Optimization problems in rearrangement classes for fractional $p$-Laplacian equations

1.   Introduction and main results

The present paper deals with some optimization problems related to elliptic equations of nonlinear, nonlocal type, with data varying in rearrangement classes. For the reader's convenience, we recall here the basic definition, referring to Section 2 for details. Given a bounded smooth domain $\Omega\subset {\mathbb R}^N$ and a non-negative function $g_0\in L^\infty(\Omega)$, we say that $g\in L^\infty(\Omega)$ lies in the rearrangement class of $g_0$, denoted $\mathcal{G}$, if for all $t \geqslant 0$

where we denote by $|\cdot|$ the $N$-dimensional Lebesgue measure of sets. We may define several functionals $\Phi: \mathcal{G}\to {\mathbb R}$ corresponding to variational problems, and study the optimization problems

We note that, since $\mathcal{G}$ is not a convex set, the problems above do not fall in the familiar case of convex optimization, whatever the nature of $\Phi$. The following is a classical example. For all $g\in \mathcal{G}$ consider the Dirichlet problem

which, by classical results in the calculus of variations, admits a unique weak solution $u_g\in H^1_0(\Omega)$. So set

The existence of a maximizer for $\Phi$, i.e., of a datum $\hat g\in \mathcal{G}$ s.t. for all $g\in \mathcal{G}$

was proved in ^[3,4], while the existence of a minimizer was investigated in ^[5]. One challenging feature of such problem is that, in general, the functional $\Phi$ turns out to be continuous (in a suitable sense) but the class $\mathcal{G}$ fails to be compact. Therefore, a possible strategy consists in optimizing $\Phi$ over the closure $\overline{\mathcal{G}}$ of $\mathcal{G}$ in the sequential weak^* topology of $L^\infty(\Omega)$ (a much larger, and convex, set), and then proving that the maximizers and minimizers actually lie in $\mathcal{G}$ (which is far from being trivial).

In addition, due to the nature of the rearrangement equivalence and some functional inequalities, the maximizer/minimizer $g$ may show some structural connection to the solution $u_g$ of the corresponding variational problem. This has interesting consequences, for instance let $g_0$ be the characteristic function of some subdomain $D_0\subset\Omega$, then the optimal $g$ is as well the characteristic function of some $D\subset\Omega$ with $|D| = |D_0|$. Moreover, it is proved that $g = \eta\circ u_g$ in $\Omega$ for some nondecreasing $\eta$, while by the Dirichlet condition we have $u_g = 0$ on $\partial\Omega$. Therefore, any optimal domain $D$ has a positive distance from $\partial\Omega$.

A similar approach applies to several variational problems and functionals. For instance, in ^[8] the authors consider the following $p$-Laplacian equation with $p > 1$, $q\in[0, p)$:

which admits a unique non-negative solution $u_g\in W^{1, p}_0(\Omega)$, and study the maximum and minimum over $\mathcal{G}$ of the functional

In ^[7], the following weighted eigenvalue problem is considered:

It is well known that the problem above admits a principal eigenvalue $\lambda(g) > 0$ (see ^[20]), and the authors prove that the functional $\lambda(g)$ has a minimizer in $\mathcal{G}$.

In recent years, several researchers have studied optimization problems related to elliptic equations of fractional order (see ^[23] for a general introduction to such problems and the related variational methods). In the linear framework, the model operator is the $s$-fractional Laplacian with $s\in(0, 1)$, defined by

where $C_{N, s} > 0$ is a normalization constant. In ^[26], the following problem is examined:

where $h(x, \cdot)$ is nondecreasing and grows sublinearly in the second variable. The solution $u_g\in H^s_0(\Omega)$ is unique and is the unique minimizer of the energy functional

(where $H(x, \cdot)$ denotes the primitive of $h(x, \cdot)$), so the authors investigate the minimization of $\Phi(g)$ over $\mathcal{G}$. Besides, in ^[1], the following nonlocal eigenvalue problem is considered:

The authors prove, among other results, the existence of a minimizer $g\in \mathcal{G}$ for the principal eigenvalue $\lambda(g)$. Optimization of the principal eigenvalue of fractional operators has significant applications in biomathematics, see ^[25]. In all the aforementioned problems, optimization in $\mathcal{G}$ also yields representation formulas and qualitative properties (e.g., Steiner symmetry over convenient domains) of the optimal data.

In the present paper, we focus on the following nonlinear, nonlocal operator:

Here $N \geqslant 2$, $p > 1$, $s\in(0, 1)$, and $K: {\mathbb R}^N\times {\mathbb R}^N\to {\mathbb R}$ is a measurable kernel s.t. for a.e. $x, y\in {\mathbb R}^N$,

$(K_1)$ $K(x, y) = K(y, x)$;

$(K_2)$ $C_1 \leqslant K(x, y)|x-y|^{N+ps} \leqslant C_2$ ($0 < C_1 \leqslant C_2$).

