D s,2 ( R N ) versus C ( R N ) local minimizers

Abstract

Let $s \in (0, 1)$ , $N > 2 s$ and $\begin{matrix} D^{s, 2} (R^{N}) : = {u \in L^{2_{s}^{*}} (R^{N}) : ‖ u ‖_{D^{s, 2} (R^{N})} : = {(\frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y)}^{\frac{1}{2}} < \infty}, \end{matrix}$ where $2_{s}^{*} : = \frac{2 N}{N - 2 s}$ is the fractional critical exponent and $C_{N, s}$ is a positive constant. We consider functionals $J : D^{s, 2} (R^{N}) \to R$ of the type $\begin{array}{l} J (u) : = \frac{1}{2} ‖ u ‖_{D^{s, 2} (R^{N})}^{2} - \int_{R^{N}} b (x) G (u) d x, \end{array}$ where $G (t) : = \int_{0}^{t} g (τ) d τ$ , $g : R \to R$ is a continuous function with subcritical growth at infinity, and $b : R^{N} \to R$ is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace $\begin{matrix} V_{s} : = {u \in D^{s, 2} (R^{N}) : u \in C (R^{N}) with sup_{x \in R^{N}} (1 + | x |^{N - 2 s}) | u (x) | < \infty} \end{matrix}$ must be a local minimizer of J in the $D^{s, 2} (R^{N})$ -topology.

Keywords

Fractional operators variational methods

1. Introduction

In the seminal paper [5], Brezis and Nirenberg showed that for functionals of the type $\begin{matrix} I (u) : = \frac{1}{2} \int_{Ω} | \nabla u |^{2} d x - \int_{Ω} F (x, u) d x, with F (x, t) : = \int_{0}^{t} f (x, τ) d τ, \end{matrix}$ where $Ω \subset R^{N}$ is a smooth bounded domain, $N ⩾ 3$ , $f : Ω \times R \to R$ is a Carathéodory function such that $| f (x, t) | ⩽ C (1 + | t |^{p})$ for a.e. $x \in Ω$ and all $t \in R$ , where $p ⩽ \frac{N + 2}{N - 2}$ , any local minimizer of I in the $C_{0}^{1} (\bar{Ω})$ topology is a local minimizer in the $H_{0}^{1} (Ω)$ topology. This result plays a fundamental role in studying the existence and multiplicity of solutions for semilinear elliptic boundary value problems of the type $\begin{array}{l} \{\begin{matrix} - Δ u = f (x, u) & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{array}$ via variational methods. Later, the result in [5] has been extended for different nonlinear operators on bounded domains by several authors; see [2,9–11,16]. Recently, Carl, Costa and Tehrani [7] established a Brezis-Nirenberg type result for functionals of the type $\begin{array}{l} J (u) : = \frac{1}{2} \int_{R^{N}} | \nabla u |^{2} d x - \int_{R^{N}} b (x) G (u) d x, \end{array}$ whose critical points are the weak solutions of $\begin{array}{l} \{\begin{matrix} - Δ u = b (x) g (u) in R^{N}, \\ u \in D^{1, 2} (R^{N}), \end{matrix} \end{array}$ where $N ⩾ 3$ , $g : R \to R$ is a continuous subcritical nonlinearity, which is superlinear at $t = 0$ and at infinity, and $b : R^{N} \to R$ is an appropriate weight function. More precisely, in [7] the authors proved that a local minimizer of $J$ in the topology of the subspace V must be a local minimizer of $J$ in the $D^{1, 2} (R^{N})$ -topology, where $\begin{matrix} D^{1, 2} (R^{N}) : = {u \in L^{\frac{2 N}{N - 2}} (R^{N}) : \int_{R^{N}} | \nabla u |^{2} d x < \infty}, \end{matrix}$ and $\begin{matrix} V : = {v \in D^{1, 2} (R^{N}) : v \in C (R^{N}) with sup_{x \in R^{N}} (1 + | x |^{N - 2}) | v (x) | < \infty} . \end{matrix}$ On the other hand, in the fractional context, Iannizzotto, Mosconi and Squassina [12] obtained an analogue of the result in [5] by considering the following class of fractional boundary value problems $\begin{array}{l} \{\begin{matrix} {(- Δ)}^{s} u = f (x, u) & in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{array}$ where $s \in (0, 1)$ , $N > 2 s$ , $Ω \subset R^{N}$ is a smooth bounded open set, $f : Ω \times R \to R$ is a Carathéodory function such that $| f (x, t) | ⩽ C (1 + | t |^{p - 1})$ for a.e. $x \in Ω$ and all $t \in R$ , with $1 ⩽ p ⩽ 2_{s}^{*} : = \frac{2 N}{N - 2 s}$ . Here the fractional Laplacian operator ${(- Δ)}^{s}$ is defined for smooth functions $u : R^{N} \to R$ by $\begin{array}{l} {(- Δ)}^{s} u (x) : = C_{N, s} lim_{ζ \to 0} \int_{R^{N} ∖ B_{ζ} (x)} \frac{(u (x) - u (y))}{| x - y |^{N + 2 s}} d y, x \in R^{N}, \\ C_{N, s} : = π^{- \frac{N}{2}} 2^{2 s} \frac{Γ (\frac{N + 2 s}{2})}{Γ (2 - s)} s (1 - s); \end{array}$ see [1,8,14] for more details and applications. In [12] the authors demonstrated that for functionals of the form $\begin{array}{l} I (u) : = \frac{1}{2} \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y - \int_{Ω} F (x, u) d x, \end{array}$ local minimizers with respect to $C_{δ}^{0} (\bar{Ω})$ -topology are actually local minimizers in the natural $H_{0}^{s} (Ω)$ -topology, where $C_{δ}^{0} (\bar{Ω}) : = {u \in C^{0} (\bar{Ω}) : \frac{u}{δ^{s}} admits a continuous extension to \bar{Ω}}$ , $δ (x) : = dist (x, R^{N} ∖ Ω)$ , $x \in \bar{Ω}$ , and $H_{0}^{s} (Ω) : = {u \in H^{s} (R^{N}) : u = 0 in R^{N} ∖ Ω}$ . A similar result for the spectral fractional Laplacian can be found in [3].

