On a Differential Inclusion Involving Dirichlet–Laplace Operators of Fractional Orders

Abstract

In this paper, we investigate a nonlinear differential inclusion with Dirichlet boundary conditions containing a weak Laplace operator of fractional orders (defined via the spectral decomposition of the Laplace operator $-{\varDelta }$ under Dirichlet boundary conditions). Using variational methods, we characterize solutions of such a problem. Our approach is based on tools from convex analysis (properties of a Legendre–Fenchel transform).

1 Introduction

In recent years, a fractional Laplacian (more precisely, the fractional powers of the Laplacian) attracted attention of many scientists due to its applications in various areas. It appears including in probability (cf. [1, 6, 7, 13]), economics and finance (cf. [1, 14]), mechanics (cf. [5, 7]), optimal control theory (cf. [9, 20]), fluid mechanics and hydrodynamics (cf. [8, 10,11,12, 28,29,30]). There exist many definitions of such operators (e.g., Fourier transform [15, 17], hypersingular integral [15], Riesz potential operator [24], Bochner’s definition [27], spectral decomposition (cf. [4, 14, 18]).

In this paper, we are concerned with the study of solutions to the following differential inclusion

$$\begin{aligned} \sum \limits _{i=0}^k\alpha _i[(-{\varDelta })_\omega ]^{\beta _i}u(x)\in \partial _uF(x,u(x)),\quad x\in {\varOmega }\,\,\hbox {a.e.}, \end{aligned}$$

(1)

where $\alpha _i>0$, $i=0,\dots ,k$ ($k\in {\mathbb {N}}\cup \{0\}$), $0\le \beta _0<\beta _1<\dots <\beta _k$, ${\varOmega }\subset {\mathbb {R}}^N$ is an open and bounded set, $F:{\varOmega }\times {\mathbb {R}}\rightarrow [0,\infty )$, $\partial _uF$ denotes a subgradient of F with respect to u, $[(-{\varDelta }_\omega )]^{\gamma }$ denotes a weak fractional Laplace operator of order $\gamma >0$ with zero Dirichlet boundary values on $\partial {\varOmega }$ (the term “weak” is explained in Sect. 2). This work is based on the spectral definition of the mentioned operator (we shall call it a weak fractional Dirichlet–Laplace operator). More precisely, the definition of the Dirichlet Laplacian comes from the functional calculus for unbounded self-adjoint operators in a Hilbert space [16, 21] and is based on the spectral integral representation theorem for such operators [23, 25].

Let us note that if $F(t,\cdot )$ is differentiable on ${\mathbb {R}}$, then (1) reduces to the following boundary value problem

$$\begin{aligned} \sum \limits _{i=0}^k\alpha _i[(-{\varDelta })_\omega ]^{\beta _i}u(x)= D_uF(x,u(x)),\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

(2)

Motivated by the paper [3], we characterize solutions of the problem (1). They are minimizers of a some integral functional J related to (1) due to the Legendre–Fenchel transform of the function F and the Fenchel–Young inequality (cf. Theorem 3). Using a standard method of the calculus of variations, we prove the existence and uniqueness of minimizers of J on an appropriate space of functions, being the space of solutions of (1) (cf. Theorem 4). Finally, obtained results are illustrated by some examples. In particular, we show that if F is adequately chosen, then the minimizer of J is a solution of (2). To the best of our knowledge, the problem of a type (1) was not investigated by other authors.

2 Preliminaries

In this section, we give some preliminary notions and results that will be used in the remaining part of this paper.

2.1 Weak Fractional Dirichlet–Laplace Operator

In a whole paper we assume that ${\varOmega }\subset {\mathbb {R}}^N$ is an open and bounded set.

Let $T_0=-{\varDelta }:C^\infty _c({\varOmega },{\mathbb {R}})\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})$ be the classical Dirichlet–Laplace operator.

Definition 1

(cf. [21]) We say that $u:{\varOmega }\rightarrow {\mathbb {R}}$ has a weak (minus) Dirichlet–Laplacian if $u\in H^1_0({\varOmega },{\mathbb {R}})$ and there exists a function $g\in L^2({\varOmega },{\mathbb {R}})$ such that

$$\begin{aligned} \int \limits _{{\varOmega }}\nabla u(x)\nabla v(x)\hbox {d}x=\int \limits _{{\varOmega }}g(x)v(x)\hbox {d}x \end{aligned}$$

for any $v\in H^1_0({\varOmega },{\mathbb {R}})$. The function g will be called the weak Dirichlet–Laplacian and denoted by $(-{\varDelta })_\omega u$.

The operator $(-{\varDelta })_\omega$ is called the weak Dirichlet–Laplace operator and the set

$$\begin{aligned} \begin{aligned}&D((-{\varDelta })_\omega ) \\&\quad :=\left\{ u\in H^1_0({\varOmega },{\mathbb {R}});\quad \exists _{g\in L^2},\,\,\forall _{v\in H^1_{0}}\quad \int \limits _{{\varOmega }}\nabla u(x)\nabla v(x)\hbox {d}x=\int \limits _{{\varOmega }}g(x)v(x)\hbox {d}x\right\} \end{aligned} \end{aligned}$$

is named the domain of the operator $(-{\varDelta })_\omega$.

Using the Friedrich’s extension procedure, we obtain (cf. [21, Theorem 3.1], [23])

Theorem 1

The operator

$$\begin{aligned} (-{\varDelta })_\omega : D((-{\varDelta })_\omega )\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}}) \end{aligned}$$

is the self-adjoint extension of the operator $T_0$ and

$$\begin{aligned} T_0 \subset -{\varDelta }\subset (-{\varDelta })_\omega , \end{aligned}$$

where $-{\varDelta }:H^1_0({\varOmega },{\mathbb {R}})\cap H^2({\varOmega },{\mathbb {R}})\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})$ is the (strong) Dirichlet–Laplace operator.