If $C_1 = C_2 = C_{N, p, s} > 0$ (a normalization constant varying from one reference to the other), $\mathcal{L}_K$ reduces to the $s$-fractional $p$-Laplacian

which in turn coincides with the $s$-fractional Laplacian seen above for $p = 2$. The nonlinear operator $\mathcal{L}_K$ arises from problems in game theory (see ^[2,6]). Besides, the special case $(-\Delta)_p^s$ can be seen as either an approximation of the classical $p$-Laplace operator for fixed $p$ and $s\to 1^-$ (see ^[19]), or an approximation of the fractional $\infty$-Laplacian for fixed $s$ and $p\to\infty$, with applications to the problem of Hölder continuous extensions of functions (see ^[22]). Equations driven by the fractional $p$-Laplacian are the subject of a vast literature, dealing with existence, qualitative properties, and regularity of the solutions (see for instance ^{[13,14,15,24]}).

Inspired by the cited references, we will examine two variational problems driven by $\mathcal{L}_K$, set on a bounded domain $\Omega$ with $C^{1, 1}$-smooth boundary, with a datum $g$ varying in a rearrangement class $\mathcal{G}$, and optimize the corresponding functionals. First, we consider the following nonlinear, nonlocal eigenvalue problem:

Let $\lambda(g)$ be the principal eigenvalue of (1.1), defined by

and $u_g$ be the (unique) associated eigenfunction s.t. $u_g > 0$ in $\Omega$ and

With such definitions, we will study the following optimization problem:

Precisely, we will prove that such problem admits at least one solution, that any solution actually minimizes $\lambda(g)$ over the larger set $\overline{\mathcal{G}}$, while all minimizers over $\overline{\mathcal{G}}$ lie in $\mathcal{G}$, and finally that any minimal weight can be represented as a nondecreasing function of the corresponding eigenfunction:

Theorem 1.1. Let $\Omega\subset {\mathbb R}^N$ be a bounded domain with $C^{1, 1}$-boundary, $p > 1$, $s\in(0, 1)$, $K: {\mathbb R}^N\times {\mathbb R}^N\to {\mathbb R}$ be measurable satisfying $(K_1)$, $(K_2)$, $g_0\in L^\infty(\Omega)_+\setminus\{0\}$, $\mathcal{G}$ be the rearrangement class of $g_0$. For all $g\in \overline{\mathcal{G}}$, let $\lambda(g)$ be the principal eigenvalue of (1.1). Then,

$(i)$ there exists $\hat g\in \mathcal{G}$ s.t. $\lambda(\hat g) \leqslant\lambda(g)$ for all $g\in \mathcal{G}$;

$(ii)$ for all $\hat g$ as in $(i)$ and $g\in \overline{\mathcal{G}}\setminus \mathcal{G}$, $\lambda(\hat g) < \lambda(g)$;

$(iii)$ for all $\hat g$ as in $(i)$ there exists a nondecreasing map $\eta: {\mathbb R}\to {\mathbb R}$ s.t. $\hat g = \eta\circ u_{\hat g}$ in $\Omega$.

Then, we will focus on the following general nonlinear Dirichlet problem:

where, in addition to the previous hypotheses, we assume that $h:\Omega\times {\mathbb R}\to {\mathbb R}_+$ is a Carathéodory mapping satisfying the following conditions:

$(h_1)$ $h(x, \cdot)$ is nondecreasing in ${\mathbb R}$ for a.e. $x\in\Omega$;

$(h_2)$ $h(x, t) \leqslant C_0(1+|t|^{q-1})$ for a.e. $x\in\Omega$ and all $t\in {\mathbb R}$, with $C_0 > 0$ and $q\in(1, p)$.

For all $g\in \overline{\mathcal{G}}$ problem (1.2) has a unique solution $u_g$, with associated energy

(where $H(x, \cdot)$ denotes the primitive of $h(x, \cdot)$). Our second result deals with following optimization problem:

and is stated as follows:

Theorem 1.2. Let $\Omega\subset {\mathbb R}^N$ be a bounded domain with $C^{1, 1}$-boundary, $p > 1$, $s\in(0, 1)$, $K: {\mathbb R}^N\times {\mathbb R}^N\to {\mathbb R}$ be measurable satisfying $(K_1)$, $(K_2)$, $h:\Omega\times {\mathbb R}\to {\mathbb R}_+$ be a Carathéodory mapping satisfying $(h_1)$, $(h_2)$, $g_0\in L^\infty(\Omega)_+\setminus\{0\}$, $\mathcal{G}$ be the rearrangement class of $g_0$. For all $g\in \overline{\mathcal{G}}$, let $u_g$ be the solution of (1.2) and $\Psi(g)$ be the associated energy. Then,

$(i)$ there exists $\hat g\in \mathcal{G}$ s.t. $\Psi(\hat g) \leqslant\Psi(g)$ for all $g\in \mathcal{G}$;

$(ii)$ for all $\hat g$ as in $(i)$ and $g\in \overline{\mathcal{G}}\setminus \mathcal{G}$, $\Psi(\hat g) < \Psi(g)$;

$(iii)$ for all $\hat g$ as in $(i)$ there exists a nondecreasing map $\eta: {\mathbb R}\to {\mathbb R}$ s.t. $\hat g = \eta\circ u_{\hat g}$ in $\Omega$.

Theorem 1.1 above extends [1, Theorem 1.1] to the nonlinear framework, which requires some delicate arguments due to the non-Hilbertian structure of the problem. Similarly, Theorem 1.2 extends [26, Theorem 3.2], also introducing the structure property of minimizers in $(iii)$.