Particularly motivated by [7,12], the aim of this paper is to extend the main result in [7] by providing a $D^{s, 2} (R^{N})$ versus $C (R^{N})$ local minimizer principle, where $\begin{matrix} D^{s, 2} (R^{N}) : = {u \in L^{2_{s}^{*}} (R^{N}) : ‖ u ‖_{D^{s, 2} (R^{N})} : = {(\frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{N + 2 s}} d x d y)}^{\frac{1}{2}} < \infty} . \end{matrix}$ More precisely, we focus on the following class of fractional problems $\begin{array}{l} (1.1) & \{\begin{matrix} {(- Δ)}^{s} u = b (x) g (u) in R^{N}, \\ u \in D^{s, 2} (R^{N}), \end{matrix} \end{array}$ where $s \in (0, 1)$ , $N > 2 s$ , $g : R \to R$ satisfies the following assumptions:

$g \in C (R)$ and ${lim}_{| t | \to 0} \frac{g (t)}{| t |} = 0$ ,

there exist $C > 0$ and $p \in (2, 2_{s}^{*})$ such that $| g (t_{1}) - g (t_{2}) | ⩽ C | t_{1} - t_{2} | [1 + max {| t_{1} |, | t_{2} |}^{p - 2}]$ for all $t_{1}, t_{2} \in R$ ,

and

b : R^{N} \to R

fulfills the following conditions:

$b \in L^{1} (R^{N}) \cap L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})$ ,

there exists $C^{'} > 0$ such that $\begin{matrix} | x |^{\frac{N}{r}} ‖ b ‖_{L^{q} (R^{N} ∖ B_{| x |} (0))} ⩽ C^{'} with q > \frac{N}{2 s}, \frac{1}{r} + \frac{1}{q} = 1 . \end{matrix}$

The weak solutions of (1.1) can be obtained as critical points of the functional

J : D^{s, 2} (R^{N}) \to R

given by

\begin{array}{l} J (u) : = \frac{1}{2} ‖ u ‖_{D^{s, 2} (R^{N})}^{2} - \int_{R^{N}} b (x) G (u) d x . \end{array}

Let

V_{s}

be the subspace of

D^{s, 2} (R^{N})

defined by

\begin{matrix} V_{s} : = {v \in D^{s, 2} (R^{N}) : v \in C (R^{N}) with ‖ v ‖_{V_{s}} : = sup_{x \in R^{N}} (1 + | x |^{N - 2 s}) | v (x) | < \infty} . \end{matrix}

The main result of this paper can be stated as follows.

Theorem 1.1.
Assume that $(G 1)$ – $(G 2)$ and $(B 1)$ – $(B 2)$ hold. Let $\hat{u} \in H$ be a weak solution of ( 1.1 ) and a local minimizer of J in the $V_{s}$ -topology, that is $\begin{array}{l} \exists ε_{1} > 0 : J (\hat{u} + h) ⩾ J (\hat{u}) \forall h \in V_{s}, ‖ h ‖_{V_{s}} < ε_{1} . \end{array}$ Then $\hat{u}$ is a local minimizer of J in the $D^{s, 2} (R^{N})$ -topology, that is $\begin{array}{l} \exists ε_{2} > 0 : J (\hat{u} + h) ⩾ J (\hat{u}) \forall h \in D^{s, 2} (R^{N}), ‖ h ‖_{D^{s, 2} (R^{N})} < ε_{2} . \end{array}$

The proof of Theorem 1.1 is inspired by the approach in [7]. However, due to the nonlocal character of ${(- Δ)}^{s}$ , it will be needed to prove some regularity results for weak solutions of (1.1) and show that any weak solution of (1.1) belongs to the subspace $V_{s}$ . These facts will be crucial in implementing our minimization argument. Finally, we believe that Theorem 1.1 may play a central role in studying multiple solutions for fractional elliptic problems in $R^{N}$ .
2. Proof of Theorem 1.1

We start by establishing some fruitful regularity results.

