Differential inclusion systems with fractional competing operator and multivalued fractional convection term

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, $N\geq 3$, with a smooth boundary $\partial\Omega$, let $\mu_{1},\mu_{2}\in\mathbb{R}$, and let $F_{1},F_{2}:\Omega\times\mathbb{R}^{2}\times\mathbb{R}^{2N}\to 2^{\mathbb{R}}$ be two compact convex-valued multifunctions. Consider the differential inclusion system

$$\left\{\begin{array}[]{lll}(-\Delta)_{p_{1}}^{s_{1}}u_{1}+\mu_{1}(-\Delta)_{q_{1}}^{t_{1}}u_{1}\in F_{1}(x,u_{1},u_{2},D^{r_{1}}u_{1},D^{r_{2}}u_{2})&\mbox{ in $\Omega$},\\ (-\Delta)_{p_{2}}^{s_{2}}u_{2}+\mu_{2}(-\Delta)_{q_{2}}^{t_{2}}u_{2}\in F_{2}(x,u_{1},u_{2},D^{r_{1}}u_{1},D^{r_{2}}u_{2})&\mbox{ in $\Omega$},\\ u_{1}=u_{2}=0&\mbox{ in $\mathbb{R}^{N}\backslash\Omega$},\end{array}\right.$$

(1.1)

where

• $({\rm H}_{1})$

  $0<t_{i}<r_{i}<s_{i}\leq 1$ and $1<q_{i}<p_{i}<\frac{N}{s_{i}}$ for each $i=1,2$.

The symbol $(-\Delta)_{p}^{s}$, with $p>1$ and $0<s<1$, denotes the fractional $p$-Laplacian, defined by setting, provided $u$ is smooth enough,

$$(-\Delta)^{s}_{p}u(x):=2\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}\setminus B_{\varepsilon}(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\,{\rm d}y,\quad x\in\mathbb{R}^{N}$$

with $B_{\varepsilon}(x):=\{z\in\mathbb{R}^{N}\,\mid\,\|z-x\|_{\mathbb{R}^{N}}<\varepsilon\}$. When $s=1$ it becomes the classical $p$-Laplacian, namely

$$-\Delta_{p}u:=-\mathop{\rm div}\nolimits(|\nabla u|^{p-2}\nabla u).$$

Moreover, $D^{s}u$ indicates the distributional Riesz fractional gradient of $u$ in the sense of [19, 20]. If $u$ appropriately decays and is sufficiently smooth then, setting

$$c_{N,s}:=-\frac{2^{s}\,\Gamma\left(\frac{N+s+1}{2}\right)}{\pi^{\frac{N}{2}}\,\Gamma\left(\frac{1-s}{2}\right)}\,,$$

one has [20, pp. 289 and 298]

$$D^{s}u(x):=c_{N,s}\lim_{\varepsilon\to 0^{+}}\int_{\mathbb{R}^{N}\setminus B_{\varepsilon}(x)}\frac{u(x)-u(y)}{|x-y|^{N+s}}\,\frac{x-y}{|x-y|}{\rm d}y,\quad x\in\mathbb{R}^{N}.$$

The right-hand sides $F_{1}$ and $F_{2}$ satisfy the conditions below, where, to avoid cumbersome formulae, we shall write

$$y:=(y_{1},y_{2}),\quad z:=(z_{1},z_{2}),\quad p_{i}^{*}:=\frac{Np_{i}}{N-s_{i}p_{i}},\quad i=1,2.$$

(1.2)

• $({\rm H}_{2})$

  $x\mapsto F_{i}(x,y,z)$ is measurable on $\Omega$ for all $(y,z)\in\mathbb{R}^{2}\times\mathbb{R}^{2N}$ and $(y,z)\mapsto F_{i}(x,y,z)$ is upper semi-continuous for a.e. $x\in\Omega$.