Remark 1

In [2, Section 8.2] $(-{\varDelta })_\omega$ is called the Laplace–Dirichlet operator (without “weak”) and denoted by $-{\varDelta }$.

It is known (cf. [21]) that the spectrum $\sigma ((-{\varDelta })_\omega )$ of $(-{\varDelta })_\omega$ contains only the eigenvalues of $(-{\varDelta })_\omega$ that can be written in a non-decreasing sequence, repeating each eigenvalue according to its multiplicity $0<\lambda _1\le \lambda _2\le \dots \le \lambda _j\rightarrow \infty$. Moreover, a system $\{e_j\}$ of eigenfunctions of the operator $(-{\varDelta })_\omega$, corresponding to $\lambda _j$, is a hilbertian basis in $L^2({\varOmega },{\mathbb {R}})$.

Remark 2

In paper [21], the author proved that if ${\varOmega }$ is an open and bounded set of class $C^{1,1}$ or a convex polygon in ${\mathbb {R}}^2$, then the weak and strong Dirichlet–Laplace operators coincide.

Now, let $\beta >0$. By the weak fractional Dirichlet–Laplace operator of order $\beta$ we mean the operator $[(-{\varDelta })_{\omega }]^\beta :D([(-{\varDelta })_{\omega }]^\beta )\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})$ defined in the following way (cf. [21, Section 3]):

$$\begin{aligned} \left( [(-{\varDelta })_{\omega }]^\beta u\right) (t)=\left( \left( \int \limits _{\sigma ((-{\varDelta })_\omega )}\lambda ^\beta E(\hbox {d}\lambda )\right) u\right) (t)=\sum \limits _{j=1}^\infty (\lambda _j)^\beta a_j e_j(t), \end{aligned}$$

where

$$\begin{aligned} D\left( [(-{\varDelta })_{\omega }]^\beta \right)&=\left\{ u\in L^2({\varOmega },{\mathbb {R}});\quad \int \limits _{\sigma ((-{\varDelta })_\omega )}|\lambda ^\beta |^2\Vert E(\hbox {d}\lambda )u\Vert ^2=\sum \limits _{j=1}^\infty ((\lambda _j)^\beta )^2 a^2_j<\infty ,\right. \\&\quad \quad \hbox {where}\,\, a_j\text {-s}\,\, \hbox {is}\,\, \hbox {such}\,\, \hbox {that}\,\, u(t)=\left( \Bigg (\int \limits _{\sigma ((-{\varDelta })_\omega )}1 E(\hbox {d}\lambda )\Bigg )u\right) (t) \\&\left. =\sum \limits _{j=1}^\infty a_j e_j(t)\right\} \end{aligned}$$

(here E is the spectral measure given by $(-{\varDelta })_\omega$ and the convergence of the series is meant in $L^2({\varOmega },{\mathbb {R}})$).

It is well known that the operator $[(-{\varDelta })_{\omega }]^\beta$ is self-adjoint, its spectrum $\sigma ([(-{\varDelta })_{\omega }]^\beta )$ contains only proper values $(\lambda _j)^\beta$, $j\in {\mathbb {N}}$. Moreover, eigenspaces, corresponding to $(\lambda _j)^\beta$-s and eigenspaces for $[(-{\varDelta })_{\omega }]$, corresponding to $\lambda _j$-s are the same.

In the space $D([(-{\varDelta })_{\omega }]^\beta )$, we define two scalar products:

$$\begin{aligned} \langle u,v\rangle _{\beta }&:=\langle u,v\rangle _{L^2}+\left\langle [(-{\varDelta })_\omega ]^\beta u,[(-{\varDelta })_\omega ]^\beta v\right\rangle _{L^2}, \\ \langle u,v\rangle _{\sim \beta }&:=\left\langle [(-{\varDelta })_\omega ]^\beta u,[(-{\varDelta })_\omega ]^\beta v\right\rangle _{L^2} \end{aligned}$$

which generate norms:

$$\begin{aligned} \Vert u\Vert _{\beta }&=\left( \Vert u\Vert ^2_{L^2}+\Vert [(-{\varDelta })_\omega ]^\beta u\Vert ^2_{L^2}\right) ^{\frac{1}{2}}, \end{aligned}$$

(3)

$$\begin{aligned} \Vert u\Vert _{\sim \beta }&=\Vert [(-{\varDelta })_\omega ]^\beta u\Vert _{L^2}, \end{aligned}$$

(4)

respectively. Norms (3) and (4) are equivalent due to the following Poincaré inequality in $D([(-{\varDelta })_{\omega }]^\beta )$ (cf. [21, inequality (3.2)]):

$$\begin{aligned} \Vert u\Vert ^2_{L^2}\le M_\beta \Vert u\Vert ^2_{\sim \beta }, \end{aligned}$$

(5)

where

$$\begin{aligned} M_\beta ={\left\{ \begin{array}{ll} 1&{}\hbox {if}\quad \lambda _1\ge 1\\ \tfrac{1}{((\lambda _1)^\beta )^2}&{}\hbox {if}\quad \lambda _1<1 \end{array}\right. } \end{aligned}$$

(here $\lambda _1>0$ is the first (the smallest) eigenvalue of the operator $(-{\varDelta })_\omega$).