Proposition 2.1.

Let $s \in (0, 1)$ and $N > 2 s$ . Let $u \in D^{s, 2} (R^{N})$ be a weak solution of ${(- Δ)}^{s} u = f (x, u)$ in $R^{N}$ , i.e. $\begin{array}{l} (2.1) & \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + 2 s}} d x d y = \int_{R^{N}} f (x, u) φ d x for all φ \in C_{c}^{\infty} (R^{N}), \end{array}$ where $f : R^{N} \times R \to R$ is a Caratheodory function satisfying the following condition:

$| f (x, t) | ⩽ c (x) | t | + d (x)$ for a.e. $x \in R^{N}$ and all $t \in R$ , for some $c \in L^{\frac{N}{2 s}} (R^{N})$ , $d \in L^{\frac{N}{2 s}} (R^{N}) \cap L^{1} (R^{N})$ such that $c (x), d (x) ⩾ 0$ for a.e. $x \in R^{N}$ .

Then we have:

$u \in L^{τ} (R^{N})$ for all $τ \in [2_{s}^{*}, \infty)$ and $\begin{matrix} ‖ u ‖_{L^{τ} (R^{N})} ⩽ C_{1} + C_{2} ‖ u ‖_{L^{2_{s}^{*}} (R^{N})}, \end{matrix}$ where $C_{1} ⩾ 0$ and $C_{2} > 0$ depend on N, s, τ, c and d. In particular, $C_{1} = 0$ if $d \equiv 0$ .

If $f (\cdot, u) \in L^{\bar{p}} (R^{N})$ for some $\bar{p} > \frac{N}{2 s}$ , then $u \in L^{\infty} (R^{N}) \cap C (R^{N})$ and $\begin{matrix} ‖ u ‖_{L^{\infty} (R^{N})} ⩽ C_{3} (‖ u ‖_{L^{2_{s}^{*}} (R^{N})} + {‖ f (\cdot, u) ‖}_{L^{\bar{p}} (R^{N})}), \end{matrix}$ where $C_{3} > 0$ depends only on N, s and $\bar{p}$ .

Proof.