• $({\rm H}_{3})$

  There exist $m_{i}>0$, $\delta_{i}\in L^{(p_{i}^{*})^{\prime}}(\Omega)$, $i=1,2$, such that

  $\displaystyle\sup_{w_{i}\in F_{i}(x,y,z)}|w_{i}|\leq$

  $\displaystyle m_{1}\left(|y_{1}|^{\frac{p_{1}^{*}}{(p_{i}^{*})^{\prime}}}+|y_{2}|^{\frac{p_{2}^{*}}{(p_{i}^{*})^{\prime}}}+|z_{1}|^{\frac{p_{1}}{(p_{i}^{*})^{\prime}}}+|z_{2}|^{\frac{p_{2}}{(p_{i}^{*})^{\prime}}}\right)+\delta_{i}(x)$

  a.e. in $\Omega$ and for all $(y,z)\in\mathbb{R}^{2}\times\mathbb{R}^{2N}$.

• $({\rm H}_{4})$

  There are $M_{i},M_{i}^{\prime}>0$, $\sigma_{i}\in L^{1}(\Omega)_{+}$, $i=1,2$, fulfilling

  $\displaystyle w_{i}y_{i}\leq M_{i}(|y_{1}|^{p_{1}}+|y_{2}|^{p_{2}})+M_{i}^{\prime}(|z_{1}|^{p_{1}}+|z_{2}|^{p_{2}})+\sigma_{i}(x)$

  a.e. in $\Omega$ and for all $(y,z)\in\mathbb{R}^{2}\times\mathbb{R}^{2N}$, $w_{i}\in F_{i}(x,y,z)$.

The involved differential operators are of the type

$$A_{\mu}(u):=(-\Delta)_{p}^{s}u+\mu(-\Delta)_{q}^{t}u,\quad u\in W^{s,p}_{0}(\Omega),$$

where $\mu\in\mathbb{R}$, $0<t\leq r\leq s\leq 1$, $1<q<p<\frac{N}{s}$, while convection comes from the presence of fractional gradients $D^{r}u$ at right-hand sides. $A_{\mu}$ exhibits different behaviors depending on the values of $t,s\in(0,1]$. Precisely, if $t=1$, then $t=r=s=1$, Problem (1.1) falls inside the local framework, and has been already investigated in some recent works; see, e.g., [9] for single-valued reactions and [18, 4] as regards multi-valued ones. Moreover, the nature of $A_{\mu}$ drastically changes depending on $\mu$. In fact, when $\mu>0$, the operator $A_{\mu}$ is basically patterned after the (possibly) fractional $(p,q)$-Laplacian, which turns out non-homogeneous because $p\neq q$. If $\mu=0$ it coincides with the fractional $p$-Laplacian. Both cases have been widely investigated and meaningful results are by now available in the literature. On the contrary, for $\mu<0$ the operator $A_{\mu}$ contains the difference between the fractional $p$- and $q$-Laplacians. It is usually called competitive and, as already pointed out in [14, 17], doesn’t comply with any ellipticity or monotonicity condition. In fact, given $u_{0}\in W^{s,p}_{0}(\Omega)\setminus\{0\}$ and chosen $u:=\tau u_{0}$, $\tau>0$, the expression

$$\langle A_{\mu}(u),u\rangle=\tau^{p}\|\nabla u_{0}\|_{s,p}^{p}+\mu\tau^{q}\|u_{0}\|_{s,q}^{q}$$

turns out negative for $\tau$ small and positive when $\tau$ is large. Therefore, nonlinear regularity theory, comparison principles, as well as existence theorems for pseudo-monotone maps cannot be employed. Moreover, since the reactions are multi-valued and contain the fractional gradient of the solutions, also variational techniques are no longer directly usable. To overcome these difficulties we first exploit Galerkin’s method, thus working with a sequence $\{E_{n}\}$ of finite dimensional functional spaces. For each $n\in\mathbb{N}$, an approximate solution $(u_{1,n},u_{2,n})\in E_{n}$ to (1.1) is obtained via a suitable version (cf. Proposition 3) of a classical surjectivity result. Next, letting $n\to+\infty$ yields a solution in a generalized sense (cf. Definitions 8 and 10), which turns out weak sense once $\mu_{1}\wedge\mu_{2}\geq 0$.

