatorname{Re}\lambda + \sigma(v)}. \qquad (4)$$
**Proof.** Let $\varphi \in \mathcal{X}_p$, we have

$$ \begin{aligned} \|(\lambda - T)^{-1}\varphi\|_{\mathcal{X}_p^{\sigma}}^{p} &= \int_{D \times V} \left| \int_{0}^{t^{-}(x,v)} e^{-t(\lambda+\sigma(v))} \varphi(x - tv, v) dt \right|^p \sigma(v) dx dv \\ &\leq \int_{V} \frac{\sigma(v)}{(\mathrm{Re}\lambda + \sigma(v))^{\frac{p}{q}}} dv \int_{0}^{t^{-}(x,v)} e^{-t(\mathrm{Re}\lambda+\sigma(v))} dt \int_{D} |\varphi(x - tv, v)|^p dx \\ &\leq \int_{D \times V} \frac{\sigma(v)}{(\mathrm{Re}\lambda + \sigma(v))^p} |\varphi(y, v)|^p dy dv \\ &\leq \sup_{v \in V} \frac{\sigma(v)}{(\mathrm{Re}\lambda + \sigma(v))^p} \|\varphi\|_{\mathcal{X}_p}^{p}. \end{aligned} $$

This yields that

$$ \|(\lambda - T)^{-1}\|_{\mathcal{L}(\mathcal{X}_p, \mathcal{X}_p^{\sigma})} \leq \sup_{v \in V} \frac{\sigma(v)^{\frac{1}{p}}}{\mathrm{Re}\lambda + \sigma(v)}. \qquad \text{Q.E.D.} $$

**Remark 2.1.** (a) For $p = 1$, one can derive from the estimate (4) that

$$ \|(\lambda - T)^{-1}\|_{\mathcal{L}(\mathcal{X}_1, \mathcal{X}_1^{\sigma})} \leq 1 \qquad (5) $$

because, for any $\lambda \in G_{\sigma_0}$, we have $\sup_{v \in V} \frac{\sigma(v)}{\mathrm{Re}\lambda + \sigma(v)} \leq 1$ regardless of $\lambda$.

(b) Note that, for any real $\lambda$ such that $\lambda > -\sigma_0$, the operator $(\lambda - T)^{-1}$ is positive (in the lattice sense) on $\mathcal{X}_p$, that is, it maps the positive cone of $\mathcal{X}_p$ into the positive cone of $\mathcal{X}_p^\sigma$. $\square$

We shall now recall some fact about Nemytskii (or superposition) operators on $L^p$ spaces.

**Definition 2.1.** A function $f : \Omega \times \mathbb{C} \to \mathbb{C}$ is said to satisfy the Carathéodory conditions on $\Omega \times \mathbb{C}$ if

(a) the function $t \mapsto f(t, u)$ is measurable on $\Omega$ for all $u \in \mathbb{C}$,

(b) the function $u \mapsto f(t, u)$ is continuous on $\mathbb{C}$ for almost all $t \in \Omega$.

If $f$ satisfies the Carathéodory conditions, we can define the operator $\mathcal{N}_f$ on the set of functions $\psi : \Omega \to \mathbb{C}$ by

$$ (\mathcal{N}_f \psi)(x, v) := f(x, v, \psi(x, v)) $$

for every $(x, v) \in D \times V$. The operator $\mathcal{N}_f$ is called the Nemytskii operator generated by $f$. In $L_p$-spaces the Nemytskii operator has been extensively investigated (see [3, 4, 19] and the references therein). The first results on continuity and boundedness of this operator in Lebesgue spaces over bounded domains $\Omega$ in Euclidean space are due to M.A. Krasnosel'skii (see for example [4, Theorem 3.1]).

We recall the following result (established in a more general setting in [12] (see also [13])) which provides necessary and sufficient conditions under which the Nemytskii operator maps continuously a weighted $L^p$-space into itself. (The statement of this result is adapted to our context.)
**Lemma 2.2.** Assume that $f$ satisfies the Carathéodory conditions. Then $N_f$ maps continuously $X_p^\sigma$ into itself, if and only if, the function $f$ satisfies the following growth condition

$$|f(x, v, u)| \leq \frac{\alpha(x, v)}{\sigma(v)^{\frac{1}{p}}} + \beta|u|, \qquad (6)$$

for a.e. $(x, v) \in D \times V$, and every $u \in \mathbb{C}$ with a fixed nonnegative function $\alpha \in X_p$ and a fixed nonnegative constant $\beta$. Further, it maps bounded sets of $X_p^\sigma$ into bounded sets of $X_p^\sigma$.

As it was recalled above, the continuity of $N_f$ is established in [12, 13]. The fact that it maps bounded sets of $X_p^\sigma$ into bounded sets of $X_p^\sigma$ follows immediately from the estimate (6).

Let $X, Y$ be two Banach spaces and $U: X \to Y$ a linear bounded operator. We recall that $U$ is called weakly compact if, for each bounded subset $B$ of $X$, $U(B)$ is a relatively weakly compact subset of $Y$.

**Definition 2.2.** A Banach space $X$ has the Dunford-Pettis property if, for every sequence $(x_n)_{n \in \mathbb{N}}$ in $X$ converging weakly to 0 and every sequence $(x_n^*)_{n \in \mathbb{N}}$ in $X^*$ (the dual space of $X$) converging weakly to 0, the sequence $(x_n^*(x_n))_{n \in \mathbb{N}}$ converges to 0.