To prove $(i)$ , we combine a Brezis-Kato type argument [4] with a Moser iteration [15]. Let $β > 0$ and $L > 1$ , and define $x_{L} : = min {| x |, L}$ for $x \in R$ . Choosing $φ : = u u_{L}^{2 β} \in D^{s, 2} (R^{N}) \cap L^{\infty} (R^{N})$ as test function in (2.1), we find $\begin{array}{l} (2.2) & \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u (x) - u (y)) ((u u_{L}^{2 β}) (x) - (u u_{L}^{2 β}) (y))}{| x - y |^{N + 2 s}} d x d y = \int_{R^{N}} f (x, u) u u_{L}^{2 β} d x . \end{array}$ Using Lemma 3.1 in [12], we know that $\begin{array}{l} (u (x) - u (y)) ((u u_{L}^{2 β}) (x) - (u u_{L}^{2 β}) (y)) ⩾ \frac{2 β + 1}{{(β + 1)}^{2}} {((u u_{L}^{β}) (x) - (u u_{L}^{β}) (y))}^{2}, \end{array}$ which together with the fractional Sobolev inequality (see [8]) $\begin{array}{l} (2.3) & S_{*} ‖ v ‖_{L^{2_{s}^{*}} (R^{N})}^{2} ⩽ ‖ v ‖_{D^{s, 2} (R^{N})}^{2} for all v \in D^{s, 2} (R^{N}), \end{array}$ allows us to estimate the left-hand side of (2.2) from below as follows $\begin{array}{l} (2.4) & \frac{C}{β + 1} ‖ w_{L} ‖_{L^{2_{s}^{*}} (R^{N})}^{2} ⩽ \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u (x) - u (y)) ((u u_{L}^{2 β}) (x) - (u u_{L}^{2 β}) (y))}{| x - y |^{N + 2 s}} d x d y, \end{array}$ where $w_{L} : = u u_{L}^{β}$ , and $C > 0$ depends only on N and s and is independent of β and L. On the other hand, thanks to $(F)$ , we see that $\begin{array}{l} \int_{R^{N}} f (x, u) u u_{L}^{2 β} d x & ⩽ \int_{R^{N}} c (x) u^{2} u_{L}^{2 β} d x + \int_{R^{N}} d (x) | u | u_{L}^{2 β} d x \\ (2.5) & ⩽ \int_{R^{N}} c (x) u^{2} u_{L}^{2 β} d x + \int_{R^{N}} d (x) u^{2} u_{L}^{2 β} d x + \int_{R^{N}} d (x) d x, \end{array}$ where we used the fact that $| x | x_{L}^{2 β} ⩽ x^{2} x_{L}^{2 β} + 1$ for all $x \in R$ . Now, let $M > 0$ be fixed. The first term on the right-hand side of (2.5) can be estimated in the following way $\begin{array}{l} \int_{R^{N}} c (x) u^{2} u_{L}^{2 β} d x & = \int_{{c (x) ⩽ M}} c (x) u^{2} u_{L}^{2 β} d x + \int_{{c (x) > M}} c (x) u^{2} u_{L}^{2 β} d x \\ (2.6) & ⩽ M \int_{R^{N}} u^{2} u_{L}^{2 β} d x + {(\int_{{c (x) > M}} {| c (x) |}^{\frac{N}{2 s}} d x)}^{\frac{N - 2 s}{N}} ‖ w_{L} ‖_{L^{2_{s}^{*}} (R^{N})}^{2} . \end{array}$ In a similar manner, for the second term on the right-hand side of (2.5), we have that $\begin{array}{l} (2.7) & \int_{R^{N}} d (x) u^{2} u_{L}^{2 β} d x ⩽ M \int_{R^{N}} u^{2} u_{L}^{2 β} d x + {(\int_{{d (x) > M}} {| d (x) |}^{\frac{N}{2 s}} d x)}^{\frac{N - 2 s}{N}} ‖ w_{L} ‖_{L^{2_{s}^{*}} (R^{N})}^{2} . \end{array}$ Combining (2.5), (2.6) and (2.7), we get $\begin{array}{l} \int_{R^{N}} f (x, u) u u_{L}^{2 β} d x & ⩽ 2 M \int_{R^{N}} u^{2} u_{L}^{2 β} d x + {(\int_{{c (x) > M}} {| c (x) |}^{\frac{N}{2 s}} d x)}^{\frac{N - 2 s}{N}} ‖ w_{L} ‖_{L^{2_{s}^{*}} (R^{N})}^{2} \\ (2.8) & + {(\int_{{d (x) > M}} {| d (x) |}^{\frac{N}{2 s}} d x)}^{\frac{N - 2 s}{N}} ‖ w_{L} ‖_{L^{2_{s}^{*}} (R^{N})}^{2} + \int_{R^{N}} d (x) d x . \end{array}$ Set $\begin{matrix} ε_{1} (M) : = {(\int_{{c (x) > M}} {| c (x) |}^{\frac{N}{2 s}} d x)}^{\frac{N - 2 s}{N}} and ε_{2} (M) : = {(\int_{{d (x) > M}} {| d (x) |}^{\frac{N}{2 s}} d x)}^{\frac{N - 2 s}{N}}, \end{matrix}$ and note that $ε_{1} (M), ε_{2} (M) \to 0$ as $M \to \infty$ . Pick $M > 0$ such that $\begin{matrix} ε_{1} (M) + ε_{2} (M) < \frac{C}{2 (β + 1)} . \end{matrix}$ In light of (2.4) and (2.8), we arrive at $\begin{array}{l} (2.9) & \frac{C}{2 (β + 1)} ‖ w_{L} ‖_{L^{2_{s}^{*}} (R^{N})}^{2} ⩽ 2 M ‖ w_{L} ‖_{L^{2} (R^{N})}^{2} + \int_{R^{N}} d (x) d x . \end{array}$ If we assume that $u \in L^{2 (β + 1)} (R^{N})$ , letting $L \to \infty$ in (2.9), we infer that $u \in L^{2_{s}^{*} (β + 1)} (R^{N})$ . Thus a bootstrap argument can start: since $u \in L^{2_{s}^{*}} (R^{N})$ , we can apply (2.9) with $β + 1 = \frac{2_{s}^{*}}{2}$ to deduce that $u \in L^{\frac{{(2_{s}^{*})}^{2}}{2}} (R^{N})$ . We can then apply again (2.9) and, after k iterations, we obtain that $u \in L^{2 {(\frac{N}{N - 2 s})}^{k}} (R^{N})$ , and so $u \in L^{τ} (R^{N})$ for all $τ \in [2_{s}^{*}, \infty)$ . Therefore, $(i)$ is proved. Next we show $(ii)$ . Fix $x_{0} \in R^{N}$ and denote by $U : R^{N} \times (0, \infty) \to R$ the s-harmonic extension of u (see [6]), that is $U (x, y) : = (P_{s} (\cdot, y) * u) (x)$ where $\begin{matrix} P_{s} (x, y) : = p_{N, s} \frac{y^{2 s}}{{(| x |^{2} + y^{2})}^{\frac{N + 2 s}{2}}}, p_{N, s} : = \frac{Γ (\frac{N + 2 s}{2})}{π^{\frac{N}{2}} Γ (s)} . \end{matrix}$ Recall that the trace of U on $\partial R_{+}^{N + 1}$ is u and that U satisfies $\begin{array}{l} \iint_{R_{+}^{N + 1}} y^{1 - 2 s} \nabla U \nabla Φ d x d y = κ_{s} \int_{R^{N}} f (x, U (x, 0)) Φ (x, 0) d x \end{array}$ for all $Φ \in C_{c}^{\infty} (\overline{R_{+}^{N + 1}})$ , where $κ_{s} : = 2^{1 - 2 s} \frac{Γ (1 - s)}{Γ (s)}$ . Then it is easy to check that $| U |$ fulfills $\begin{array}{l} \iint_{R_{+}^{N + 1}} y^{1 - 2 s} \nabla | U | \nabla Φ d x d y ⩽ κ_{s} \int_{R^{N}} | f (x, U (x, 0)) | Φ (x, 0) d x \end{array}$ for all $Φ \in C_{c}^{\infty} (\overline{R_{+}^{N + 1}})$ such that $Φ ⩾ 0$ . Using Young’s inequality for convolution and $‖ P_{s} (\cdot, y) ‖_{L^{1} (R^{N})} = 1$ for all $y > 0$ , we see that $\begin{array}{l} \iint_{B_{1} (x_{0}) \times (0, 1)} {| U (x, y) |}^{2_{s}^{*}} y^{1 - 2 s} d x d y & ⩽ \iint_{R^{N} \times (0, 1)} {| U (x, y) |}^{2_{s}^{*}} y^{1 - 2 s} d x d y \\ = \int_{0}^{1} (\int_{R^{N}} {| \int_{R^{N}} P_{s} (x - z, y) u (z) d z |}^{2_{s}^{*}} d x) y^{1 - 2 s} d y \\ ⩽ \int_{0}^{1} {‖ P_{s} (\cdot, y) ‖}_{L^{1} (R^{N})}^{2_{s}^{*}} ‖ u ‖_{L^{2_{s}^{*}} (R^{N})}^{2_{s}^{*}} y^{1 - 2 s} d y \\ = ‖ u ‖_{L^{2_{s}^{*}} (R^{N})}^{2_{s}^{*}} \int_{0}^{1} y^{1 - 2 s} d y = \frac{‖ u ‖_{L^{2_{s}^{*}} (R^{N})}^{2_{s}^{*}}}{2 - 2 s} . \end{array}$ Hence, applying Proposition 2.6- $(i)$ in [13] with $ν : = 2_{s}^{*}$ , $p : = \bar{p}$ , $a (x) : = 0$ and $b (x) : = | f (x, U (x, 0)) |$ , we get $\begin{array}{l} sup_{(x, y) \in B_{\frac{1}{2}} (x_{0}) \times (0, \frac{1}{2})} | U (x, y) | ⩽ C (‖ u ‖_{L^{2_{s}^{*}} (R^{N})} + {‖ f (\cdot, u) ‖}_{L^{\bar{p}} (R^{N})}), \end{array}$ for some $C > 0$ depending only on N, s, $\bar{p}$ , and independent of $x_{0}$ . In addition, by Proposition 2.6- $(iii)$ in [13], $U \in C^{0, α} (\overline{B_{\frac{1}{2}} (x_{0})} \times [0, \frac{1}{2}])$ . As a consequence, u is continuous at $x_{0}$ and $\begin{matrix} | u (x_{0}) | = | U (x_{0}, 0) | ⩽ C (‖ u ‖_{L^{2_{s}^{*}} (R^{N})} + {‖ f (\cdot, u) ‖}_{L^{\bar{p}} (R^{N})}) . \end{matrix}$ From the arbitrariness of $x_{0} \in R^{N}$ , we derive that $u \in L^{\infty} (R^{N}) \cap C (R^{N})$ and that the estimate in $(ii)$ is true. □

