The Existence of a Nontrivial Weak Solution to a Double Critical Problem Involving a Fractional Laplacian in ℝ^N with a Hardy Term

Abstract

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:

$${( - \Delta )^s}u - \gamma {u \over {{{\left| x \right|}^{2s}}}} = {{{{\left| u \right|}^{2_s^ * (\beta ) - 2}}u} \over {{{\left| x \right|}^\beta }}} + \left[ {{I_\mu } * {F_\alpha }( \cdot ,u)} \right](x){f_\alpha }(x,u),\;\;\;\;u \in {\dot H^{^s}}({\mathbb{R}^n}),$$

(0.1)

where $s \in (0,1),0 \le \alpha ,\beta < 2c < n,\mu \in (0,n),\gamma < {\gamma _H},{I_\mu }(x) = {\left| x \right|^{ - \mu }},{F_\alpha }(x,u) = {{{{\left| {u(x)} \right|}^{2_\mu ^\# (\alpha )}}} \over {{{\left| x \right|}^{{\delta _\mu }(\alpha )}}}},\;{f_\alpha }(x,u) = {{{{\left| {u(x)} \right|}^{2_\mu ^\# (\alpha ) - 2}}u(x)} \over {{{\left| x \right|}^{{\delta _\mu }(\alpha )}}}},2_\mu ^\# (\alpha ) = (1 - {\textstyle{\mu \over {2n}}}) \cdot 2_s^ * (\alpha ),\;{\delta _\mu }(\alpha ) = (1 - {\textstyle{\mu \over {2n}}})\alpha ,\;2_s^ * (\alpha ) = {{2(n - \alpha )} \over {n - 2s}}$ and ${\gamma _H} = {4^s}{{{\Gamma ^2}({\textstyle{{n + 2s} \over 4}})} \over {{\Gamma ^2}({\textstyle{{n - 2s} \over 4}})}}.$. We show that problem (0.1) admits at least a weak solution under some conditions.

To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings

$${\dot H^s}({\mathbb{R}^n})\alpha {L^{2_s^ * (\alpha )}}({\mathbb{R}^n},{\left| y \right|^{ - \alpha }})\alpha {L^{p,{\textstyle{{n - 2s} \over 2}}p + pr}}({\mathbb{R}^n},{\left| y \right|^{ - pr}}),$$

(0.2)

where s ∈ (0, 1), 0 < α < 2s < n, p ∈ [1, 2_s*(α)) and $r = {\textstyle{\alpha \over {2_s^ * (\alpha )}}}.$. We also establish an improved Sobolev inequality,

$${\left( {\int_{{\mathbb{R}^n}} {{{{{\left| {u(y)} \right|}^{2_s^ * (\alpha )}}} \over {{{\left| y \right|}^\alpha }}}} {\rm{d}}y} \right)^{{\textstyle{1 \over {2_s^ * (\alpha )}}}}} \le C\left\| u \right\|_{{{\dot H}^s}({\mathbb{R}^n})}^\theta \left\| u \right\|_{{L^{p,{\textstyle{{n - 2s} \over 2}}p + pr}}({\mathbb{R}^n},{{\left| y \right|}^{ - pr}})}^{1 - \theta },\;\;\;\;\;\;\forall u \in {\dot H^s}({\mathbb{R}^n}),$$

(0.3)

where $s \in (0,1),\;0 \le \alpha < 2s < n,p \in [1,2_s^ * (\alpha )),\;r = {\alpha \over {2_s^ * (\alpha )}},\;0 < \max {\rm{\{ }}{2 \over {2_s^ * (\alpha )}}{\rm{,}}{{2_s^ * - 1} \over {2_s^ * (\alpha )}}{\rm{\} }} < \theta < 1,\;2_s^ * = {{2n} \over {n - 2s}}$ and C= C(n, s, α) 0 is a constant. Inequality (0.3) is a more general form of Theorem 1 in Palatucci, Pisante [1].

By using the mountain pass lemma along with (0.2) and (0.3), we obtain a nontrivial weak solution to problem (0.1) in a direct way. It is worth pointing out that (0.2) and (0.3) could be applied to simplify the proof of the existence results in [2] and [3].