Fractional gradients were first introduced more than sixty years ago by Horváth [12], but took a great interest especially after the works of Shieh and Spector [19, 20, 5]. The operator $D^{s}u$ appears as a natural non-local version of $\nabla u$, to whom it formally tends when $s\to 1^{-}$. It enjoys good geometric and physical properties [2, 21], like invariance under translations or rotations, homogeneity of order $s$, continuity, etc.

Section 2 contains some auxiliary results and the functional framework needed for handling both fractional gradients and the fractional $p$-Laplacian. The existence of (generalized, strong generalized, or weak) solutions to (1.1) is established in Section 3.

Let $X,Y$ be two nonempty sets. A multifunction $\Phi:X\to 2^{Y}$ is a map from $X$ into the family of all nonempty subsets of $Y$. A function $\varphi:X\to Y$ is called a selection of $\Phi$ when $\varphi(x)\in\Phi(x)$ for every $x\in X$. Given $B\subseteq Y$, put $\Phi^{-}(B):=\{x\in X\,\mid\,\Phi(x)\cap B\neq\emptyset\}.$ If $X,Y$ are topological spaces and $\Phi^{-}(B)$ turns out closed in $X$ for all closed sets $B\subseteq Y$ then we say that $\Phi$ is upper semi-continuous. Suppose $(X,\mathcal{F})$ is a measurable space and $Y$ is a topological space. The multifunction $\Phi$ is called measurable when $\Phi^{-}(B)\in\mathcal{F}$ for every open set $B\subseteq Y$. The result below, stated in [1, p. 215], will be repeatedly useful.

Let $(X,\|\cdot\|)$ be a real normed space with topological dual $X^{*}$ and duality brackets $\langle\cdot,\cdot\rangle$. Given a nonempty set $A\subseteq X$, define $|A|:=\sup_{x\in A}\|x\|$. We say that $\varphi:X\to X^{*}$ is monotone when

$$\langle\varphi(x)-\varphi(z),x-z\rangle\geq 0\quad\forall\,x,z\in X,$$

and of type $(\mathrm{S})_{+}$ provided

$$x_{n}\rightharpoonup x\;\;\mbox{in $X$,}\;\;\limsup_{n\to+\infty}\langle\varphi(x_{n}),x_{n}-x\rangle\leq 0\implies x_{n}\to x\;\;\mbox{in $X$.}$$

The next elementary result [8, Proposition 2.1] ensures that condition $(\mathrm{S})_{+}$ holds true for the fractional $(p,q)$-Laplacian.

A multifunction $\Phi:X\to 2^{X^{*}}$ is called coercive provided

$$\lim_{\|x\|\to\infty}\frac{\inf\{\langle x^{*},x\rangle\,\mid\,x\in X,\,x^{*}\in\Phi(x)\}}{\|x\|}=+\infty\,.$$

The following result is a direct consequence of [11, Proposition 3.2.33].

Hereafter, if $X$ and $Y$ are two topological spaces, the symbol $X\hookrightarrow Y$ means that $X$ continuously embeds in $Y$. Given $p>1$, put $p^{\prime}:=\frac{p}{p-1}$, denote by $\|\cdot\|_{p}$ the usual norm of $L^{p}(\Omega)$, and indicate with $\|\cdot\|_{1,p}$ the norm on $W^{1,p}_{0}(\Omega)$ arising from Poincaré’s inequality, namely

$$\|u\|_{1,p}:=\|\nabla u\|_{p}\,,\quad u\in W^{1,p}_{0}(\Omega).$$

If $u\in W^{1,p}_{0}(\Omega)$, we set $u(x)=0$ on $\mathbb{R}^{N}\setminus\Omega$; cf. [6, Section 5]. Fix $s\in(0,1)$. The Gagliardo seminorm of a measurable function $u:\mathbb{R}^{N}\to\mathbb{R}$ is