Remark 2.1.

Reasoning as in the proof of Proposition 2.1- $(ii)$ , we can see that $\begin{matrix} | u (x_{0}) | ⩽ C (‖ u ‖_{L^{2_{s}^{*}} (B_{1} (x_{0}))} + {‖ f (\cdot, u) ‖}_{L^{\bar{p}} (B_{1} (x_{0}))}) for all x_{0} \in R^{N}, \end{matrix}$ which ensures that $| u (x) | \to 0$ as $| x | \to \infty$ .

Lemma 2.1.

Assume that $(G 1)$ – $(G 2)$ and $(B 1)$ – $(B 2)$ hold. Let u be a weak solution of ( 1.1 ). Then $u \in V_{s}$ .

Proof.

Let $f (x, t) : = b (x) g (t)$ . By $(G 2)$ , there exists $C > 0$ such that $\begin{matrix} | g (t) | ⩽ C (1 + | t |^{p - 1}) for all t \in R, \end{matrix}$ and so $\begin{matrix} | f (x, t) | ⩽ C | b (x) | + C | b (x) | | t |^{p - 2} | t | for a.e. x \in R^{N} and all t \in R . \end{matrix}$ Set $c (x) : = C | b (x) | | u (x) |^{p - 2}$ and $d (x) : = C | b (x) |$ . Clearly, $d \in L^{1} (R^{N}) \cap L^{\frac{N}{2 s}} (R^{N})$ thanks to $(B 1)$ . On the other hand, $u \in L^{2_{s}^{*}} (R^{N})$ and $b \in L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})$ imply that $| b |^{\frac{N}{2 s}} \in L^{\frac{2_{s}^{*}}{(2_{s}^{*} - p) \frac{N}{2 s}}} (R^{N})$ and that $| u |^{(p - 2) \frac{N}{2 s}} \in L^{\frac{2_{s}^{*}}{(p - 2) \frac{N}{2 s}}} (R^{N})$ . Thus, by Hölder’s inequality, we have $| b | | u |^{p - 2} \in L^{\frac{N}{2 s}} (R^{N})$ . This fact and Proposition 2.1- $(ii)$ yield $u \in L^{\infty} (R^{N}) \cap C (R^{N})$ . Hence, $x \mapsto g (u (x))$ belongs to $L^{\infty} (R^{N})$ and $f (x) : = b (x) g (u (x))$ satisfies $(B 1)$ – $(B 2)$ . Applying the Riesz potential formula, we know that $\begin{matrix} u (x) = {(- Δ)}^{- s} f (x) = C_{N, - s} \int_{R^{N}} \frac{f (y)}{| x - y |^{N - 2 s}} d y . \end{matrix}$ Using the following identity $\begin{matrix} \int_{0}^{\infty} χ_{B_{t} (x)} (y) \frac{1}{t^{N - 2 s + 1}} d t = \frac{| x - y |^{- (N - 2 s)}}{N - 2 s} for all x, y \in R^{N} such that x \neq y, \end{matrix}$ and Fubini’s theorem, we see that $\begin{array}{l} | u (x) | & ⩽ C_{N, - s} \int_{R^{N}} \frac{| f (y) |}{| x - y |^{N - 2 s}} d y \\ = C_{N, - s} (N - 2 s) \int_{0}^{\infty} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t . \end{array}$ Next we prove that, for some $C > 0$ , $\begin{array}{l} (2.10) & \int_{0}^{\infty} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t ⩽ C | x |^{2 s - N} for | x | large. \end{array}$ We may write $\begin{array}{l} \int_{0}^{\infty} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t = \int_{0}^{\frac{| x |}{2}} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t + \int_{\frac{| x |}{2}}^{\infty} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t . \end{array}$ Clearly, $\begin{matrix} \int_{0}^{\frac{| x |}{2}} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t ⩽ ‖ f ‖_{L^{1} (R^{N})} \int_{0}^{\frac{| x |}{2}} \frac{1}{t^{N - 2 s + 1}} d t ⩽ C | x |^{2 s - N} ‖ f ‖_{L^{1} (R^{N})} . \end{matrix}$ On the other hand, exploiting Hölder’s inequality and $(B_{2})$ , $\begin{array}{l} \int_{\frac{| x |}{2}}^{\infty} \frac{1}{t^{N - 2 s + 1}} ‖ f ‖_{L^{1} (B_{t} (x))} d t & ⩽ C \int_{\frac{| x |}{2}}^{\infty} \frac{t^{\frac{N}{r}}}{t^{N - 2 s + 1}} ‖ f ‖_{L^{q} (B_{t} (x))} d t \\ ⩽ C ‖ f ‖_{L^{q} (B_{\frac{| x |}{2}} (x))} {(\frac{| x |}{2})}^{2 s - N + \frac{N}{r}} \\ ⩽ C ‖ f ‖_{L^{q} (R^{N} ∖ B_{\frac{| x |}{2}} (0))} | x |^{\frac{N}{r}} | x |^{2 s - N} \\ ⩽ C | x |^{2 s - N} . \end{array}$ Consequently, (2.10) holds and this ensures that $u \in V_{s}$ . The proof of the lemma is then complete. □

Now we are ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1.