Now, let us consider the function

$$\begin{aligned} w:{\mathbb {R}}\ni \lambda \rightarrow {\left\{ \begin{array}{ll} 0&{}\lambda <0\\ \alpha _k\lambda ^{\beta _k}+\dots +\alpha _1\lambda ^{\beta _1}+ \alpha _0\lambda ^{\beta _0}&{}\lambda \ge 0, \end{array}\right. } \end{aligned}$$

(6)

where $0\le \beta _0<\beta _1<\dots <\beta _k$, $k\in {\mathbb {N}}\cup \{0\}$, $\alpha _i>0$, $i=0,\dots ,k$. Then, we can define the operator $w((-{\varDelta })_{\omega }):D(w((-{\varDelta })_{\omega }))\subset L^2({\varOmega },{\mathbb {R}})\rightarrow L^2({\varOmega },{\mathbb {R}})$ as follows:

$$\begin{aligned} w((-{\varDelta })_{\omega })u:=\sum \limits _{i=0}^k\alpha _i((-{\varDelta })_\omega )^{\beta _i}u. \end{aligned}$$

The operator $w((-{\varDelta })_{\omega })$ is self-adjoint and $D(w((-{\varDelta })_{\omega }))=D(((-{\varDelta })_{\omega })^{\beta _k})$ (cf. [21]). Moreover, in $D([(-{\varDelta })_{\omega }]^{\beta _k})$ one can introduce a new scalar product of the form

$$\begin{aligned} \langle u,v\rangle _{w((-{\varDelta })_{\omega })}:=\langle w((-{\varDelta })_{\omega }) u,w((-{\varDelta })_{\omega }) v\rangle _{L^2} \end{aligned}$$

(7)

which generates a norm

$$\begin{aligned} \Vert u\Vert _{w((-{\varDelta })_{\omega })}=\Vert w((-{\varDelta })_{\omega }) u\Vert _{L^2}. \end{aligned}$$

(8)

Norms (4) and (8) (the norm (8) will be used in the main part of this article) are equivalent in $D([(-{\varDelta })_{\omega }]^{\beta _k})$ (cf. [21, Lemma 3.6]) and

$$\begin{aligned} \Vert u\Vert ^2_{\sim \beta _k}\le \frac{1}{\alpha _k^2}\Vert u\Vert ^2_{w((-{\varDelta })_{\omega })} \le \frac{C_1}{\alpha _k^2}\left( 1+kC_2^2\right) \Vert u\Vert ^2_{\sim \beta _k}, \end{aligned}$$

(9)

where $C_1, C_2>0$ are constants from [21, Lemma 3.6]. Hence and from (5) we obtain the following Poincaré inequality in $D([(-{\varDelta })_{\omega }]^{\beta _k})$:

$$\begin{aligned} \Vert u\Vert ^2_{L^2}\le \frac{1}{ M_\beta \alpha _k^2}\Vert u\Vert ^2_{w((-{\varDelta })_{\omega })}. \end{aligned}$$

(10)

Moreover, the space $D(w((-{\varDelta })_{\omega }))=D(((-{\varDelta })_{\omega })^{\beta _k})$ with the scalar product (7) is complete (cf. [21, Lemma 3.6]).

2.2 Basic Facts from Convex Analysis

In this part we recall some basic definitions and facts concerning the convex analysis. More details can be found in [19].

Let $f : {\mathbb {R}}^n\rightarrow {\mathbb {R}}$. We shall say that a vector $y\in {\mathbb {R}}^n$ is a subgradient of f at $x\in {\mathbb {R}}^n$ if

$$\begin{aligned} f(z)-f(x)\geqslant \langle y,z-x\rangle _{{\mathbb {R}}^n},\quad \hbox {for}\,\, \hbox {every}\,\, z\in {\mathbb {R}}^n. \end{aligned}$$

The set of all subgradients of f at x is called a subdifferential and is denoted by $\partial f(x)$. If $\partial f(x)\ne \emptyset$ then the function f is called subdifferentiable at x. If f is a convex function, then the subdifferential is a nonempty, convex and compact set. Moreover, if f is Gateaux differentiable at x, then $\partial f(x)=\{\nabla f(x)\}$.

The function $f^*:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\cup \{\infty \}$ defined by

$$\begin{aligned} f^*(y):=\underset{x\in {\mathbb {R}}^n}{\sup }\{\langle y,x\rangle _{{\mathbb {R}}^n}-f(x)\} \end{aligned}$$

is called the Legendre–Fenchel transform of the function f.

In the main part of this paper we use the following facts:

$$\begin{aligned} f(x)\le g(x), \,\,x\in {\mathbb {R}}^n\quad \Longrightarrow \quad f^*(y)\ge g^*(y),\,\,y\in {\mathbb {R}}^n. \end{aligned}$$

(11)

Theorem 2

(Fenchel–Young inequality) Let $f : {\mathbb {R}}^n\rightarrow {\mathbb {R}}$. Any points $x,y\in {\mathbb {R}}^n$ satisfy the inequality

$$\begin{aligned} f(x)+f^*(y)\geqslant \langle y,x\rangle . \end{aligned}$$

Equality holds if and only if $y\in \partial f(x)$.

In conclusion of this section we give an another useful fact (cf. [22, Part II, Lemma 4.3.1]).

Lemma 1

Let ${\varOmega }\subset {\mathbb {R}}^d$ be open, $f:{\varOmega }\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}$, with $f(\cdot ,v)$ measurable for all $v\in {\mathbb {R}}^d$, $f(x,\cdot )$ continuous for almost all $x\in {\varOmega }$, and

$$\begin{aligned} f(x,v)\ge -a(x)+b|v|^p,\quad x\in {\varOmega }\,\,\mathrm{a.e.},\,\,v\in {\mathbb {R}}^d, \end{aligned}$$

where $a\in L^1({\varOmega },{\mathbb {R}})$, $b\in {\mathbb {R}}$, $p\ge 1$. Then, the functional ${\varPhi }:L^p({\varOmega },{\mathbb {R}}^d)\rightarrow {\mathbb {R}}\cup \{\infty \}$,

$$\begin{aligned} {\varPhi }(v):=\int \limits _{\varOmega }f(x,v(x))\mathrm{d}x \end{aligned}$$

is a sequentially lower semicontinuous on $L^p({\varOmega },{\mathbb {R}}^d)$.

Remark 3

The above lemma can be also proved in case of the function f defined on ${\varOmega }\times {\mathbb {R}}^n$, where $n\ne d$.

3 Differential Inclusion with Fractional Dirichlet–Laplace Operators

In this section we prove the main result of this paper, namely a theorem on the existence of minimizers of a some functional associated with the problem (1).