The dual problems, i.e., maximization of $\lambda(g)$ and $\Psi(g)$ respectively, remain open for now. The reason is easily understood, as soon as we recall that both $\lambda(g)$ and $\Psi(g)$ admit variational characterizations as minima of convenient functions on the Sobolev space $W^{s, p}_0(\Omega)$, so further minimizing with respect to $g$ conjures a 'double minimization' problem. On the contrary, maximizing $\lambda(g)$, $\Psi(g)$, respectively, would result in a min-max problem, which requires a different approach.

The structure of the paper is the following: In Section 2 we recall some preliminaries on rearrangement classes and fractional order equations; in Section 3 we deal with the eigenvalue problem (1.1); and in Section 4 we deal with the general Dirichlet problem (1.2).

Notation. For all $\Omega\subset {\mathbb R}^N$, we denote by $|\Omega|$ the $N$-dimensional Lebesgue measure of $\Omega$ and $\Omega^c = {\mathbb R}^N\setminus\Omega$. For all $x\in {\mathbb R}^N$, $r > 0$ we denote by $B_r(x)$ the open ball centered at $x$ with radius $r$. When we say that $g \geqslant 0$ in $\Omega$, we mean $g(x) \geqslant 0$ for a.e. $x\in\Omega$, and similar expressions. Whenever $X$ is a function space on the domain $\Omega$, $X_+$ denotes the positive order cone of $X$. In any Banach space we denote by $\to$ strong (or norm) convergence, by $\rightharpoonup$ weak convergence, and by $\overset{*}{\rightharpoonup}$ weak^* convergence. For all $q\in[1, \infty]$, we denote by $\|\cdot\|_q$ the norm of $L^q(\Omega)$. Finally, $C$ denotes several positive constants, varying from line to line.

2.   Preliminaries

In this section we collect some necessary preliminary results on rearrangement classes and fractional Sobolev spaces.

2.1.   Rearrangement classes

Let $\Omega\subset {\mathbb R}^N$ ($N \geqslant 2$) be a bounded domain, $g_0\in L^\infty(\Omega)$ be s.t. $0 \leqslant g_0 \leqslant M$ in $\Omega$ ($M > 0$), and $g_0 > 0$ on some subset of $\Omega$ with positive measure. We say that a function $g\in L^\infty(\Omega)$ is a rearrangement of $g_0$, denoted $g\sim g_0$, if for all $t \geqslant 0$

Also, we define the rearrangement class

Clearly, $0 \leqslant g \leqslant M$ in $\Omega$ for all $g\in \mathcal{G}$. Recalling that $L^\infty(\Omega)$ is the topological dual of $L^1(\Omega)$, we can endow such space with the weak^* topology, characterized by the following type of convergence:

We denote by $\overline{\mathcal{G}}$ the closure of $\mathcal{G}$ in $L^\infty(\Omega)$ with respect to such topology. It is proved in ^[3,4] that $\overline{\mathcal{G}}$ is a sequentially weakly* compact convex set, and that $0 \leqslant g \leqslant M$ in $\Omega$ for all $g\in \overline{\mathcal{G}}$. Therefore, given a sequentially weakly* continuous functional $\Phi: \overline{\mathcal{G}}\to {\mathbb R}$, there exist $\check g, \hat g\in \overline{\mathcal{G}}$ s.t. for all $g\in \overline{\mathcal{G}}$

In general, the extrema are not attained at points of $\mathcal{G}$. As usual, we say that $\Phi$ is Gâteaux differentiable at $g\in \overline{\mathcal{G}}$, if there exists a linear functional $\Phi'(g)\in L^\infty(\Omega)^*$ s.t. for all $h\in \overline{\mathcal{G}}$

We remark that $g\in \overline{\mathcal{G}}$ being a minimizer (or maximizer) of $\Phi$ does not imply $\Phi'(g) = 0$ in general. Nevertheless, if $\Phi$ is convex, then for all $h\in \overline{\mathcal{G}}$

with strict inequality if $\Phi$ is strictly convex and $h\neq g$ (see ^[27] for an introduction to convex functionals and variational inequalities). Finally, let us recall a technical lemma on optimization of linear functionals over $\overline{\mathcal{G}}$, which also provides a representation formula:

Lemma 2.1. Let $h\in L^1(\Omega)$. Then,

$(i)$ there exists $\hat g\in \mathcal{G}$ s.t. for all $g\in \overline{\mathcal{G}}$

$(ii)$ if $\hat g$ is unique, then there exists a nondecreasing map $\eta: {\mathbb R}\to {\mathbb R}$ s.t. $\hat g = \eta\circ h$ in $\Omega$.

