Existence and multiplicity results for Dirichlet problem with fractional Laplacian and nonlinearity

Abstract

The existence and multiplicity results for Dirichlet BVPs with the fractional Laplacian are established depending on the range of parameter and behavior of the nonlinearity at zero and at infinity.

Appendix—technicalities: fractional Laplacian

We follow the lines of [18]. Let $\alpha \in (0,2)$, $u:{{\mathbb {R}}}^d \rightarrow {{\mathbb {R}}}$ be a measurable and

$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^d} \frac{|u(x)|}{(1+|x|)^{d+\alpha }} \, dx < \infty . \end{aligned}$$

(25)

For such a function the fractional Laplacian can be defined as in [6]

$$\begin{aligned} \left( -\Delta \right) ^{\alpha /2}u(x)=c_{d,-\alpha }\lim _{\varepsilon \rightarrow 0^+} \int \limits _{\{y\in {\mathbb {R}}^d:|x-y|>\varepsilon \}}\frac{u(x)-u(y)}{|x-y|^{d+\alpha }}dy, \end{aligned}$$

whenever the limit exists while the constant $c_{d,-\alpha }$ can be defined for $\gamma =-\alpha$ as

$$\begin{aligned} c_{d,\gamma }=\Gamma ((d-\gamma )/2)/(2^\gamma \pi ^{d/2}|\Gamma (\gamma /2)|). \end{aligned}$$

One can see that if u satisfies (25) and $u \in C^2(D)$ for some open set $D \subset {{\mathbb {R}}}^d$ then $\left( -\Delta \right) ^{\alpha /2}u(x)$ is well defined for any $x \in D$, which can be justified by Taylor series of u. The fractional Laplacian may be defined in a weak sense, see [6].

Moreover, for $D\subset {{\mathbb {R}}}^d$ there exist the Green operator ${{\mathcal {G}}}_D$ being the inverse $\left( (-\Delta )^{\alpha /2}\right) ^{-1}$. The Green function $G_D(x,y)$ corresponding to the problem (1)–(2) with $D=(-1,1)$ is the kernel of ${{\mathcal {G}}}_D$. Namely, if $g \in L^{\infty }$ then the unique (weak) solution of this problem is given by

$$\begin{aligned} u(x) = {{\mathcal {G}}}_D g(y) = \int \limits _D G_{D}(x,y) g(y) \, dy. \end{aligned}$$

(26)

One can notice that u defined by (26) is in fact in $C^{\gamma }$ with $\gamma >0$, cf. [23], whence also follows that ${{\mathcal {G}}}_D$ increases interior regularity by $\alpha$ in the Hölder sense. The theory on the Green operator and the Green function may be found e.g. in [6] or [7]. Furthermore, for any $\alpha \in (0,2)$ the Green function for the ball B(0, 1) is given by an explicit formula [5]

$$\begin{aligned} G_{B(0,1)}(x,y)=c^d_{\alpha }|x-y|^{\alpha -d}\int \limits _0^{w(x,y)}r^{\alpha /2-1}(r+1)^{-d/2}\,dr, \quad x,y \in B(0,1), \end{aligned}$$

where $w(x,y)=(1-|x|^2)(1-|y|^2)|x-y|^{-2}$ and $c^d_{\alpha }=\Gamma (d/2)/(2^{\alpha }\pi ^{d/2}\Gamma ^2(\alpha /2)).$ We have $G_{B(0,1)}(x,y) = 0$ if $x \notin B(0,1)$ or $y \notin B(0,1)$. Estimates on the regularity of solutions to the equations involving fractional Laplacian were provided by Ros-Oton and Serra [23]. The regularity and the existence and uniqueness issues for the problems connected with the fractional Laplacian were also considered by Cabré and Sire [9, 10]. For any open bounded $C^{1,1}$ domain D, $g\in L^{\infty }$ and a distance function $\delta (x)=\text {dist}(x,\partial D)$ if u is the solution of the Dirichlet problem (1)–(2) with $D=(-1,1)$ then $u/\delta ^{\alpha /2}|_{D}$ can be continuously extended to ${\overline{D}}$. Moreover, we have $u/\delta ^{\alpha /2} \in C^{\gamma }(\overline{D})$ and we control the norm

$$\begin{aligned} ||u/\delta ^{\alpha /2}||_{C^{\gamma }({\overline{D}})} \le C |g| \end{aligned}$$

for some $\gamma <\min \{\alpha /2,1-\alpha /2\}.$ It is thus sufficient, by the compact embedding $C^{\gamma }(\overline{D})\subset C(\overline{D})$, for proving the compactness of the operator ${{\mathcal {G}}}_D:C(\overline{D})\rightarrow C(\overline{D}).$ Moreover, once Hölder continuity is established bootstrap arguments can be used to ascertain the existence of classical $C^2$ or even more regular solutions. Since $(-\Delta )^{\alpha /2}(1-|x|^2)_+^{\alpha /2}=1$ then $(-\Delta )^{-\alpha /2}1(0)=1$ and due to unimodality $|(-\Delta )^{-\alpha /2}1|=1$.