By a solution of inclusion (1) we mean a function $u\in D(w((-{\varDelta })_\omega ))$ satisfying (1) a.e. on ${\varOmega }$.

Let us assume that F satisfies the following conditions:

(F1)::

$F(\cdot ,{u})$ is measurable on ${\varOmega }$ for all $u\in {\mathbb {R}}$ and $F(x,\cdot )$ is convex on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$.

(F2)::

$F(x,\cdot )$ is coercive on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$, i.e.

$$\begin{aligned} \underset{|u|\rightarrow \infty }{\lim }\tfrac{F(x,u)}{|u|}=\infty ,\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

(12)

(F3)::

there exist a function $v\in L^1({\varOmega },{\mathbb {R}})$ and a constant $c>0$ such that

$$\begin{aligned} F(x,u)\le v(x)+c|u|^2,\quad x\in {\varOmega }\,\,\hbox {a.e.}, u\in {\mathbb {R}}. \end{aligned}$$

(13)

Remark 4

Of course, since $F(x,\cdot )$ is convex, therefore it is continuous on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$ (cf. [26, Corollary 10.1.1]).

Let us consider the function $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} f(u):=v+c|u|^2, \end{aligned}$$

(14)

where $c>0$ and $v\in {\mathbb {R}}$ are fixed. Then, it easy to check that

$$\begin{aligned} f^*(z)= -v+\frac{1}{4c}|z|^2,\quad z\in {\mathbb {R}}. \end{aligned}$$

(15)

In the next result we shall use the following

Lemma 2

Let $G:{\varOmega }\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ be a Carathéodory function and the function $H:{\varOmega }\rightarrow {\mathbb {R}}\cup \{\infty \}$ is defined by

$$\begin{aligned} H(x):=\underset{w\in {\mathbb {R}}}{\sup }\{G(x,w)\},\quad x\in {\varOmega }\,\,\mathrm{a.e.} \end{aligned}$$

Then H is measurable on ${\varOmega }$.

Proof

From the fact that $G(\cdot ,w)$ is continuous on ${\mathbb {R}}$ it follows that

$$\begin{aligned} H(x)=\underset{w\in {\mathbb {Q}}}{\sup }\{G(x,w)\},\quad x\in {\varOmega }\,\,\hbox {a.e.}, \end{aligned}$$

where ${\mathbb {Q}}$ denotes the set of rational numbers. Since the supremum of the countable set of measurable functions is a measurable set, therefore H is measurable on ${\varOmega }$.$\square$ $\square$

Now, we formulate and prove the following result.

Proposition 1

Let us assume that conditions (F1)–(F3) are satisfied. Then, the Legendre–Fenchel transform of the function F with respect to u $F^*:{\varOmega }\times {\mathbb {R}}\rightarrow (-\infty ,\infty ]$ given by

$$\begin{aligned} F^*(x,z):=\underset{u\in {\mathbb {R}}}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-F(x,u)\right] ,\quad x\in {\varOmega }\,\,\mathrm{a.e.},\,\,z\in {\mathbb {R}} \end{aligned}$$

has the following properties:

• (a) $F^*(x,z)$ is finite for a.e. $x\in {\varOmega }$ and all $z\in {\mathbb {R}}$,

• (b) $F^*(\cdot ,z)$ is measurable on ${\varOmega }$ for all $z\in {\mathbb {R}}$ and $F^*(x,\cdot )$ is continuous on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$,

• (c) for a.e. $x\in {\varOmega }$ and all $z\in {\mathbb {R}}$

  $$\begin{aligned} F^*(x,z)\ge -v(x)+\frac{1}{4c}|z|^2, \end{aligned}$$

  (16)

  where the constant c and the function v are from assumption (F3).

Proof

proof of the property (a): Let us fix any $z\in {\mathbb {R}}$ and $x\in S\subset {\varOmega }$, where $\mu ({\varOmega }{\setminus } S)=0$ ($\mu$ denotes the Lebesque measure on ${\mathbb {R}}^N$). From (F2) it follows that for any $R>0$ there exists $r>0$ such that for $|u|>r$ we have $F(x,u)\ge R|u|$. In particular, putting $R=|z|$, we get

$$\begin{aligned} F(x,u)\ge |z||u|,\quad |u|>r. \end{aligned}$$

Consequently,

$$\begin{aligned} \langle u,z\rangle _{{\mathbb {R}}}-F(x,u)\le |u||z|-R|u|=0,\quad |u|>r. \end{aligned}$$

Hence and from continuity of the function $F(\cdot ,u)$ we obtain

$$\begin{aligned} F^*(x,z)&=\underset{u\in {\mathbb {R}}}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-F(x,u)\right] \\&=\max \left\{ \underset{|u|\le r}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-F(x,u)\right] ,\underset{|u|>r}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-F(x,u)\right] \right\} \\&\le \max \{\hbox {const},0\}<\infty . \end{aligned}$$

This means that $F^*$ is finite for a.e. $x\in {\varOmega }$ and all $z\in {\mathbb {R}}$.

proof of the property (b): Measurability of $F^*(\cdot ,z)$ on ${\varOmega }$ for any fixed $z\in {\mathbb {R}}$ follows from Lemma 2. From the definition of the Legendre–Fenchel transform it follows that $F^*(x,\cdot )$ is convex on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$. Thus and from the proved condition (a) of this proposition we obtain continuity of $F^*$ with respect to z for a.e. $x\in {\varOmega }$ (cf. [26, Corollary 10.1.1]).

proof of the property (c): The condition (16) follows from assumption (F3), property (11) and equality (15). $\square$

Now, let us consider the functional $J:D(w((-{\varDelta })_\omega ))\longrightarrow {\mathbb {R}}\cup \{\infty \}$ given by:

$$\begin{aligned} J(u):=\int \limits _{\varOmega }\big (F(x,u(x))+F^*(x,w((-{\varDelta })_\omega )u(x))-\langle u(x),w((-{\varDelta })_\omega )u(x)\rangle _{\mathbb {R}}\big )\hbox {d}x. \end{aligned}$$

(17)

Let us note that since all terms in the above integral are measurable on ${\varOmega }$ (the first and third terms are even summable on ${\varOmega }$), therefore J is well defined.