The following proposition gives a characterization of Banach spaces with the Dunford-Pettis property (see, for example, [15, p. 596] or [11]).

**Proposition 2.1.** Let $X$ be a Banach space. Then the following are equivalent.

1. $X$ has the Dunford-Pettis property.

2. Every weakly compact operator from $X$ into any Banach space $Y$ maps weakly compact sets of $X$ into norm compact sets of $Y$.

**Definition 2.3.** Let $X$ be a Banach space and let $J: D(J) \subseteq X \to X$ a nonlinear operator.

(a) We say that $J$ is ws-compact if $J$ is continuous and if $(x_n)_{n \in \mathbb{N}} \subseteq D(J)$ is a weakly convergent sequence in $X$, then $(J(x_n))_{n \in \mathbb{N}}$ has a strongly convergent subsequence in $X$.

(b) We say that $J$ is ww-compact if $J$ is continuous and if $(x_n)_{n \in \mathbb{N}} \subseteq D(J)$ is a weakly convergent sequence in $X$, then $(J(x_n))_{n \in \mathbb{N}}$ has a weakly convergent subsequence in $X$.

**Remark 2.2.** (1) It is clear that each compact operator is ws-compact and when $X$ is reflexive, these two notions are equivalent. However, the fact that $J$ is ws-compact does not necessarily imply the compactness of $J$ even if $J$ is linear. It is well known that a compact linear operator from a Banach space $X$ into a Banach space $Y$ transforms weakly convergent sequence into norm convergent sequences. If $X$ is not reflexive, the converse of this assertion is, in general, not true even if the space $Y$ is reflexive.

(2) Note that linear weakly compact operators acting on Banach spaces with the Dunford-Pettis property are ws-compact. (A Banach space $X$ has the Dunford-Pettis property if every weakly compact operator defined on $X$ takes weakly compact sets into norm compact sets.) Indeed, if $X$ is a Banach space with the Dunford-Pettis property, then every weakly compact linear operator from $X$ into an arbitrary Banach space $Y$ maps weakly convergent sequences in $X$ onto norm convergent sequences in $Y$.
(3) ws-compact and ww-compact operators are not necessarily weakly continuous (see, for example, Theorem 2.6 in [3]).

□

We close this section by recalling two fixed point theorems. The following one is the classical Schauder fixed point theorem (see, for example, [35] or [23]).

**Theorem 2.1.** Let $X$ be a Banach space and let $M \subset X$ be nonempty, convex, closed and bounded. If $A: M \to M$ is a compact operator, then $A$ has a fixed point in $M$.

The next one is a new variant of Schauder fixed point theorem for weak topology (see [27, Theorem 2.1] or [23, Theorem 4.3.1, p. 222]).

**Theorem 2.2.** Let $M$ be a nonempty closed convex subset of a Banach space $X$ and assume that $A: M \to M$ is a ws-compact operator. If $A(M)$ is relatively weakly compact, then $A$ has a fixed point in $M$.

# 3 Preparatory results

Now, let $K$ be the linear operator defined by

$$K: X_p^\sigma \to X_p, \quad \psi \mapsto \int_V \kappa(v, v')\psi(x, v')dv',$$

where the kernel $\kappa(\cdot, \cdot)$ is a measurable nonnegative real valued function on $V \times V$.

We introduce the following hypothesis

(A2) $K$ viewed as an operator from $L^{p,\sigma}$ to $L^p$ is compact, that is $K \in \mathcal{K}(L^{p,\sigma}, L^p)$,

where $\mathcal{K}(L^{p,\sigma}, L^p)$ stands for the set of all compact operators from $L^{p,\sigma}$ into $L^p$.

**Lemma 3.1.** The operator $T$ is dissipative on $X_2$.

This result is well known in neutron transport theory. It is valid on $X_p$ with $1 \le p < +\infty$ even if $\sigma(\cdot)$ satisfies the assumption (A1). We give here a proof for completeness.

**Proof.** For $\psi \in D(T)$, we can write

$$ \begin{aligned} \langle T\psi, \psi \rangle &= \int_{D \times V} \psi(-v \cdot \nabla_x \psi) dxdv = \frac{-1}{2} \int_{D \times V} v \cdot \nabla_x |\psi|^2 dxdv = \frac{-1}{2} \int_{\partial D \times V} |\psi|^2 v \cdot \nu_x d\gamma_x dv \\ &= \frac{1}{2} [\|\psi_-\|_{L^{2,-}}^2 - \|\psi_+\|_{L^{2,+}}^2] = -\frac{1}{2} \|\psi_+\|_{L^{2,+}}^2 \\ &\le 0, \end{aligned} $$

where $\|\psi_\pm\|_{L^{2,\pm}}^2 = \int_{\Gamma_\pm} |\psi(x,v)|^2 |v \cdot \nu_x| d\gamma_x dv$. Hence $\langle T\psi, \psi \rangle \le 0$ for every $\psi \in D(T)$. Then $T$ is dissipative. Q.E.D.
**Lemma 3.2.** Let K be an operator from $X_2^\sigma$ into $X_2$ with kernel $\kappa(v, v') = k_1(v)k_2(v')$ with $k_1(\cdot) \in L^2$ and $\frac{k_2(\cdot)}{\sigma(\cdot)^{\frac{1}{2}}} \in L^2$. For all $\varphi \in X_2$ and $\psi \in X_2^\sigma$, we have

$$ \langle K\psi, \varphi \rangle_{X_2} = \langle \psi, K^*\varphi \rangle_{X_2^\sigma}, $$

where $\langle \cdot, \cdot \rangle_{X_2^\sigma}$ (respectively $\langle \cdot, \cdot \rangle_{X_2}$) denotes the inner product on $X_2^\sigma$ (respectively on $X_2$) and

$$ (K^*\varphi)(x, v) := \int_V \frac{k_2(v)k_1(v')}{\sigma(v)}\varphi(x, v')dv'. $$