$$[u]_{s,p}:=\left(\int_{R^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}{{\rm d}x}{{\rm d}y}\right)^{\frac{1}{p}},$$

while $W^{s,p}(\mathbb{R}^{N})$ denotes the fractional Sobolev space

$$W^{s,p}(\mathbb{R}^{N}):=\left\{u\in L^{p}(\mathbb{R}^{N}):\ [u]_{s,p}<+\infty\right\},$$

endowed with the norm

$$\|u\|_{W^{s,p}(\mathbb{R}^{N})}:=\left(\|u\|^{p}_{L^{p}(\mathbb{R}^{N})}+[u]_{s,p}^{p}\right)^{\frac{1}{p}}.$$

As usual, on the space

$$W^{s,p}_{0}(\Omega):=\{u\in W^{s,p}(\mathbb{R}^{N}):u=0\;\mbox{a.e. in}\;\mathbb{R}^{N}\setminus\Omega\}$$

we will consider the equivalent norm

$$\|u\|_{s,p}:=[u]_{s,p},\quad u\in W^{s,p}_{0}(\Omega).$$

Let $W^{-s,p^{\prime}}(\Omega):=(W^{s,p}_{0}(\Omega))^{*}$ and let $p^{*}_{s}$ be the fractional Sobolev critical exponent, i.e., $p^{*}_{s}=\frac{Np}{N-sp}$ when $sp<N$, $p^{*}_{s}=+\infty$ otherwise. Thanks to Propositions 2.1–2.2, Theorem 6.7, and Corollary 7.2 of [6] one has

However, contrary to the non-fractional case, we know [15] that

$$1\leq q<p\leq+\infty\;\;\centernot\implies W^{s,p}_{0}(\Omega)\subseteq W^{s,q}_{0}(\Omega).$$

Define, for every $u,v\in W^{s,p}_{0}(\Omega)$,

$$\langle(-\Delta)^{s}_{p}u,v\rangle:=\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\,.$$

The operator $(-\Delta)_{p}^{s}$ is called (negative) $s$-fractional $p$-Laplacian. It possesses the following properties.

• $({\rm p}_{1})$

  $(-\Delta)^{s}_{p}:W^{s,p}_{0}(\Omega)\rightarrow W^{-s,p^{\prime}}(\Omega)$ turns out monotone, continuous, and of type $({\rm S})_{+}$; vide, e.g., [7, Lemma 2.1].

• $({\rm p}_{2})$

  One has

  $$\|(-\Delta)^{s}_{p}u\|_{W^{-s,p^{\prime}}(\Omega)}\leq\|u\|_{s,p}^{p-1}\quad\forall\,u\in W^{s,p}_{0}(\Omega).$$

  Hence, $(-\Delta)^{s}_{p}$ maps bounded sets into bounded sets.

• $({\rm p}_{3})$

  The first eigenvalue $\lambda_{1,p,s}$ of $(-\Delta)^{s}_{p}$ is given by (cf. [13])

  $$\lambda_{1,p,s}=\inf_{u\in W_{0}^{s,p}(\Omega),u\neq 0}\frac{\|u\|_{s,p}^{p}}{\|u\|_{p}^{p}}\,.$$

To deal with distributional fractional gradients, we first introduce the Bessel potential spaces $L^{\alpha,p}(\mathbb{R}^{N})$, where $\alpha>0$. Set, for every $x\in\mathbb{R}^{N}$,

$$g_{\alpha}(x):=\frac{1}{(4\pi)^{\frac{\alpha}{2}}\Gamma\left(\frac{\alpha}{2}\right)}\int_{0}^{+\infty}e^{\frac{-\pi|x|^{2}}{\delta}}e^{\frac{-\delta}{4\pi}}\delta^{\frac{\alpha-N}{2}}\frac{{\rm d}\delta}{\delta}\,.$$

On account of [16, Section 7.1] one can assert that:

• 1)

  $g_{\alpha}\in L^{1}(\mathbb{R}^{N})$ and $\|g_{\alpha}\|_{L^{1}(\mathbb{R}^{N})}=1$.