In the proof of the main result we use the following fact.

Lemma 3

If assumptions (F1)–(F3) are satisfied, then the functional

$I:D(w((-{\varDelta })_{\omega }))\rightarrow {\mathbb {R}}\cup \{\infty \}$ given by

$$\begin{aligned} I(u)=\int \limits _{{\varOmega }}F^*(x,w((-{\varDelta })_{\omega })u(x)))\mathrm{d}x \end{aligned}$$

(18)

is sequentially weakly lower semicontinuous on $D(w((-{\varDelta })_{\omega }))$.

Proof

Let us consider the functional $I_1:L^2({\varOmega },{\mathbb {R}})\rightarrow {\mathbb {R}}\cup \{\infty \}$ given by

$$\begin{aligned} I_1(z)=\int \limits _{{\varOmega }}F^*(x,z(x))\hbox {d}x. \end{aligned}$$

From Proposition 1, Lemma 1 and Remark 3 it follows that $I_1$ is a sequentially lower semicontinuous functional on $L^2({\varOmega },{\mathbb {R}})$.

Let $(u_n)_{n\in {\mathbb {N}}}\subset D(w((-{\varDelta })_{\omega }))$ be a sequence convergent to a some element u in the space $D(w((-{\varDelta })_{\omega }))$. Then, the sequence $(w((-{\varDelta })\omega )u_n)_{n\in {\mathbb {N}}}\subset L^2({\varOmega },{\mathbb {R}})$ is convergent to $w((-{\varDelta })\omega )u$ in $L^2({\varOmega },{\mathbb {R}})$. From the proved part of this Lemma it follows that the functional I is sequentially lower semicontinuous on $D(w((-{\varDelta })_{\omega }))$. Since I is convex on $D(w((-{\varDelta })_{\omega }))$ ($F^*(x,\cdot )$ is convex on ${\mathbb {R}}$), therefore it is sequential weak lower semicontinuous on $D(w((-{\varDelta })_{\omega }))$.

The proof is completed. $\square$

The next theorem plays a key role in the study of the existence of solutions to problem (1).

Theorem 3

Assume that conditions (F1)–(F3) are satisfied. A function $u_*\in D(w((-{\varDelta })_{\omega }))$ is a solution of the inclusion (1) if and only if $u_*$ is a minimizer of J on $D(w((-{\varDelta })_{\omega }))$ and $J(u_*)=0$.

Proof

Let $u_*\in D(w((-{\varDelta })_{\omega }))$ be a solution of the problem (1). From Theorem 2 it follows that

$$\begin{aligned}&F(x,u_*(x))+F^*(x,w((-{\varDelta })_{\omega })u_*(x))-\langle u_*(x),w((-{\varDelta })_\omega )u_*(x)\rangle _{\mathbb {R}}=0, \\&\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

Consequently, $J(u_*)=0$. On the other hand, for every $u\in D(w((-{\varDelta })_{\omega }))$, using once again Theorem 2, we have

$$\begin{aligned} F(x,u(x))+F^*(x,w((-{\varDelta })_{\omega })u(x))-\langle u(x),w((-{\varDelta })_\omega )u(x)\rangle _{\mathbb {R}}\ge 0,\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

(19)

This means that $J(u)\ge 0$ for every $u\in D(w((-{\varDelta })_{\omega }))$, so $u_*$ is a minimizer of J on $D(w((-{\varDelta })_{\omega }))$.

Now, we assume that $u_*$ is a minimizer of J on $D(w((-{\varDelta })_{\omega }))$ and $J(u_*)=0$. This means in particular that

$$\begin{aligned} 0= & {} J(u_*) \\= & {} \int \limits _{\varOmega }\big (F(x,u_*(x))+F^*(x,w((-{\varDelta })_\omega ) u_*(x))-\langle u_*(x),w((-{\varDelta })_\omega )u_*(x)\rangle _{\mathbb {R}}\big )\hbox {d}x. \end{aligned}$$

On the other hand, from Theorem 2 it follows that

$$\begin{aligned}&F(x,u_*(x))+F^*(x,w((-{\varDelta })_{\omega })u_*(x))-\langle u_*(x),w((-{\varDelta })_\omega )u_*(x)\rangle _{\mathbb {R}}\ge 0, \\&\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

Consequently,

$$\begin{aligned}&F(x,u_*(x))+F^*(x,w((-{\varDelta })_{\omega })u_*(x))-\langle u_*(x),w((-{\varDelta })_\omega )u_*(x)\rangle _{\mathbb {R}}=0, \\&\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

The second part of Theorem 2 guaranties that $u_*\in D(w((-{\varDelta })_{\omega }))$ is a solution of the inclusion (1).

The proof is completed. $\square$

Remark 5

From the above theorem it follows that if $u_*\in D(w((-{\varDelta })_{\omega }))$ is a solution of the problem (1) then it is a minimizer of J. Consequently, only minimizers of J can be solutions of (1) (they are in effect solutions of (1) if additionally $J(u_*)=0$, where $u_*$ denotes a minimizer of J).

Now, we formulate and prove the main result of this paper, namely a theorem on the existence of a minimizer of the functional J on $D(w((-{\varDelta })_{\omega }))$. We have

Theorem 4

Assume that conditions (F1)–(F3) are satisfied and

$$\begin{aligned} c<\frac{\sqrt{M_{\beta _k}}\alpha _k}{4}, \end{aligned}$$

(20)

where $\lambda _1$ is the first eigenvalue of $(-{\varDelta })_\omega$ and c is a constant from the assumption (F3). Then there exists a minimizer $u_*\in D(w((-{\varDelta })_{\omega }))$ of the functional J. Moreover, if $F(x,\cdot )$ is strictly convex on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$, then the minimizer is unique.