**Proof.** Let $\varphi \in X_2$ and $\psi \in X_2^\sigma$. By Fubini's theorem, we have

$$ \begin{aligned} \langle K\psi, \varphi \rangle_{X_2} &= \int_{D \times V} K\psi(x, v)\varphi(x, v)dxdv \\ &= \int_{D \times V} \left[ \int_V \frac{1}{\sigma(v')} k_1(v)k_2(v')\varphi(x, v)dv \right] \psi(x, v')\sigma(v')dxdv'. \end{aligned} $$

It suffices to prove that $f \in X_2^\sigma$, where

$$ f(x, v') := \int_V \frac{k_1(v)k_2(v')}{\sigma(v')}\varphi(x, v)dv. $$

By the Hölder inequality, we have

$$ \begin{aligned} \|f\|_{X_2^\sigma}^2 &\le \int_{D \times V} \left[ \int_V \frac{1}{\sigma(v')} |k_1(v)||k_2(v')||\varphi(x, v)|dv \right]^2 \sigma(v')dxdv' \\ &\le \left\| \frac{k_2}{\sqrt{\sigma(\cdot)}} \right\|_{L^2}^2 \|k_1\|_{L^2}^2 \int_{D \times V} |\varphi(x, v)|^2 dxdv \\ &< \infty. \end{aligned} $$

Q.E.D.

**Lemma 3.3.** Let $\lambda \in G_{\sigma_0}$ and denote by $K^*$ the dual operator of $K$ defined in lemma 3.2. Assume that the hypotheses (A1) and (A2) are satisfied. Then $K^*(\lambda - T)^{-1}K$ is a compact operator on $X_2^\sigma$.

**Proof.** Let $\lambda \in G_{\sigma_0}$. Since K is assumed to be a compact from $X_2^\sigma$ into $X_2$, then, approximating $K \in K(L^{2, \sigma}, L^2)$ by a sequence of finite rank operators and by linearity, we may restrict ourselves to collision operators K with kernels

$$ \kappa(v, v') = k_1(v)k_2(v'), $$

where $k_1 \in L^2$ and $h_2 = \frac{k_2}{\sigma^{\frac{1}{2}}} \in L^2$. By density, we can assume that $k_1, h_2 \in C_c(V \setminus S)$. According to Lemma 3.2

$$ \left\{ \begin{array}{rcl} K^* : X_2 & \to & X_2^\sigma \\ \psi & \to & K^*\psi(x, v) = \int_V h_2(v)k_1(v')\psi(x, v')dv'. \end{array} \right. $$
For $\varphi \in X_2^\sigma$, we have

$$
\begin{align*}
(K^*(\lambda - T)^{-1} K \varphi)(x, v) &= \int_V h_2(v) k_1(v') ((\lambda - T)^{-1} K \varphi)(x, v') dv' \\
&= \int_V h_2(v) k_1(v') \int_0^{t^{-}(x,v')} e^{-t(\lambda+\sigma(v'))} K \varphi(x - tv', v') dt dv' \\
&= \int_V h_2(v) k_1(v') dv' \int_0^{t^{-}(x,v')} e^{-t(\lambda+\sigma(v'))} dt \\
&\qquad \times \int_V k_1(v') k_2(v'') \varphi(x - tv', v'') dv''.
\end{align*}
$$

The change of variables $y = x - tv'$ leads to

$$
(K^*(\lambda - T)^{-1} K \varphi)(x, v) = \int_{D \times V} h_2(v) \int_0^\infty k_1^2\left(\frac{x-y}{t}\right) k_2(v'') e^{-t(\lambda + \sigma\left(\frac{x-y}{t}\right))} \varphi(y, v'') \frac{dt}{t^n} dy dv''.
$$

Thus the operator $K^*(\lambda - T)^{-1}K$ can be factorized as $K^*(\lambda - T)^{-1}K = \mathcal{A}\mathcal{N}_\lambda\mathcal{F}$, where

$$
\left\{
\begin{array}{l}
\mathcal{A} : X_2^\sigma \to L^2(D), \\
\mathcal{A} \varphi(x, v) := \int_V k_2(v) \varphi(x, v) dv,
\end{array}
\right.
$$

$$
\left\{
\begin{array}{l}
\mathcal{F} : L^2(D) \to X_2^\sigma, \\
\mathcal{F} \varphi(x) := h_2(v) \varphi(x),
\end{array}
\right.
$$

and

$$
\left\{
\begin{array}{l}
\mathcal{N}_{\lambda} : L^2(D) \to L^2(D), \\
\mathcal{N}_{\lambda} \varphi(x) := \displaystyle\int_{D} n_{\lambda}(x, y) \varphi(y) dy,
\end{array}
\right.
$$

where

$$
n_{\lambda}(x, y) := \int_{0}^{\infty} k_{1}^{2}\left(\frac{x-y}{t}\right) e^{-t\left(\lambda+\sigma\left(\frac{x-y}{t}\right)\right)} \frac{dt}{t^{n}}.
$$