Proof. By [3, Theorems 1, 4], there exists $\hat g\in \mathcal{G}$ which maximizes the linear functional

over $\mathcal{G}$. Given $g\in \overline{\mathcal{G}}\setminus \mathcal{G}$, we can find a sequence $(g_n)$ in $\mathcal{G}$ s.t. $g_n \overset{*}{\rightharpoonup} g$. For all $n\in {\mathbb N}$ we have

so passing to the limit we get

thus proving $(i)$. From [3, Theorem 5] we have $(ii)$. □

2.2.   Fractional Sobolev spaces

We recall some basic notions about the variational formulations of problems (1.1) and (1.2). For $p > 1$, $s\in(0, 1)$, all open $\Omega\subseteq {\mathbb R}^N$, and all measurable $u:\Omega\to {\mathbb R}$ we define the Gagliardo seminorm

The corresponding fractional Sobolev space is defined by

If $\Omega$ is bounded and with a $C^{1, 1}$-smooth boundary, we incorporate the Dirichlet conditions by defining the space

endowed with the norm $\|u\|_{ W^{s, p}_0(\Omega)} = [u]_{s, p, {\mathbb R}^N}$. This is a uniformly convex, separable Banach space with dual $W^{-s, p'}(\Omega)$, s.t. $C^\infty_c(\Omega)$ is a dense subset of $W^{s, p}_0(\Omega)$, and the embedding $W^{s, p}_0(\Omega)\hookrightarrow L^q(\Omega)$ is compact for all $q\in[1, p^*_s)$, where

For a detailed account on fractional Sobolev spaces, we refer the reader to ^[10,21]. Now let $K: {\mathbb R}^n\times {\mathbb R}^N\to {\mathbb R}$ be a measurable kernel satisfying $(K_1)$ and $(K_2)$. We introduce an equivalent norm on $W^{s, p}_0(\Omega)$ by setting

We can now rephrase more carefully the definitions given in Section 1, by defining the operator $\mathcal{L}_K: W^{s, p}_0(\Omega)\to W^{-s, p'}(\Omega)$ as the gradient of the $C^1$-functional

Equivalently, for all $u, \varphi\in W^{s, p}_0(\Omega)$ we set

Both problems that we are going to study belong to the following class of nonlinear, nonlocal Dirichlet problems:

where $f:\Omega\times {\mathbb R}\to {\mathbb R}$ is a Carathéodory mapping subject to the following subcritical growth conditions: there exist $C > 0$, $r\in(1, p^*_s)$ s.t. for a.e. $x\in\Omega$ and all $t\in {\mathbb R}$

We say that $u\in W^{s, p}_0(\Omega)$ is a weak solution of (2.1), if for all $\varphi\in W^{s, p}_0(\Omega)$

There is a wide literature on problem (2.1), especially for the model case $\mathcal{L}_K = (-\Delta)_p^s$, see for instance ^{[9,14,16,17,24]}. We will only need to recall the following properties, which can be proved adapting [16, Proposition 2.3] and [9, Theorem 1.5], respectively:

Lemma 2.2. Let $f$ satisfy (2.2), $u\in W^{s, p}_0(\Omega)$ be a weak solution of (2.1). Then, $u\in L^\infty(\Omega)$.

Lemma 2.3. Let $f$ satisfy (2.2), and $\delta > 0$, $c\in C(\overline\Omega)_+$ be s.t. for a.e. $x\in\Omega$ and all $t\in[0, \delta]$

Also, let $u\in W^{s, p}_0(\Omega)_+$ be a weak solution of (2.2). Then, either $u = 0$, or $u > 0$ in $\Omega$.

We will not cope with regularity of the weak solutions here. In the model case of the fractional $p$-Laplacian, under hypothesis (2.2), using Lemma 2.2 above and [15, Theorems 1.1, 2.7], it can be seen that whenever $u\in W^{s, p}_0(\Omega)$ solves (2.1), we have $u\in C^s({\mathbb R}^N)$ and there exist $\alpha\in(0, s)$ depending only on the data of the problem, s.t. the function

admits a $\alpha$-Hölder continuous extension to $\overline\Omega$. The same result is not known for the general operator $\mathcal{L}_K$, except the linear case $p = 2$ with a special anisotropic kernel, see ^[28].

For future use, we prove here a technical lemma:

Lemma 2.4. Let $(g_n)$ be a sequence in $\overline{\mathcal{G}}$ s.t. $g_n \overset{*}{\rightharpoonup} g$, $(u_n)$ be a bounded sequence in $W^{s, p}_0(\Omega)$, $r\in[1, p^*_s)$. Then, there exists $u\in W^{s, p}_0(\Omega)$ s.t. up to a subsequence

Proof. By the compact embedding $W^{s, p}_0(\Omega)\hookrightarrow L^r(\Omega)$, passing if necessary to a subsequence we have $u_n\to u$ in $L^r(\Omega)$ and $u_n(x)\to u(x)$ for a.e. $x\in\Omega$, as $n\to\infty$. In particular $|u|^r\in L^1(\Omega)$, so

Besides, recalling that $0 \leqslant g_n \leqslant M$ in $\Omega$ for all $n\in {\mathbb N}$, we have by Hölder's inequality

and the latter tends to $0$ as $n\to\infty$. □

3.   Optimization of the principal eigenvalue

In this section we consider the eigenvalue problem (1.1) and prove Theorem 1.1. Let $\Omega$, $p$, $s$, $K$, $g_0$ be as in Section 1. For any $g\in \overline{\mathcal{G}}$, as in Subsection 2.2 we say that $u\in W^{s, p}_0(\Omega)$ is a (weak) solution of (1.1) if for all $\varphi\in W^{s, p}_0(\Omega)$