Proof

Since the space $D(w((-{\varDelta })_{\omega }))$ is reflexive (as the Hilbert space), therefore it is sufficient to show that the functional J is coercive and sequentially weakly lower semicontinuous on $D(w((-{\varDelta })_{\omega }))$.

Coercivity: Let $(u_l)_{l\in {\mathbb {N}}}\subset D(w((-{\varDelta })_{\omega }))$ be any sequence such that

$$\begin{aligned} \Vert u_l\Vert _{w((-{\varDelta })_{\omega })} \underset{l\rightarrow \infty }{\longrightarrow }\infty . \end{aligned}$$

From the proof of Theorem 3 it follows that $J(u_l)\ge 0$. Moreover, using nonnegativity of F, conditions (16), (20), the Hölder inequality and the Poincaré inequality (10) we obtain

$$\begin{aligned} J(u_l)=&\int \limits _{\varOmega }\big (F(x,u_l(x))+F^*(x,w((-{\varDelta })_\omega ) u_l(x))-\langle u_l(x),w((-{\varDelta })_\omega )u_l(x)\rangle _{\mathbb {R}}\big )\hbox {d}x\\ \ge&\int \limits _{\varOmega }F^*(x,w((-{\varDelta })_\omega )u_l(x))\hbox {d}x-\int \limits _{\varOmega }\langle u_l(x),w((-{\varDelta })_\omega )u_l(x)\rangle _{\mathbb {R}}\hbox {d}x\\ \ge&-\int \limits _{\varOmega }v(x)\hbox {d}x+\frac{1}{4c}\Vert w((-{\varDelta })_\omega )u_l \Vert ^2_{L^2}-\Vert u_l\Vert _{L^2}\Vert w((-{\varDelta })_\omega )u_l\Vert _{L^2}\\ \ge&\,C+\frac{1}{4c}\Vert u_l\Vert ^2_{w((-{\varDelta })_{\omega })}- \frac{1}{\sqrt{M_{\beta _k}}\alpha _k}\Vert u_l\Vert ^2_{w((-{\varDelta })_{\omega })}\\ =&\,C+\left( \frac{1}{4c}-\frac{1}{\sqrt{M_{\beta _k}}\alpha _k}\right) \Vert u_l\Vert ^2_{w((-{\varDelta })_{\omega })} \underset{l\rightarrow \infty }{\longrightarrow }\infty , \end{aligned}$$

where $C=-\int \limits _{\varOmega }v(x)\hbox {d}x$. This means that J is coercive.

Sequential weak lower semicontinuity of J :  Assume that $(u_l)_{l\in {\mathbb {N}}}\subset D(w((-{\varDelta })_{\omega }))$ is a sequence such that

$$\begin{aligned} u_{l}\underset{l\rightarrow \infty }{\rightharpoonup }u_*\quad \hbox {weakly}\,\,\hbox {in}\,\, D(w((-{\varDelta })_{\omega })) \end{aligned}$$

and let

$$\begin{aligned} J(u)=J_1(u)+J_2(u)-J_3(u), \end{aligned}$$

where

$$\begin{aligned} J_1(u)= & {} \int \limits _{\varOmega }F(x,u(x))\hbox {d}x,\quad J_2(u)=\int \limits _{\varOmega }F^*(x,w((-{\varDelta })_\omega )u(x))\hbox {d}x, \\ J_3(u)= & {} \int \limits _{\varOmega }\langle u(x),w((-{\varDelta })_\omega )u(x)\rangle _{\mathbb {R}}\hbox {d}x. \end{aligned}$$

First, let us note that since F is a Carathéodory function, therefore $(F(\cdot ,u_{l}(\cdot )))_{l\in {\mathbb {N}}}$ is the sequence of measurable functions. Moreover all terms of this sequence are nonnegative. Consequently, using Fatou’s Lemma, we conclude

$$\begin{aligned} \int \limits _{{\varOmega }}\underset{l\rightarrow \infty }{\liminf }F(x,u_{l}(x)) \hbox {d}x\le \underset{l\rightarrow \infty }{\liminf }\int \limits _{{\varOmega }}F(x,u_{l}(x))\hbox {d}x. \end{aligned}$$

(21)

[21, Proposition 3.10] implies

$$\begin{aligned} u_{l}\underset{l\rightarrow \infty }{\longrightarrow }u_*\quad \hbox {strongly}\,\,\hbox {in}\,\,L^2({\varOmega },{\mathbb {R}}). \end{aligned}$$

(22)

So, there exists a subsequent $(u_{l_j})_{j\in {\mathbb {N}}}$ such that

$$\begin{aligned} u_{l_j}(x)\underset{j\rightarrow \infty }{\longrightarrow }u_*(x),\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

Hence and from the fact that F is continuous with respect to the second variable we get

$$\begin{aligned} \underset{j\rightarrow \infty }{\lim }F(x,u_{l_j}(x))=F(x,u_{*}(x)),\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

Supposing contrary and repeating the above argumentation we assert that

$$\begin{aligned} \underset{l\rightarrow \infty }{\lim }F(x,u_{l}(x))=F(x,u_{*}(x)),\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

Consequently, the inequality (21) can be written as

$$\begin{aligned} \int \limits _{{\varOmega }}F(x,u_{*}(x))\hbox {d}x\le \underset{l\rightarrow \infty }{\liminf }\int \limits _{{\varOmega }}F(x,u_{l}(x))\hbox {d}x. \end{aligned}$$

This means that $J_1$ is sequentially weakly lower semicontinuous on $D(w((-{\varDelta })_{\omega }))$. The functional $J_2$ has also such a property due to Lemma 3.