Because the operators $\mathcal{A}$ and $\mathcal{F}$ are bounded, it suffices to establish that $\mathcal{N}_\lambda$ is a compact operator.
Indeed, let $\varphi \in L^2(D)$. According to the Hölder inequality, we have

$$
\begin{align*}
\|\mathcal{N}_\lambda \varphi\|_{L^2(D)}^2 &= \left| \int_{\mathbb{R}^n} \left| \int_{\mathbb{R}^n} \int_0^{+\infty} \tilde{k}_1\left(\frac{x-y}{t}\right)^2 e^{-t\left(\lambda+\sigma\left(\frac{x-y}{t}\right)\right)} \tilde{\varphi}(y) \frac{dt}{t^n} dy \right| \right|^2 \\
&\le \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} \int_0^{+\infty} \left| \tilde{k}_1\left(\frac{x-y}{t}\right) \right|^2 e^{-t\left(\mathrm{Re}\lambda+\sigma\left(\frac{x-y}{t}\right)\right)} \frac{dt}{t^n} dy \right) \\
&\quad \times \left( \int_{\mathbb{R}^n} \int_0^{+\infty} \left| \tilde{k}_1\left(\frac{x-y}{t}\right) \right|^2 e^{-t\left(\mathrm{Re}\lambda+\sigma\left(\frac{x-y}{t}\right)\right)} \left| \tilde{\varphi}(y) \right|^2 \frac{dt}{t^n} dy \right) dx,
\end{align*}
$$

where $\tilde{k}_1$ (respectively, $\tilde{\varphi}$) denotes the zero extension of $k_1$ outside $V$ (respectively, the zero extension of $\varphi$ outside $D$).
Using the change of variables, we get

$$
\begin{align*}
\int_{\mathbb{R}^n} \int_0^{+\infty} |\tilde{k}_1(\frac{x-y}{t})|^2 e^{-t(\text{Re}\lambda + \sigma(\frac{x-y}{t}))} \frac{dt}{t^n} dy &= \int_{\mathbb{R}^n} \int_0^{+\infty} |\tilde{k}_1(\frac{z}{t})|^2 e^{-t(\text{Re}\lambda + \sigma(\frac{z}{t}))} \frac{dt}{t^n} dz \\
&= \int_{\mathbb{R}^n} \int_0^{+\infty} |\tilde{k}_1(v)|^2 e^{-t(\text{Re}\lambda + \sigma(v))} dtdv \\
&\leq \frac{1}{\text{Re}\lambda + \sigma_0} \|k_1\|_2^2.
\end{align*}
$$

Therefore

$$
\begin{align*}
\|\mathcal{N}_\lambda \varphi\|_{L^2(D)}^2 &\le \frac{1}{\text{Re}\lambda + \sigma_0} \|k_1\|_2^2 \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \int_0^{+\infty} |\tilde{k}_1(v)|^2 e^{-t(\text{Re}\lambda + \sigma(v))} |\tilde{\varphi}(x - tv)|^2 dt dv dx \\
&= \frac{1}{\text{Re}\lambda + \sigma_0} \|k_1\|_2^2 \int_{\mathbb{R}^n} \int_0^{+\infty} |\tilde{k}_1(v)|^2 e^{-t(\text{Re}\lambda + \sigma(v))} \int_{\mathbb{R}^n} |\tilde{\varphi}(x - tv)|^2 dx dv dt \\
&= \frac{1}{\text{Re}\lambda + \sigma_0} \|k_1\|_2^2 \int_{\mathbb{R}^n} \int_0^{+\infty} |\tilde{k}_1(v)|^2 e^{-t(\text{Re}\lambda + \sigma(v))} dt dv \int_D |\varphi(x)|^2 dx \\
&= \frac{\|\varphi\|_{L^2(D)}^2}{(\text{Re}\lambda + \sigma_0)^2} \|k_1\|_2^4.
\end{align*}
$$

Then

$$
\|\mathcal{N}_{\lambda}\|_{L(L^2(D))} \leq \frac{\|k_1\|_{L^2}}{\text{Re}\lambda + \sigma_0}.
$$

So, $\mathcal{N}_\lambda$ depends continuously on $k_1 \in L^2$. Now, since $\mathcal{N}_\lambda$ is an integral operator, and the set of bounded functions which vanish in a neighborhood of $v=0$ is dense in $L^2$, it follows that $\mathcal{N}_\lambda$ is a limit, in the uniform topology, of integral operators with bounded kernels. Hence $\mathcal{N}_\lambda$ is a compact operator, which completes the proof.

Q.E.D.

**Theorem 3.1.** Let $p \in [1, +\infty)$ and assume that the hypotheses (A1) and (A2) are satisfied. Then, for any $\lambda \in G_{\sigma_0}$, the following two assertions hold true:

(a) $(\lambda - T)^{-1}K$ is compact on $X_p^\sigma$, for all $p \in (1, +\infty)$,

(b) $(\lambda - T)^{-1}K$ is a Dunford-Pettis operator from $X_1^\sigma$ into itself.

**Proof.** (a) We know from Lemma 3.3 that the operator $K^*(\lambda - T)^{-1}K$ is compact in $X_2^\sigma$. We shall now prove the compactness of $(\lambda - T)^{-1}K$ in $X_2^\sigma$. To this end, let $(\psi_n)_{n \in \mathbb{N}}$ be a sequence of $X_2^\sigma$ which converges weakly to 0, then it suffices to establish that the sequence $((\lambda - T)^{-1}K\psi_n)_{n \in \mathbb{N}}$ converges strongly in $X_2^\sigma$ to 0. Arguing as in the proof of Lemma 3.3 we may restrict ourselves to collision operators $K$ with kernels

$$
\kappa(v, v') = k_1(v)k_2(v'),
$$

where $k_1, h_2 \in C_c(V \setminus S)$.