We say that $\lambda\in {\mathbb R}$ is an eigenvalue if (1.1) admits a solution $u\neq 0$, which is then called a $\lambda$-eigenfunction. Though a full description of the eigenvalues of (1.1) is missing, from ^[11,12,22] we know that for all $g\in L^\infty(\Omega)_+$ there exists a principal eigenvalue $\lambda(g) > 0$, namely the smallest positive eigenvalue, which admits the following variational characterization:

In addition, from ^[11] we know that $\lambda(g)$ is an isolated eigenvalue, simple, with constant sign eigenfunctions, while for any eigenvalue $\lambda > \lambda(g)$ the associated $\lambda$-eigenfunctions change sign in $\Omega$. So, recalling Lemma 2.3, there exists a unique normalized positive $\lambda(g)$-eigenfunction $u_g\in W^{s, p}_0(\Omega)$ s.t.

In particular $g\mapsto\lambda(g)$ defines a real-valued functional defined in the rearrangement class of weights $\mathcal{G}$ (or in $\overline{\mathcal{G}}$), and we are interested in the minimizers of such functional. Equivalently, we may set for all $g\in \overline{\mathcal{G}}$

and consider the maximization problem

First, we want to maximize $\Phi(g)$ over $\overline{\mathcal{G}}$, which is possible due to the following lemma:

Lemma 3.1. The functional $\Phi(g)$ is sequentially weakly* continuous in $\overline{\mathcal{G}}$.

Proof. Let $(g_n)$ be a sequence in $\overline{\mathcal{G}}$ s.t. $g_n \overset{*}{\rightharpoonup} g$, and for simplicity denote $u_n = u_{g_n}$ for all $n\in {\mathbb N}$, and $u = u_g$. We need to prove that $\Phi(g_n)\to\Phi(g)$. Since $u^p\in L^1(\Omega)$, we have

Also, by definition of $\Phi$ we have for all $n\in {\mathbb N}$

and the latter tends to $\Phi(g)$ as $n\to\infty$. Therefore

In particular, for all $n\in {\mathbb N}$ we have

so $(u_n)$ is bounded in $W^{s, p}_0(\Omega)$. By reflexivity and the compact embedding $W^{s, p}_0(\Omega)\hookrightarrow L^p(\Omega)$, passing to a subsequence we have $u_n\rightharpoonup v$ in $W^{s, p}_0(\Omega)$, $u_n\to v$ in $L^p(\Omega)$, and $u_n(x)\to v(x)$ for a.e. $x\in\Omega$, as $n\to\infty$. In particular, $v \geqslant 0$ in $\Omega$. By convexity we have

By Lemma 2.4 (with $r = p$) we also have

So we get

This, besides (3.2), concludes the proof. □

Lemma 3.1, along with the compactness of $\overline{\mathcal{G}}$, proves that $\Phi(g)$ admits a minimizer and a maximizer in $\overline{\mathcal{G}}$. We next need to ensure that at least one maximizer lies in the smaller set $\mathcal{G}$. In the next lemmas we will investigate further properties of $\Phi$.

Lemma 3.2. The functional $\Phi$ is strictly convex in $\overline{\mathcal{G}}$.

Proof. We introduce an alternative expression for $\Phi$. For all $g\in \overline{\mathcal{G}}$, $u\in W^{s, p}_0(\Omega)_+$ set

We fix $g\in \overline{\mathcal{G}}$ and maximize $F(g, \cdot)$ over positive functions. For all $u\in W^{s, p}_0(\Omega)_+\setminus\{0\}$ and $\tau > 0$, the function

is differentiable in $\tau$ with derivative

So the maximum of $\tau\mapsto F(g, \tau u)$ is attained at

and amounts at

Maximizing further over $u$, we obtain

Noting that $[|u|]_K \leqslant[u]_K$ for all $u\in W^{s, p}_0(\Omega)$, and recalling (3.1), we have for all $g\in \overline{\mathcal{G}}$

We claim that the supremum in (3.3) is attained at the unique function

Indeed, by normalization of $u_g$ we have

For uniqueness, first consider a function $u = \tau u_g$ with $\tau\neq\tau_0(u_g)$. By unique maximization in $\tau$ we have

Besides, for all $v\in W^{s, p}_0(\Omega)_+\setminus\{0\}$ which is not a $\lambda(g)$-eigenfunction, arguing as above with $v$ replacing $u$, and recalling that the infimum in (3.1) is attained only at principal eigenfunctions, we have

So, $\tilde u_g$ is the unique maximizer of (3.3).