• 2)

  $g_{\alpha}$ enjoys the semi-group property, i.e., $g_{\alpha}\ast g_{\beta}=g_{\alpha+\beta}$ for any $\alpha,\beta>0$, with $*$ being the convolution operator.

Now, put

$$L^{\alpha,p}(\mathbb{R}^{N}):=\{u:\,u=g_{\alpha}\ast\tilde{u}\;\mbox{for some}\;\tilde{u}\in L^{p}(\mathbb{R}^{N})\}$$

as well as

$$\|u\|_{L^{\alpha,p}(\mathbb{R}^{N})}=\|\tilde{u}\|_{L^{p}(\mathbb{R}^{N})}\;\;\mbox{whenever}\;\;u=g_{\alpha}\ast\tilde{u}.$$

Using 1) and 2) we easily get

$$0<\alpha<\beta\;\implies\;L^{\beta,p}(\mathbb{R}^{N})\subseteq L^{\alpha,p}(\mathbb{R}^{N})\subseteq L^{p}(\mathbb{R}^{N}).$$

Moreover, by [19, Theorem 2.2], one has

Finally, given $s\in(0,1)$, define

$$L^{s,p}_{0}(\Omega):=\{u\in L^{s,p}(\mathbb{R}^{N}):u=0\;\text{a.e. in}\;\mathbb{R}^{N}\setminus\Omega\}.$$

Thanks to Theorem 5 we infer

$$L^{s+\varepsilon,p}_{0}(\Omega)\hookrightarrow W^{s,p}_{0}(\Omega)\hookrightarrow L^{s-\varepsilon,p}_{0}(\Omega)\quad\forall\,\varepsilon\in(0,s).$$

(2.1)

The next basic notion is taken from [19]. For $0<\alpha<N$, let

$$\gamma(N,\alpha):=\frac{\Gamma\left(\frac{N-\alpha}{2}\right)}{\pi^{\frac{N}{2}}2^{\alpha}\Gamma\left(\frac{\alpha}{2}\right)},\quad I_{\alpha}(x):=\frac{\gamma(N,\alpha)}{|x|^{N-\alpha}},\quad x\in\mathbb{R}^{N}\setminus\{0\}.$$

If $u\in L^{p}(\mathbb{R}^{N})$ and $I_{1-s}\ast u$ makes sense then the vector

$$D^{s}u:=\left(\frac{\partial}{\partial x_{1}}(I_{1-s}\ast u),\ldots,\frac{\partial}{\partial x_{N}}(I_{1-s}\ast u)\right),$$

where partial derivatives are understood in a distributional sense, is called distributional Riesz $s$-fractional gradient of $u$. Theorem 1.2 in [19] ensures that

$$D^{s}u=I_{1-s}\ast Du\quad\forall\,u\in C^{\infty}_{c}(\mathbb{R}^{N}).$$

Further, $D^{s}u$ looks like the natural extension of $\nabla u$ to the fractional framework, In fact, it exhibits analogous properties and, roughly speaking, $D^{s}u\to\nabla u$ when $s\to 1^{-}$; see, e.g., [10, Section 2].

According to [19, Definition 1.5], $X^{s,p}(\mathbb{R}^{N})$ denotes the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ with respect to the norm

$$\|u\|_{X^{s,p}(\mathbb{R}^{N})}:=\left(\|u\|_{L^{p}(\mathbb{R}^{N})}^{p}+\|D^{s}u\|_{L^{p}(\mathbb{R}^{N})}^{p}\right)^{\frac{1}{p}}.$$

Since, by [19, Theorem 1.7], $X^{s,p}(\mathbb{R}^{N})=L^{s,p}(\mathbb{R}^{N})$ we can deduce many facts about $X^{s,p}(\mathbb{R}^{N})$ from the existing literature on $L^{s,p}(\mathbb{R}^{N})$. Moreover, if

$$X^{s,p}_{0}(\Omega):=\{u\in X^{s,p}(\mathbb{R}^{N}):u=0\;\text{a.e. in}\;\mathbb{R}^{N}\setminus\Omega\},$$

then $X^{s,p}_{0}(\Omega)=L^{s,p}_{0}(\Omega)$.