Putting, for each $n \in \mathbb{N}$, $\varphi_n = (\lambda - T)^{-1}K\psi_n$.
We claim that $(\varphi_n)_{n \in \mathbb{N}}$ converges strongly to 0 in $X_2^\sigma$. Indeed, using the fact that $X_2^\sigma \subset X_2$ and the dissipativity of $T$, for each $n \in \mathbb{N}$, we get

$$|\langle(\lambda - T)\varphi_n, \varphi_n\rangle_{X_2}| \geq \operatorname{Re} \langle(\lambda - T)\varphi_n, \varphi_n\rangle_{X_2} \geq \operatorname{Re}\lambda \|\varphi_n\|_{X_2}^2.$$

Next applying Lemma 3.2, one can write

$$ \begin{aligned} \operatorname{Re}\lambda \|\varphi_n\|_{X_2} & \leq |\langle(\lambda - T)\varphi_n, \varphi_n\rangle_{X_2}| \\ &= |\langle K\psi_n, (\lambda - T)^{-1}K\psi_n\rangle_{X_2}| \\ &= |\langle \psi_n, K^*(\lambda - T)^{-1}K\psi_n\rangle_{X_2}|. \end{aligned} $$

Since, the operator $K^*(\lambda - T)^{-1}K$ is compact in $X_2^\sigma$, then $\lim_{n \to +\infty} \|K^*(\lambda - T)^{-1}K\psi_n\|_{X_2^\sigma} = 0$, therefore

$$\lim_{n \to +\infty} \varphi_n = 0 \quad \text{in } X_2.$$

On the other hand, we have

$$ \begin{aligned} \|\varphi_n\|_{X_2^\sigma}^2 &= \int_{D \times V} |\varphi_n(x, v)|^2 \sigma(v) dx dv \\ &= \int_{D \times V} \left| \int_0^{t^{-}(x,v)} (K\varphi_n)(x - tv, v) dt \right|^2 \sigma(v) dx dv \\ &= \int_{D \times V} \left| \int_0^{t^{-}(x,v)} \int_V k_1(v) k_2(v') \varphi_n(x - tv, v') dv' dt \right|^2 \hat{\sigma}(v) dx dv \\ &= \|\varphi_n\|_{X_2^\sigma}^2 \\ &\leq \|\sigma\|_{L^\infty(\text{supp}(k_1))}^2 \|\varphi_n\|_{X_2^\sigma}^2. \end{aligned} $$

where

$$\hat{\sigma}(v) = \begin{cases} \sigma(v) & \text{if } v \in \text{supp}(k_1) \\ 0 & \text{if } v \notin \text{supp}(k_1) \end{cases}$$

Consequently, $(\varphi_n)_{n \in \mathbb{N}}$ converge strongly to 0 in $X_2^\sigma$. This proves that the operator $(\lambda - T)^{-1}K$ is compact in $X_2^\sigma$. Now using an interpolation argument (cf. [19, Theorem 3.10]), we conclude that $(\lambda - T)^{-1}K$ is compact in $X_2^\sigma$ with $1 < p < +\infty$.

(b) We shall now prove that $(\lambda - T)^{-1}K$ is a Dunford-Pettis operator on $X_1^\sigma$. Let $\mathcal{O}$ be a weakly compact subset of $X_1^\sigma$. Our goal is to establish that $(\lambda - T)^{-1}K\mathcal{O}$ is a compact subset of $X_1^\sigma$ for the norm topology. To do so, let $\varphi \in \mathcal{O}$ and $c > 0$ and write

$$\varphi_c = (\lambda - T)^{-1} K(\varphi \chi_{\{|\varphi| \ge c\}}) \text{ and } \hat{\varphi}_c = (\lambda - T)^{-1} K(\varphi \chi_{\{|\varphi| < c\}})$$

where $\chi(\cdot)$ denotes the characteristic function.

For any $(y, w) \in \mathbb{R}^{2n}$, we have

$$ \int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x + y, v + w) - \tilde{\varphi}_c(x, v)|\tilde{\sigma}(v) dx dv \leq \int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x + y, v + w)|\tilde{\sigma}(v) dx dv \\ + \int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x, v)|\tilde{\sigma}(v) dx dv, $$
where $\tilde{\varphi}_c$ (resp. $\tilde{\sigma}$) denotes the zero extension of $\varphi_c$ (respectively, of $\sigma$) outside $D \times V$ (respectively, outside $V$). It follows from the estimate (5) that

$$ 
\begin{aligned}
\int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x, v)|\tilde{\sigma}(v)dx dv &\leq \|K(\varphi\chi_{\{|\varphi|\geq c\}})\|_{X^\sigma} \\
&\leq \|K\|_{\mathcal{L}(X^\sigma, X)} \int_{D \times V} |\varphi\chi_{\{|\varphi|\geq c\}}(x, v)|\sigma(v)dx dv.
\end{aligned}
$$