We now prove that $\Phi$ is convex. Let $g_1, g_2\in \overline{\mathcal{G}}$, $\tau\in(0, 1)$ and set

so $g_\tau\in \overline{\mathcal{G}}$ (a convex set, as seen in Subsection 2.1). For all $u\in W^{s, p}_0(\Omega)_+\setminus\{0\}$, we have by (3.3)

Taking the supremum over $u$ and using (3.3) again,

To prove that $\Phi$ is strictly convex, we argue by contradiction, assuming that for some $g_1\neq g_2$ as above and $\tau\in(0, 1)$

Set $\tilde u_i = \tilde u_{g_i}$ ($i = 1, 2$) and $\tilde u_\tau = \tilde u_{g_\tau}$ for brevity. Then, by (3.3) and the equality above

Recalling that $\tilde u_i$ is the only maximizer of $F(g_i, \cdot)$, the last inequality implies $\tilde u_1 = \tilde u_2 = \tilde u_\tau$, as well as

Therefore we have $\lambda(g_1) = \lambda(g_2) = \lambda$. Moreover, $\tilde u_\tau > 0$ is a $\lambda$-eigenfunction with both weights $g_1$, $g_2$, i.e., for all $\varphi\in W^{s, p}_0(\Omega)$

So $g_1\tilde u_\tau^{p-1} = g_2\tilde u_\tau^{p-1}$ in $\Omega$, which in turn, since $\tilde u_\tau > 0$, implies $g_1 = g_2$ a.e. in $\Omega$, a contradiction. □

The next lemma establishes differentiability of $\Phi$.

Lemma 3.3. The functional $\Phi$ is Gâteaux differentiable in $\overline{\mathcal{G}}$, and for all $g, h\in \overline{\mathcal{G}}$

where $\tilde u_g$ is the principal eigenfunction normalized as in (3.4).

Proof. First, let $(g_n)$ be a sequence in $\overline{\mathcal{G}}$ s.t. $g_n \overset{*}{\rightharpoonup} g$, and set for brevity $\tilde u_n = \tilde u_{g_n}$, $\tilde u = \tilde u_g$. We claim that

Indeed, by normalization we have for all $n\in {\mathbb N}$

and the latter is bounded from above, since $\Phi(g)$ has a maximizer in $\overline{\mathcal{G}}$. So, $(\tilde u_n)$ is bounded in $W^{s, p}_0(\Omega)$. By uniform convexity and the compact embedding $W^{s, p}_0(\Omega)\hookrightarrow L^p(\Omega)$, passing to a subsequence we have $\tilde u_n\rightharpoonup v$ in $W^{s, p}_0(\Omega)$, $\tilde u_n\to v$ in $L^p(\Omega)$, and $\tilde u_n(x)\to v(x)$ for a.e. $x\in\Omega$, as $n\to\infty$ (in particular $v \geqslant 0$ in $\Omega$). By convexity, we see that

By Lemma 2.4, we have

By Lemma 3.1, we have $\Phi(g_n)\to\Phi(g)$, so by (3.3) we get

Therefore $v$ is a maximizer of $F(g, \cdot)$ over $W^{s, p}_0(\Omega)_+$, hence by uniqueness $v = \tilde u$. Then we have $\tilde u_n\to\tilde u$ in $L^p(\Omega)$, which is equivalent to (3.5).

We claim that for all $n\in {\mathbb N}$

Indeed, by (3.3) we have

Now fix $g, h\in \overline{\mathcal{G}}$, $g\neq h$, and a sequence $(\tau_n)$ in $(0, 1)$ s.t. $\tau_n\to 0$. By convexity of $\overline{\mathcal{G}}$, we have for all $n\in {\mathbb N}$

Also, clearly $g_n \overset{*}{\rightharpoonup} g$. By (3.6), setting as usual $\tilde u_n = \tilde u_{g_n}$ and $\tilde u = \tilde u_g$, we have for all $n\in {\mathbb N}$

Dividing by $\tau_n > 0$ and recalling (3.5), we get

Note that $2\tilde u^p\in L^1(\Omega)\subset L^\infty(\Omega)^*$, and by the arbitrariness of the sequence $(\tau_n)$ we deduce that $\Phi$ is Gâteaux differentiable at $g$ with

which concludes the proof. □

We can now prove the main result of this section.

Proof of Theorem 1.1. We already know that $\Phi$ has a maximizer $\bar g$ over $\overline{\mathcal{G}}$. Set $\bar w = 2\tilde u_{\bar g}^p\in L^1(\Omega)$, then by Lemma 3.3 we have $\Phi'(\bar g) = \bar w$. Now we maximize on $\overline{\mathcal{G}}$ the linear functional

By Lemma 2.1 $(i)$, there exists $\hat g\in \mathcal{G}$ s.t. for all $g\in \overline{\mathcal{G}}$

In particular we have

By Lemma 3.2, the functional $\Phi$ is convex. Therefore, using also Lemma 3.3 and (3.7), we have

Thus, $\hat g\in \mathcal{G}$ is as well a maximizer of $\Phi$ over $\overline{\mathcal{G}}$, which proves $(i)$ since maximizers of $\Phi$ and minimizers of $\lambda(g)$ coincide. In addition, by the relation above we have

We will now prove that $\hat g = \bar g$, arguing by contradiction. Assume $\hat g\neq\bar g$, then by the strict convexity of $\Phi$ (Lemma 3.2 again) we have

against the maximality of $\bar g$. So, any maximizer of $\Phi$ over $\overline{\mathcal{G}}$ actually lies in $\mathcal{G}$, which proves $(ii)$. Finally, let $\hat g\in \mathcal{G}$ be any maximizer of $\Phi$ and set $\hat w = 2\tilde u_{\hat g}^p\in L^1(\Omega)$. By Lemmas 3.2 and 3.3, for all $g\in \overline{\mathcal{G}}\setminus\{\hat g\}$ we have

hence

Equivalently, $\hat g$ is the only maximizer over $\overline{\mathcal{G}}$ of the linear functional above, induced by the function $\hat w$. By Lemma 2.1 $(ii)$, there exists a nondecreasing map $\tilde\eta: {\mathbb R}\to {\mathbb R}$ s.t. in $\Omega$