To shorten notation, for $i=1,2$, we set $U_{i}:=W^{s_{i},p_{i}}_{0}(\Omega)$ and denote by $\langle\cdot,\cdot\rangle_{i}$ the duality brackets of $U_{i}$. Lemma 2.6 in [3] guarantees that

$$U_{i}:=W^{s_{i},p_{i}}_{0}(\Omega)\hookrightarrow W^{t_{i},q_{i}}_{0}(\Omega).$$

(3.1)

Hence, the differential operator $u\mapsto(-\Delta)^{s_{i}}_{p_{i}}u+\mu_{i}(-\Delta)^{t_{i}}_{q_{i}}u$ turns out well-defined on $U_{i}$. Let $A_{s_{i},t_{i}}:U_{i}\to U^{*}_{i}$ be given by

$$\begin{split}\langle A_{s_{i},t_{i}}(u),v\rangle_{i}&:=\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{p_{i}-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+p_{i}s_{i}}}{\rm d}x\,{\rm d}y\\ &+\mu_{i}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{q_{i}-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+q_{i}t_{i}}}{\rm d}x\,{\rm d}y\end{split}$$

for every $u,v\in U_{i}$. Thanks to properties $({\rm p}_{1})$–$({\rm p}_{2})$ stated in Section 2, $A_{s_{i},t_{i}}$ is bounded and continuous. Consequently,

Next, put, provided $(u_{1},u_{2})\in U_{1}\times U_{2}$,

$$\begin{split}\mathcal{S}_{F_{1},F_{2}}(u_{1},u_{2}):=\{(w_{1},w_{2})&\in L^{({p_{1}^{*}})^{\prime}}(\Omega)\times L^{(p^{*}_{2})^{\prime}}(\Omega):\\ &w_{i}(\cdot)\in F_{i}(\cdot,u_{1},u_{2},D^{r_{1}}u_{1},D^{r_{2}}u_{2})\mbox{ a.e. in }\Omega,\;i=1,2\},\end{split}$$

with $p^{*}_{i}$ as in (1.2).

Our existence result can be established after introducing some suitable constants and the notion of generalized solution to (1.1). Since $r_{i}<s_{i}$, $i=1,2$, embeddings (2.1) produce

$$\|D^{r_{1}}u_{1}\|_{p_{1}}^{p_{1}}\leq\hat{c}_{1}\|u_{1}\|_{s_{1},p_{1}}^{p_{1}}\;\;\forall\,u_{1}\in U_{1},\quad\|D^{r_{2}}u_{2}\|_{p_{2}}^{p_{2}}\leq\hat{c}_{2}\|u_{2}\|_{s_{2},p_{2}}^{p_{2}}\;\;\forall\,u_{2}\in U_{2},$$

(3.2)

with appropriate $\hat{c}_{i}>0$. Via (3.1) and its analogue for couples $(s_{2},p_{2})$–$(t_{2},q_{2})$ we next have

$$\|u_{1}\|_{t_{1},q_{1}}^{q_{1}}\leq\tilde{c}_{1}\|u_{1}\|_{s_{1},p_{1}}^{p_{1}}\;\;\forall\,u_{1}\in U_{1},\quad\|u_{2}\|_{t_{2},q_{2}}^{q_{2}}\leq\tilde{c}_{2}\|u_{2}\|_{s_{2},p_{2}}^{p_{2}}\;\;\forall\,u_{2}\in U_{2},$$

(3.3)

where $\tilde{c}_{i}>0$. Finally, set

$$\langle(u_{1},u_{2}),(v_{1},v_{2})\rangle:=\langle u_{1},v_{1}\rangle_{1}+\langle u_{2},v_{2}\rangle_{2},\quad(u_{1},u_{2})\in U_{1}\times U_{2},\;(v_{1},v_{2})\in U^{*}_{1}\times U^{*}_{2}.$$