On the other hand, taking into account of the estimate (5), one can write

$$ 
\begin{aligned}
\int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x + y, v + w)|\tilde{\sigma}(v)dx dv &= \int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x, v)|\tilde{\sigma}(v - w)dx dv \\
&= \int_{D \times V} |\varphi_c(x, v)|\tilde{\sigma}(v - w)dx dv \\
&\leq \sup_{v \in V} \frac{\sigma(v)}{\mathrm{Re}\lambda + \sigma(v)} \int_{D \times V} |K(\varphi\chi_{\{|\varphi|\geq c\}})(x, v)|\tilde{\sigma}(v - w)dx dv \\
&\leq \int_V |k_1(v)|\tilde{\sigma}(v - w)dv \\
&\qquad \times \int_{D \times V} |k_2(v')||\varphi\chi_{\{|\varphi|\geq c\}}(x, v')|dx dv' \\
&\leq \int_{(\mathrm{supp}(k_1) \setminus \{w\}) \cap V} |k_1(v + w)|\sigma(v)dv \\
&\qquad \times \|h_2\|_{L^\infty(V)} \int_{D \times V} |\varphi\chi_{\{|\varphi|\geq c\}}(x, v')|\sigma(v')dx dv'.
\end{aligned}
$$

There exists $\eta > 0$ such that $(\mathrm{supp}(k_1) \setminus \bar{B}(0, \eta)) \subset V \setminus S$. Then, we get

$$ 
\begin{aligned}
\int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x + y, v + w)|\tilde{\sigma}(v)dx dv &\leq \|\sigma\|_{L^\infty(\mathrm{supp}(k_1) \setminus \bar{B}(0, \eta))} \|k_1\|_{L^1} \\
&\qquad \times \|h_2\|_{L^\infty(V)} \int_{D \times V} |\varphi\chi_{\{|\varphi|\geq c\}}(x, v')|\sigma(v')dx dv'.
\end{aligned}
$$

Consequently

$$ \int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x + y, v + w) - \tilde{\varphi}_c(x, v)|\tilde{\sigma}(v)dx dv \leq C \int_{D \times V} |\varphi\chi_{\{|\varphi|\geq c\}}(x, v')|\sigma(v)dx dv, $$

where

$$ C = \left[ \|K\|_{\mathcal{L}(X_1^\sigma, X_1)} + \|\sigma\|_{L^\infty(\mathrm{supp}(k_1) \setminus \bar{B}(0, \eta))} \|k_1\|_{L^1} \|h_2\|_{L^\infty(V)} \right]. \quad (7) $$

Since $\mathcal{O}$ is weakly compact and the measure of the set $\{(x, v) \in D \times V : |\varphi(x, v)| \geq c\}$ goes to 0 as $c \to +\infty$, it follows from [11, Theorem 15, p. 76] that, for any $\varepsilon > 0$, there exist numbers $c > 0$ and $\eta > 0$ such that for all $w \in \bar{B}(0, \eta)$

$$ \int_{\mathbb{R}^n \times \mathbb{R}^n} |\tilde{\varphi}_c(x + y, v + w) - \tilde{\varphi}_c(x, v)|\sigma(v)dx dv \leq \varepsilon $$
uniformly in $\varphi \in \mathcal{O}$.

For $c > 0$ chosen as above, according to [1, Theorem 2.21], it suffices to prove that

$$ \mathcal{O}_c = \{(\lambda - T)^{-1} K(\varphi \chi_{\{|\varphi|<c\}}); \varphi \in \mathcal{O}\} $$

is relatively compact in $X_1^\sigma$. To do so, we first observe that the set $\mathcal{O}_c$ is bounded in $X_2^\sigma$ because $\mathcal{O}$ is bounded in $X_1^\sigma$ and $\varphi\chi_{\{|\varphi|<c\}} \in X_2^\sigma$. So it suffices to prove that $(\lambda - T)^{-1}K$ is compact in $X_2^\sigma$. This was established in the part (a) of the proof. Q.E.D.

# 4 Existence results

In this section, we discuss existence results for the boundary value problem (1)-(2). We shall introduce the following assumptions

(A3) $f(\cdot, \cdot, \cdot)$ is a Carathéodory function satisfying the growth condition (6),

(A4) $\beta\sigma_0^{-\frac{1}{q}} \|K\|_{\mathcal{L}(X_p^\sigma, X_p)} < 1,$

where $\beta$ is the constant appearing in the growth condition (6) and $q$ is the conjugate of $p$.

Now, we are in a position to announce the main result of this paper. For any $r > 0$, we denote by $\mathbb{B}_r^\sigma$ the set

$$ \mathbb{B}_r^{\sigma,p} = \{ \psi \in X_p^\sigma : \|\psi\|_{X_p^\sigma} \le r \}. $$

**Theorem 4.1.** Let $1 \le p < \infty$, $K \in \mathcal{L}(X_p^\sigma, X_p)$ and assume that conditions (A1) – (A4) are satisfied. Then, for each $\lambda \in G_{\sigma_0}$ and $r \in \mathbb{R}$ satisfying

$$ r \ge \frac{\sigma_0^{-\frac{1}{q}} \|K\|_{\mathcal{L}(X_p^\sigma, X_p)} \|\alpha\|_{X_p}}{1 - \beta \sigma_0^{-\frac{1}{q}} \|K\|_{\mathcal{L}(X_p^\sigma, X_p)}} \quad (8) $$

the problem

$$ \left\{ \begin{array}{l} v \cdot \nabla_x \psi(x, v) + \sigma(v) \psi(x, v) + \lambda \psi(x, v) = \displaystyle\int_V \kappa(x, v, v') f(x, v', \psi(x, v')) dv' \\ \psi|_{\Gamma_-} = 0, \end{array} \right. \quad (9) $$

has at least one solution on $\mathbb{B}_r^{\sigma,p}$.