Now we recall (3.4) and the definition of $\hat w$, and by setting for all $t \geqslant 0$

while $\eta(t) = \eta(0)$ for all $t < 0$, we immediately see that $\eta: {\mathbb R}\to {\mathbb R}$ is a nondecreasing map s.t. $\hat g = \eta\circ u_{\hat g}$ in $\Omega$, thus proving $(iii)$. □

4.   Optimization of the energy functional

In this section we consider problem (1.2) and prove Theorem 1.2. Let $\Omega$, $p$, $s$, $K$, $g_0$ be as in Section 1, and $h$ satisfy $(h_1)$, $(h_2)$. For any $g\in \overline{\mathcal{G}}$, we say that $u\in W^{s, p}_0(\Omega)$ is a weak solution of (1.2) if for all $\varphi\in W^{s, p}_0(\Omega)$

By classical results (see for instance ^[18] for the fractional $p$-Laplacian), for all $g\in \overline{\mathcal{G}}$ problem (1.2) has a unique solution $u_g\in W^{s, p}_0(\Omega)$. In addition, by Lemma 2.2 we have $u_g\in L^\infty(\Omega)$. Such solution is the unique minimizer in $W^{s, p}_0(\Omega)$ of the energy functional associated to (1.2). The corresponding energy, depending on $g\in \overline{\mathcal{G}}$, is given by

where for all $(x, t)\in\Omega\times {\mathbb R}$ we have set

We are interested in the minimizers of $\Psi(g)$ over $\mathcal{G}$. Equivalently, we may set for all $g\in \overline{\mathcal{G}}$, $u\in W^{s, p}_0(\Omega)$

and maximize $E(g, \cdot)$ with respect to $u$, thus defining

So, as in Section 3, we are led to the maximization problem

First, we prove the continuity of $\Phi$.

Lemma 4.1. The functional $\Phi$ is sequentially weakly* continuous in $\overline{\mathcal{G}}$.

Proof. Let $(g_n)$ be a sequence in $\overline{\mathcal{G}}$ s.t. $g_n \overset{*}{\rightharpoonup} g$, and denote $u_n = u_{g_n}$, $u = u_g$. By (4.1), for all $n\in {\mathbb N}$ we have

Passing to the limit as $n\to\infty$ and using weak^* convergence, we get

From (1.2) with datum $g_n$ and solution $u_n$, multiplying by $u_n$ again, we get for all $n\in {\mathbb N}$

Since $\|g_n\|_\infty \leqslant M$ and by the continuous embedding $W^{s, p}_0(\Omega)\hookrightarrow L^1(\Omega)$, we have

with $C > 0$ independent of $n$. Also, by $(h_2)$ and the continuous embedding $W^{s, p}_0(\Omega)\hookrightarrow L^q(\Omega)$, we have

So (4.3) implies for all $n\in {\mathbb N}$

Recalling that $q < p$, we deduce that $(u_n)$ is bounded in $W^{s, p}_0(\Omega)$. Passing to a subsequence, we have $u_n\rightharpoonup v$ in $W^{s, p}_0(\Omega)$, $u_n\to v$ in $L^p(\Omega)$, and $u_n(x)\to v(x)$ for a.e. $x\in\Omega$, as $n\to\infty$. By convexity we have

By Lemma 2.4 (with $r = 1$) we find

Finally, we have

Indeed, applying $(h_2)$, Lagrange's rule, and Hölder's inequality, we get for all $n\in {\mathbb N}$

and the latter tends to $0$ as $n\to\infty$, by the continuous embeddings of $L^p(\Omega)$ into $L^1(\Omega)$, $L^q(\Omega)$, respectively, thus proving (4.5).

Next, we start from (4.1) and we apply (4.4) and (4.5):

and the latter does not exceed $\Phi(g)$, so

Comparing (4.2) and (4.6), we have $\Phi(g_n)\to\Phi(g)$, which concludes the proof. □

By Lemma 4.1, $\Phi$ has both a minimizer and a maximizer over $\overline{\mathcal{G}}$. Next we prove strict convexity:

Lemma 4.2. The functional $\Phi$ is strictly convex in $\overline{\mathcal{G}}$.

Proof. The convexity of $\Phi$ follows as in Lemma 3.2, since $\Phi(g)$ is the supremum of linear functionals (in $g$). To prove strict convexity, we argue by contradiction. Let $g_1, g_2\in \overline{\mathcal{G}}$ be s.t. $g_1\neq g_2$, set for all $\tau\in(0, 1)$

and assume that for some $\tau\in(0, 1)$

As usual, set $u_i = u_{g_i}$ ($i = 1, 2$) and $u_\tau = u_{g_\tau}$. By linearity of $E(g, u_\tau)$ in $g$ and (4.1), the relation above rephrases as

Recalling that $E(g_i, u_\tau) \leqslant E(g_i, u_i)$ ($i = 1, 2$) and the uniqueness of the maximizer in (4.1), we deduce $u_1 = u_2 = u_\tau$. Now test (1.2) with an arbitrary $\varphi\in W^{s, p}_0(\Omega)$:

So we have $g_1 = g_2$ a.e. in $\Omega$, a contradiction. Thus, $\Phi$ is strictly convex. □

The last property we need is differentiability.