Now, we show that $J_3$ is sequentially weakly continuous on $D(w((-{\varDelta })_{\omega }))$. Indeed, using once again [21, Proposition 3.10] we assert that

$$\begin{aligned} w((-{\varDelta })_\omega ) u_{l}\underset{l\rightarrow \infty }{\rightharpoonup }w ((-{\varDelta })_\omega )u_*\quad \hbox {weakly}\,\,\hbox {in}\,\, L^2({\varOmega },{\mathbb {R}}). \end{aligned}$$

This fact and convergence (22) imply

$$\begin{aligned} \underset{l\rightarrow \infty }{\lim }\int \limits _{\varOmega }\langle u_l(x),w((-{\varDelta })_\omega )u_l(x)\rangle _{\mathbb {R}}\hbox {d}x=\int \limits _{\varOmega }\langle u_*(x),w((-{\varDelta })_\omega )u_*(x)\rangle _{\mathbb {R}}\hbox {d}x. \end{aligned}$$

Finally, we conclude that J is sequentially weakly lower semicontinuous on $D(w((-{\varDelta })_{\omega }))$.

Uniqueness of a solution: It is clear that since F and $F^*$ are convex with respect to the second variable, therefore J is convex on $D(w((-{\varDelta })_{\omega }))$. Let us suppose that $u_*$ and $v_*$ are two different minimizers of J on $D(w((-{\varDelta })_{\omega }))$ and assume that $F(x,\cdot )$ is strictly convex on ${\mathbb {R}}$ for a.e. $x\in {\varOmega }$. This implies that $J_1$, so also J, are strictly convex on $D(w((-{\varDelta })_{\omega }))$. This means in particular that

$$\begin{aligned} J\left( \frac{u_*+v_*}{2}\right) <J(u_*)=J(v_*). \end{aligned}$$

This contradicts the assumption that $u_*$ ($v_*$) is a minimizer of J on $D(w((-{\varDelta })_{\omega }))$.

The proof is completed. $\square$

Remark 6

Let us note that only the order $\beta _k$ and the value $\alpha _k$ impact the existence of a minimizer of J. This is a consequence of the fact that domains $D(((-{\varDelta })_\omega )^{\beta _k})$ and $D(w((-{\varDelta })_{\omega }))$ coincide.

4 Illustrative Examples

In this section we present two simple theoretical examples which illustrate results obtained in Sect. 3.

Example 1

Let us consider the following problem:

$$\begin{aligned} ((-{\varDelta })_\omega )^{\frac{1}{3}}u(x)+ 5((-{\varDelta })_\omega )^{\frac{1}{2}}u(x)\in \partial _u F(x,u(x)),\quad x=(x_1,x_2)\in {\varOmega }\,\,\hbox {a.e.}, \end{aligned}$$

(23)

where ${\varOmega }=(0,\pi )\times (0,\pi )\subset {\mathbb {R}}^2$ and $F:{\varOmega }\times {\mathbb {R}}\rightarrow [0,\infty )$ is given by

$$\begin{aligned} F(x,u):={\left\{ \begin{array}{ll} |u|&{}\hbox {if}\,\,|u|\le 1\\ u^2&{}\hbox {if}\,\,|u|>1. \end{array}\right. } \end{aligned}$$

It is easy to check that

$$\begin{aligned} \partial _u F(x,u)={\left\{ \begin{array}{ll} {[}-2,-1]&{}\hbox {if}\,\,u=-\,1\\ \{-1\}&{}\hbox {if}\,\,u\in (-\,1,0)\\ {[}-\,1,1]&{}\hbox {if}\,\,u=0\\ \{1\}&{}\hbox {if}\,\,u\in (0,1)\\ {[}1,2]&{}\hbox {if}\,\,u=1\\ \{2u\}&{}\hbox {if}\,\,|u|>1. \end{array}\right. } \end{aligned}$$

Moreover,

$$\begin{aligned} F^*(x,z)&= \underset{u\in {\mathbb {R}}}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-F(x,u)\right] \\&=\max \left\{ \underset{|u|\le 1}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-|u|\right] ,\underset{|u|>1}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-u^2\right] \right\} . \end{aligned}$$

We check that

$$\begin{aligned} \underset{|u|\le 1}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-|u|\right] = {\left\{ \begin{array}{ll} -z-1&{}\hbox {if}\,\,z< -1\\ z-1&{}\hbox {if}\,\,z>1\\ 0&{}\hbox {if}\,\,|z|\le 1 \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \underset{|u|>1}{\sup }\left[ \langle u,z\rangle _{{\mathbb {R}}}-u^2\right] = {\left\{ \begin{array}{ll} -z-1&{}\hbox {if}\,\,z\in [-2,0]\\ z-1&{}\hbox {if}\,\,z\in [0,2]\\ \frac{1}{4}z^4&{}\hbox {if}\,\,|z|>2. \end{array}\right. } \end{aligned}$$

Consequently,

$$\begin{aligned} F^*(x,z) = {\left\{ \begin{array}{ll} 0&{}\hbox {if}\,\,|z|<1\\ -z-1&{}\hbox {if}\,\,z\in [-2,-1]\\ z-1&{}\hbox {if}\,\,z\in [1,2]\\ \frac{1}{4}z^4&{}\hbox {if}\,\,|z|>2. \end{array}\right. } \end{aligned}$$

Of course assumptions (F1)–(F3) are satisfied. In particular,

$$\begin{aligned} F(x,u)\le 1+|u|^2, \end{aligned}$$

so the condition (13) holds with $c=1$. Moreover, it is well known (cf. [2, Proposition 8.5.3]) that the first eigenvalue of $(-{\varDelta })_\omega$ equals to 2, so the assumption (20) is also satisfied. Consequently, using Theorem 4, we assert that there exists a minimizer (not necessarily unique) of the functional J given by (17) which can be the only possible solution of the problem (23) (cf. Remark 5).