**Proof.** Let $\lambda$ be a complex number such that $\mathrm{Re}\lambda > -\sigma_0$. Then according to Lemma 2.1, the operator $\lambda - T$ is invertible and therefore Problem (9) may be written in the form

$$ \psi = \Pi(\lambda)\psi, \quad \psi|_{\Gamma_-} = 0, $$

where $\Pi(\lambda) = (\lambda - T)^{-1}K\mathcal{N}_f$.

It is clear that $\Pi(\lambda)$ is continuous. We shall now check that $\Pi(\lambda)$ leaves $\mathbb{B}_r^{\sigma,p}$ invariant. According to Lemma 2.2, there exist a constant $\beta > 0$ and a function $\alpha \in X_p$ such that

$$ f(x, v, \psi(x, v)) \le \frac{\alpha(x, v)}{\sigma(v)^{\frac{1}{p}}} + \beta |\psi(x, v)| \quad \text{a.e. in } (x, v) \text{ for all } \psi \in X_p^\sigma. $$
For any $\psi \in \mathbb{B}_r^{\sigma,p}$, we have

$$
\begin{align*}
\| \Pi(\lambda)\psi \|_{X_p^\sigma} &= \|(\lambda - T)^{-1} K N_f \psi \|_{X_p^\sigma} \\
&\le \|(\lambda - T)^{-1} \|_{\mathcal{L}(X_p, X_p^\sigma)} \|K\|_{\mathcal{L}(X_p^\sigma, X_p)} \|N_f \psi \|_{X_p^\sigma} \\
&\le \sigma_0^{\frac{-1}{q}} \sup_{v \in V} \frac{\sigma(v)}{\mathrm{Re}\lambda + \sigma(v)} \|K\|_{\mathcal{L}(X_p^\sigma, X_p)} (\|\alpha\|_{X_p} + \beta r) \\
&\le \sigma_0^{\frac{-1}{q}} \|K\|_{\mathcal{L}(X_p^\sigma, X_p)} (\|\alpha\|_{X_p} + \beta r)
\end{align*}
$$

Hence, in order to show that $\Pi(\lambda)$ maps $\mathbb{B}_r^{\sigma,p}$ into itself, it suffices to choose $r$ satisfying the condition (8). Therefore, for such $r$ and for all $p \in [1, +\infty)$, we have $\Pi(\lambda)(\mathbb{B}_r^{\sigma,p}) \subset \mathbb{B}_r^{\sigma,p}$.

To complete the proof we must treat separately the cases $p \in (1, +\infty)$ and $p = 1$.

• Let $p \in (1, +\infty)$. It follows from Theorem 3.1 (a) that $(\lambda - T)^{-1}K$ is a compact operator of $\mathcal{L}(X_p^\sigma, X_p)$. Hence $\Pi(\lambda)$ is compact. Therefore, the operator $\Pi(\lambda)$ satisfies the hypotheses of Theorem 2.1. So Problem (9) has at least one solution in $\mathbb{B}_r^{\sigma,p}$.

• Let $p = 1$. We claim that the operator $\Pi(\lambda)$ is ws-compact on $X_1^\sigma$. To see this, let $(\psi_n)_{n \in \mathbb{N}}$ be a weakly convergent sequence in $X_1^\sigma$. Hence the set $\Psi = \{\psi_n : n \in \mathbb{N}\}$ is a relatively weakly compact set of $X_1^\sigma$. Let $A$ be a measurable subset of $D \times V$ with respect to the measure $\sigma(v)dxdv$. It follows from [14, Corollary 11, p. 294] that

$$
\lim_{|A| \to 0} \int_A |\psi(x, v)| dx \sigma(v) dv = 0
$$

uniformly for $\psi \in \Psi$ where $|A|$ denotes the measure of $A$ with respect to $\sigma(v)dxdv$. Using Eq. (6), we get

$$
| \int_A (N_f \psi)(x, v) dx \sigma(v) dv | \le \int_A |\alpha(x, v)| dxdv + \beta \int_A |\psi(x, v)| dxdv
$$

and therefore

$$
\lim_{|A| \to 0} \int_A (N_f \psi)(x, v) dx \sigma(v) dv = 0,
$$

uniformly for $\psi \in \Psi$. Using again [14, Corollary 11, p. 294], we conclude that $(N_f \psi_n)_n$ has a weakly convergent subsequence $(N_f \psi_{n_k})_k$. Let $\Phi$ be its weak limit. Since the set $\Psi := \{(N_f \psi_{n_k}), k \in \mathbb{N}\} \cup \{\Phi\}$ is weakly compact on $X_1^\sigma$, it follows from Theorem 3.1 (b) that $(\lambda - T)^{-1} K \Psi$ is a compact subset of $X_1^\sigma$ and therefore $(\Pi(\lambda) \psi_{n_k})_{k \in \mathbb{N}}$ has a strongly convergent subsequence in $X_1^\sigma$ which proves our claim.

Hence, for any $\lambda \in G_{\sigma_0}$ and for all $r$ satisfying the condition (8), the operator $\Pi$ satisfies the conditions of Theorem 2.2. Therefore Problem (9) has at least one solution in $\mathbb{B}_r^{\sigma,1}$. Q.E.D.

The remainder of this section is devoted to the existence of positive solutions to the boundary value problem (9). Note that $X_p^\sigma$ is a Banach lattice space; its positive cone will be denoted by $X_p^{\sigma,+}$.