Lemma 4.3. The functional $\Phi$ is Gâteaux differentiable in $\overline{\mathcal{G}}$, and for all $g, k\in \overline{\mathcal{G}}$

Proof. First, let $(g_n)$ be a sequence in $\overline{\mathcal{G}}$ s.t. $g_n \overset{*}{\rightharpoonup} g$, and let $u_n = u_{g_n}$, $u = u_g$. From Lemma 4.1 we know that $\Phi(g_n)$ tends to $\Phi(g)$, i.e.,

We further claim that

Indeed, we recall that for all $n\in {\mathbb N}$

Therefore, by (4.7), uniform boundedness of $(g_n)$, $(h_2)$, and the compact embeddings of $W^{s, p}_0(\Omega)$ into $L^1(\Omega)$, $L^q(\Omega)$, respectively, we have for all $n\in {\mathbb N}$

Since $1 < q < p$, the sequence $(u_n)$ is bounded in $W^{s, p}_0(\Omega)$. Passing to a subsequence, we have $u_n\rightharpoonup v$ in $W^{s, p}_0(\Omega)$, $u_n\to v$ in $L^p(\Omega)$, and $u_n(x)\to v(x)$ for a.e. $x\in\Omega$, as $n\to\infty$. Therefore, by convexity

Also, by Lemma 2.4 and continuous embeddings we have

So, recalling (4.7), we get

which implies $u = v$ by uniqueness of the maximizer in (4.1). So $u_n\to u$ in $L^p(\Omega)$, which yields (4.8). In addition, for all $n\in {\mathbb N}$ we have

Indeed, by definition of $\Phi(g)$ we have

Now fix $k\in \overline{\mathcal{G}}\setminus\{g\}$ and a sequence $(\tau_n)$ in $(0, 1)$ s.t. $\tau_n\to 0$ as $n\to\infty$. Set

so that $g_n \overset{*}{\rightharpoonup} g$. By (4.9) with such choice of $g_n$, we have for all $n\in {\mathbb N}$

Passing to the limit for $n\to\infty$, and noting that by (4.8) we have in particular $u_n\to u$ in $L^1(\Omega)$, we get

By arbitrariness of $(\tau_n)$, and noting that $u\in L^1(\Omega)\subset L^\infty(\Omega)^*$, we see that $\Phi$ is Gâteaux differentiable at $g$ with

which concludes the proof. □

We can now prove our optimization result, with a similar argument as in Section 3.

Proof of Theorem 1.2. By Lemma 4.1 and sequential weak^* compactness of $\overline{\mathcal{G}}$, there exists $\bar g\in \overline{\mathcal{G}}$ s.t. for all $g\in \overline{\mathcal{G}}$

Set $\bar u = u_{\bar g}\in W^{s, p}_0(\Omega)$, then by Lemma 4.3 we have for all $k\in \overline{\mathcal{G}}\setminus\{\bar g\}$

Since $\bar u\in L^1(\Omega)$, by Lemma 2.1 $(i)$ there exists $\hat g\in \mathcal{G}$ s.t. for all $g\in \overline{\mathcal{G}}$

in particular

By convexity of $\Phi$ (Lemma 4.2) and (4.10), we have

Therefore, $\hat g\in \mathcal{G}$ is a maximizer of $\Phi$ over $\overline{\mathcal{G}}$, which proves $(i)$. In fact we have $\hat g = \bar g$, otherwise by strict convexity (Lemma 4.2 again) and (4.10) we would have

against maximality of $\bar g$. Thus, any maximizer of $\Phi$ over $\overline{\mathcal{G}}$ actually lies in $\mathcal{G}$, which proves $(ii)$. Finally, let $\hat g\in \mathcal{G}$ be a maximizer of $\Phi$ and set $\hat u = u_{\hat g}\in W^{s, p}_0(\Omega)$. As we have seen before, $\hat g$ is the only maximizer in $\overline{\mathcal{G}}$ for the linear functional

hence by Lemma 2.1 $(ii)$ there exists a nondecreasing map $\eta: {\mathbb R}\to {\mathbb R}$ s.t. $\hat g = \eta\circ\hat u$ in $\Omega$, thus proving $(iii)$. □

Remark 4.4. Theorem 1.2 is analogous to Theorem 1.1 above, while in fact the problem is easier since we do not need to consider normalization to ensure uniqueness, unlike in problem (1.1). On the other hand, in this case we have no information on the sign of the solution $u_g$.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The first author is a member of GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica 'Francesco Severi'), and is partially supported by the research project Problemi non locali di tipo stazionario ed evolutivo (GNAMPA, CUP E53C23001670001) and the research project Studio di modelli nelle scienze della vita (UniSS DM 737/2021 risorse 2022-2023). We are grateful to the anonymous Referee for their careful reading of our manuscript and useful suggestions.

Conflict of interest

The authors declare no conflicts of interest.