Example 2

Let us consider the following boundary value problem:

$$\begin{aligned} \sum \limits _{i=0}^k\alpha _i((-{\varDelta })_\omega )^{\beta _i}u(x)=D_uF(x,u(x)),\quad x\in {\varOmega }\,\,\hbox {a.e.}, \end{aligned}$$

(24)

where $\alpha _i>0$, $i=0,\dots ,k$ ($k\in {\mathbb {N}}\cup \{0\}$), $0\le \beta _0<\beta _1<\dots <\beta _k$, ${\varOmega }\subset {\mathbb {R}}^N$, is any open and bounded set, $F:{\varOmega }\times {\mathbb {R}}\rightarrow [0,\infty )$ is given by

$$\begin{aligned} F(x,u):=\frac{a(x)}{p}|u|^p, \end{aligned}$$

where $1<p<2$ and $a\in L^\infty ({\varOmega },{\mathbb {R}}_+)$.

We shall show that the above problem has the only trivial solution. Indeed, first let us note that

$$\begin{aligned} F^*(x,z)=\frac{1}{qa(x)}|z|^q,\quad \tfrac{1}{p}+\tfrac{1}{q}=1 \end{aligned}$$

and the partial derivative of F with respect to u is given by

$$\begin{aligned} D_u F(x,u)=a(x)|u|^{p-2}u. \end{aligned}$$

Moreover, assumptions (F1)–(F3) hold. In particular, for any $c>0$ there exists a sufficiently large constant $R>0$ (dependent on the constants c, p and the function a) such that

$$\begin{aligned} \frac{a(x)}{p}|u|^p\le \frac{\Vert a\Vert _{L^\infty }}{p}|u|^p\le {\left\{ \begin{array}{ll} c|u|^2,&{}|u|>R\\ \frac{\Vert a\Vert _{L^\infty }}{p}R^p+c|u|^2,&{}|u|\le R, \end{array}\right. } \end{aligned}$$

so, the condition (13) is satisfied for any $c>0$.

Consequently, since all assumptions of Theorem 4 are satisfied and F is strictly convex with respect to u for a.e. $x\in {\varOmega }$, therefore the functional J given by

$$\begin{aligned} J(u)=\int \limits _{\varOmega }\left( \frac{a(x)}{p}|u(x)|^p+\frac{1}{qa(x)} \Big |[w((-{\varDelta })_\omega )u(x)]\Big |^q-u(x)w((-{\varDelta })_\omega )u(x)\right) \hbox {d}x \end{aligned}$$

(25)

has a unique minimizer $u_*\in D(w((-{\varDelta })_{\omega }))$.

On the other hand it is clear that $u_*=0$ is a solution of the problem (24), so it must minimize J on $D(w((-{\varDelta })_{\omega }))$. This means that $u_*=0$ is the only solution of (24).

Remark 7

If $p=2$ in the above example then the condition (13) holds for $c=\frac{\Vert a\Vert _{L^\infty }}{2}$. Consequently, the linear problem of the form

$$\begin{aligned} \sum \limits _{i=0}^k\alpha _i((-{\varDelta })_\omega )^{\beta _i}u(x)=a(x)u(x),\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

has the only trivial solution provided that

$$\begin{aligned} \Vert a\Vert _{L^\infty }<\frac{\sqrt{M_{\beta _k}}\alpha _k}{2}. \end{aligned}$$

(26)

Here it is worth to note that replacing the functional J given by (25) (for $p=q=2$) with the following one

$$\begin{aligned} J_1(u)&=\int \limits _{\varOmega }\left( \frac{a(x)}{2}|u(x)|^2+\frac{1}{2a(x)} \Big |[w((-{\varDelta })_\omega )u(x)-f(x)]\Big |^2 \right. \\&\qquad \left. -u(x)\big (w((-{\varDelta })_\omega ) u(x)-f(x)\big ) \right) \hbox {d}x \end{aligned}$$

we can prove (similarly as in Sect. 3) counterparts of Theorems 3 and 4 for the functional $J_1$ (in particular, under assumption (26) one can obtain the existence of a unique minimizer of $J_1$ on $D(w((-{\varDelta })_{\omega }))$) and the following, more general linear problem

$$\begin{aligned} \sum \limits _{i=0}^k\alpha _i((-{\varDelta })_\omega )^{\beta _i}u(x)=a(x)u(x)+f(x),\quad x\in {\varOmega }\,\,\hbox {a.e.}, \end{aligned}$$

(27)

where $f\in L^2({\varOmega },{\mathbb {R}})$. Additionally, if a is a constant function, then the problem (27) has a unique solution. Indeed, since $J_1$ has a unique minimizer $u_*$ on $D(w((-{\varDelta })_{\omega }))$, therefore

$$\begin{aligned} \delta J_1(u_*,h)=0.\quad h\in D(w((-{\varDelta })_{\omega })), \end{aligned}$$

(28)

where $\delta J_1(u_*,h)$ denotes the first variation of $J_1$ at $u_*$ in the direction h.

So,

$$\begin{aligned} \delta J_1(u_*,h)&= \int \limits _{\varOmega }\frac{1}{a} \Big (w((-{\varDelta })_\omega ) u_*(x)-au_*(x)-f(x)\Big ) \nonumber \\&\quad \Big (w((-{\varDelta })_\omega )h(x)-ah(x)\Big )\hbox {d}x=0 \end{aligned}$$

(29)

for any $h\in D(w((-{\varDelta })_{\omega }))$.

Since $a<\frac{\sqrt{M_{\beta _k}}\alpha _k}{2}$, therefore the constant a is not the eigenvalue of the operator $w((-{\varDelta })_{\omega })$. Consequently, the kernel of the self-adjoint operator $L{:}D(w((-{\varDelta })_{\omega }))\rightarrow L^2({\varOmega },{\mathbb {R}})$, defined as

$$\begin{aligned} Lh:=w((-{\varDelta })_\omega )h-ah,\quad a>0, \end{aligned}$$

is trivial, so the image of L satisfies the equality $R(L)=L^2({\varOmega },{\mathbb {R}})$. Hence and from (29) we conclude

$$\begin{aligned} w((-{\varDelta })_\omega )u_*(x)-au_*(x)-f(x)=0,\quad x\in {\varOmega }\,\,\hbox {a.e.} \end{aligned}$$

This means that $u_*$ is a unique solution of (27).