Let $r > 0$. We define the set $\mathbb{B}_{r,+}^{\sigma,p}$ by

$$
\mathbb{B}_{r,+}^{\sigma,p} := \mathbb{B}_r^{\sigma,p} \cap X_p^{\sigma,+}.
$$
According to Remark 2.1 (b), for any real $\lambda > \sigma_0$, $(\lambda - T)^{-1}$ is a positive operator (in the lattice sense). If further the hypotheses of Theorem 4.1 are satisfied, $\mathcal{N}_f(\mathcal{X}_p^{\sigma,+}) \subseteq \mathcal{X}_p^{\sigma,+}$ and $K$ is positive, then $(\lambda - T)^{-1}KN_f$ leaves the positive cone $\mathcal{X}_p^{\sigma,+}$ invariant. Now we replace $\mathbb{B}_{r,+}^{\sigma,p}$ by $\mathbb{B}_{r,+}^{\sigma,p}$ and the same reasoning as in the proof of Theorem 4.1, we obtain the following result.

**Corollary 4.1.** Let the hypotheses of Theorem 4.1 be satisfied. Further, if $\mathcal{N}_f(\mathcal{X}_p^{\sigma,+}) \subseteq \mathcal{X}_p^{\sigma,+}$ and $K$ is positive. Then Problem (9) has at least one solution in $\mathbb{B}_{r,+}^{\sigma,p}$.

In the following, we shall focus our attention on the existence of nontrivial positive solution.

**Corollary 4.2.** Assume that the hypotheses of Corollary 4.1 are satisfied. If there is $\rho > 0$ and $0 \neq \psi_0 \in \mathbb{B}_{r,+}^{\sigma,p}$ such that

(i) $\psi_0 \notin N(K)$ where $N(K)$ denotes the null space of $K$,

(ii) $(\mathcal{N}_f)(x, v) \geq \rho\psi_0(x, v)$ for all $\psi \in \mathbb{B}_{r,+}^{\sigma,p}$.

Then, there exist $\eta > 0$ such that the problem

$$ \left\{ \begin{array}{l} v \cdot \nabla_x \psi(x, v) + \sigma(v) \psi(x, v) + \lambda \psi(x, v) = \eta \int_V k(v, v') f(x, v', \psi(x, v')) dv', \\ \\ \psi|_{\Gamma_-} = 0, \end{array} \right. \qquad (10) $$

has at least one solution $\mathbb{B}_{r,+}^{\sigma,p}$ satisfying $\|\psi^*\|_{\mathcal{X}_p^{\sigma}} = r$.

**Proof.** Arguing as in the proofs of Theorems 4.1 and Corollary 4.1 we infer that, for all real number $\lambda$ such that $\lambda > \sigma_0$ and for all $r$ satisfying (8), the operator $\Pi(\lambda)$ is continuous and maps $\mathbb{B}_{r,+}^{\sigma,p}$ into itself. On the other hand, using the positivity of $(\lambda - T)^{-1}$ and $K$, one sees that

$$ \Pi(\lambda)(\psi) \geq \rho(\lambda - T)^{-1}K(\psi_0) \geq 0 \text{ and } (\lambda - T)^{-1}K(\psi_0) \neq 0 \text{ for all } \psi \in \mathbb{B}_{r,+}^{\sigma,p}. $$

Accordingly, for all $\psi \in \mathbb{B}_{r,+}^{\sigma,p}$, we have

$$ \|\Pi(\lambda)(\psi)\|_{\mathcal{X}_p^{\sigma}} \geq \rho \|(\lambda - T)^{-1}(\psi_0)\|_{\mathcal{X}_p^{\sigma}} > 0 $$

and consequently

$$ \inf \left\{ \| \Pi(\lambda)(\psi) \|_{\mathcal{X}_p^{\sigma}} : \psi \in \mathbb{B}_{r,+}^{\sigma,p} \right\} > 0. $$

Hence, for $\lambda \geq \sigma_0$, one can define the operator

$$ H_{\lambda}(\psi) = r \frac{\Pi(\lambda)(\psi)}{\|\Pi G(\lambda)(\psi)\|_{\mathcal{X}_p^{\sigma}}} \quad \text{for all } \psi \in \mathbb{B}_{r,+}^{\sigma,p}. $$

Using the same arguments as in the proof of the Theorem 4.1, we see that $H_{\lambda}$ is continuous, ws-compact (respectively compact) on $\mathcal{X}_1^{\sigma}$ (resp. $\mathcal{X}_p^{\sigma}$ with $1 < p < +\infty$) and maps $\mathbb{B}_{r,+}^{\sigma,1}$ (resp. $\mathbb{B}_{r,+}^{\sigma,p}$) into itself. So, applying Theorem 2.1 (respectively Theorem 2.2) we infer that $H_{\lambda}$ has a fixed point $\psi^*$ in $\mathbb{B}_{r,+}^{\sigma,1}$ (resp. $\mathbb{B}_{r,+}^{\sigma,p}$) satisfying $\|\psi^*\|_{\mathcal{X}_p^{\sigma}} = r$. Putting $\eta = \frac{r}{\|H_{\lambda}(\psi^*)\|_{\mathcal{X}_p^{\sigma}}}$, we get $H_{\lambda}(\psi^*) = \eta^{-1}\psi^*$.

Consequently, $\psi^* \in D(T) \cap \mathbb{B}_{r,+}^{\sigma,p}$ and

$$ v \cdot \nabla_x \psi^*(x, v) + \sigma(v) \psi^*(x, v) + \lambda \psi^*(x, v) = \eta \int_V \kappa(x, v, v') f(x, v', \psi^*(x, v')) dv'. $$

This completes the proof.

Q.E.D.
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