Let $\hat{u} \in D^{s, 2} (R^{N})$ be a weak solution of (1.1) and a local minimizer of J in the $V_{s}$ -topology. We aim to show that $\hat{u}$ is a local minimizer of J in the $D^{s, 2} (R^{N})$ -topology. We start by observing that, by $(G 2)$ and $(B 1)$ , J is well-defined, $J \in C^{1} (D^{s, 2} (R^{N}), R)$ and J is weakly lower semicontinuous. For all $n \in N$ , consider the following minimization problem: $\begin{array}{l} m_{n} : = inf_{u \in B_{n}} J (u) where B_{n} : = {u \in D^{s, 2} (R^{N}) : ‖ u - \hat{u} ‖_{D^{s, 2} (R^{N})} ⩽ \frac{1}{n}} . \end{array}$ Since J is weakly lower semicontinuous and $B_{n}$ is weakly closed, $m_{n}$ is achieved at some $u_{n} \in B_{n}$ . In fact, either $‖ u_{n} - \hat{u} ‖_{D^{s, 2} (R^{N})} < \frac{1}{n}$ , and so $u_{n}$ is a critical point of J, that is, $\begin{array}{l} \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u_{n} (x) - u_{n} (y)) (ϕ (x) - ϕ (y))}{| x - y |^{N + 2 s}} d x d y \\ (2.11) & - \int_{R^{N}} b (x) g (u_{n}) ϕ d x = 0 for all ϕ \in D^{s, 2} (R^{N}), \end{array}$ or $‖ u_{n} - \hat{u} ‖_{D^{s, 2} (R^{N})} = \frac{1}{n}$ and there exists a Lagrange multiplier $λ_{n} \in R$ such that $\begin{array}{l} \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u_{n} (x) - u_{n} (y)) (ϕ (x) - ϕ (y))}{| x - y |^{N + 2 s}} d x d y - \int_{R^{N}} b (x) g (u_{n}) ϕ d x \\ = λ_{n} \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{((u_{n} - \hat{u}) (x) - (u_{n} - \hat{u}) (y)) (ϕ (x) - ϕ (y))}{| x - y |^{N + 2 s}} d x d y \\ (2.12) & for all ϕ \in D^{s, 2} (R^{N}) . \end{array}$ On account of $⟨ J^{'} (u_{n}), \hat{u} - u_{n} ⟩ ⩾ 0$ , we have that $λ_{n} ⩽ 0$ . Rewriting (2.12) as $\begin{array}{l} (1 - λ_{n}) \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u_{n} (x) - u_{n} (y)) (ϕ (x) - ϕ (y))}{| x - y |^{N + 2 s}} d x d y - \int_{R^{N}} b (x) g (u_{n}) ϕ d x \\ = - λ_{n} \iint_{R^{N}} b (x) g (\hat{u}) ϕ d x for all ϕ \in D^{s, 2} (R^{N}), \end{array}$ and using the fact that $\hat{u}$ is a weak solution of (1.1), we deduce that $\begin{array}{l} \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{(u_{n} (x) - u_{n} (y)) (ϕ (x) - ϕ (y))}{| x - y |^{N + 2 s}} d x d y \\ = \frac{1}{1 - λ_{n}} \int_{R^{N}} b (x) g (u_{n}) ϕ d x \\ (2.13) & - \frac{λ_{n}}{1 - λ_{n}} \int_{R^{N}} b (x) g (\hat{u}) ϕ d x for all ϕ \in D^{s, 2} (R^{N}) . \end{array}$ Let $μ_{n} : = \frac{1}{1 - λ_{n}} \in (0, 1]$ and $μ_{n}^{'} : = \frac{- λ_{n}}{1 - λ_{n}} \in [0, 1]$ . From (2.11) and (2.13), we infer that $u_{n}$ is a weak solution of $\begin{array}{l} (2.14) & {(- Δ)}^{s} u_{n} = μ_{n} b (x) g (u_{n}) + μ_{n}^{'} b (x) g (\hat{u}) in R^{N}, \end{array}$ with $\begin{matrix} ‖ u_{n} - \hat{u} ‖_{D^{s, 2} (R^{N})} ⩽ \frac{1}{n} . \end{matrix}$ The above inequality and (2.3) guarantee that $u_{n} \to \hat{u}$ in $L^{2_{s}^{*}} (R^{N})$ . Thus there exists a subsequence of $(u_{n})$ , still denoted by itself, and a function $h \in L^{2_{s}^{*}} (R^{N})$ , such that, for all $n \in N$ , $\begin{array}{l} (2.15) & | u_{n} (x) | ⩽ h (x) for a.e. x \in R^{N} . \end{array}$ Now, for all $n \in N$ , we set $\begin{matrix} f_{n} (x, t) : = μ_{n} b (x) \frac{g (u_{n} (x))}{u_{n} (x)} t + μ_{n}^{'} b (x) g (\hat{u} (x)) for a.e. x \in R^{N} and all t \in R, \end{matrix}$ and note that $u_{n}$ satisfies ${(- Δ)}^{s} u_{n} = f_{n} (x, u_{n})$ in $R^{N}$ . From $(G 1)$ , $(G 2)$ , Lemma 2.1 and (2.15), we derive that $\hat{u} \in V_{s}$ , and there exist $α, β > 0$ such that $\begin{array}{l} (2.16) & | g (\hat{u} (x)) | ⩽ α for a.e. x \in R^{N}, \end{array}$ and, for all $n \in N$ , $\begin{array}{l} (2.17) & | \frac{g (u_{n} (x))}{u_{n} (x)} | ⩽ β (1 + {| u_{n} (x) |}^{p - 2}) ⩽ β (1 + {(h (x))}^{p - 2}) for a.e. x \in R^{N} . \end{array}$ Therefore, for all $n \in N$ , $\begin{array}{l} | f_{n} (x, t) | ⩽ c (x) | t | + d (x) for a.e. x \in R^{N} and all t \in R, \end{array}$ where $c (x) : = β b (x) (1 + {(h (x))}^{p - 2})$ and $d (x) : = α b (x)$ . Using $\frac{2_{s}^{*}}{2_{s}^{*} - p} > \frac{N}{2 s}$ and Hölder’s inequality, we see that $\begin{matrix} {‖ b h^{p - 2} ‖}_{L^{\frac{N}{2 s}} (R^{N})} ⩽ ‖ b ‖_{L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})}^{θ} ‖ h ‖_{L^{2_{s}^{*}} (R^{N})}^{1 - θ}, \end{matrix}$ for some $θ \in (0, 1)$ . Consequently, $c \in L^{\frac{N}{2 s}} (R^{N})$ and $d \in L^{1} (R^{N}) \cap L^{\frac{N}{2 s}} (R^{N})$ , and applying Proposition 2.1- $(i)$ , we obtain that, for all $τ \in [2_{s}^{*}, \infty)$ , $\begin{array}{l} (2.18) & ‖ u_{n} ‖_{L^{τ} (R^{N})} ⩽ C_{1} + C_{2} ‖ u_{n} ‖_{L^{2_{s}^{*}} (R^{N})} for all n \in N, \end{array}$ where $C_{1} ⩾ 0$ and $C_{2} > 0$ depend on N, s, τ, α, β and b. Combining $u_{n} \to \hat{u}$ in $L^{2_{s}^{*}} (R^{N})$ and (2.18), we deduce the (uniform) boundedness of $(u_{n})$ in $L^{τ} (R^{N})$ for all $τ \in [2_{s}^{*}, \infty)$ . Next, for all $n \in N$ , we put $\begin{matrix} f_{n} (x) : = μ_{n} b (x) g (u_{n} (x)) + μ_{n}^{'} b (x) g (\hat{u} (x)) for a.e. x \in R^{N} . \end{matrix}$ Choose $ε_{0} > 0$ such that $t : = \frac{2_{s}^{*}}{2_{s}^{*} - p} - ε_{0} > \frac{N}{2 s}$ . Since $μ_{n}, μ_{n}^{'} \in [0, 1]$ , it follows from Minkowski’s inequality that $\begin{matrix} ‖ f_{n} ‖_{L^{t} (R^{N})} ⩽ {‖ b g (u_{n}) ‖}_{L^{t} (R^{N})} + {‖ b g (\hat{u}) ‖}_{L^{t} (R^{N})} for all n \in N . \end{matrix}$ Exploiting (2.16), (2.17), Hölder’s inequality and the boundedness of $(u_{n})$ in $L^{τ} (R^{N})$ for all $τ \in [2_{s}^{*}, \infty)$ , we get $\begin{array}{l} {‖ b g (u_{n}) ‖}_{L^{t} (R^{N})} & ⩽ β ‖ b ‖_{L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})} (‖ u_{n} ‖_{L^{\frac{t (t + ε_{0})}{ε_{0}}} (R^{N})} + ‖ u_{n} ‖_{L^{\frac{(p - 1) t (t + ε_{0})}{ε_{0}}} (R^{N})}^{p - 1}) \\ ⩽ C ‖ b ‖_{L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})} for all n \in N, \end{array}$ and $\begin{array}{l} {‖ b g (\hat{u}) ‖}_{L^{t} (R^{N})} ⩽ C ‖ b ‖_{L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})}, \end{array}$ for some $C > 0$ independent of $n \in N$ . Invoking Proposition 2.1- $(ii)$ , we infer that $u_{n} \in L^{\infty} (R^{N}) \cap C (R^{N})$ and $\begin{array}{l} (2.19) & ‖ u_{n} ‖_{L^{\infty} (R^{N})} ⩽ C_{3} (‖ u_{n} ‖_{L^{2_{s}^{*}} (R^{N})} + ‖ f_{n} ‖_{L^{t} (R^{N})}) ⩽ C_{4} for all n \in N, \end{array}$ where $C_{3} > 0$ depends only on N, s, t, and $C_{4} > 0$ depends only on N, s, p, α, β, t and b. Now, (2.12) can be rewritten as $\begin{array}{l} (1 - λ_{n}) \frac{C_{N, s}}{2} \iint_{R^{2 N}} \frac{((u_{n} - \hat{u}) (x) - (u_{n} - \hat{u}) (y)) (ϕ (x) - ϕ (y))}{| x - y |^{N + 2 s}} d x d y \\ = \int_{R^{N}} b (x) (g (u_{n}) - g (\hat{u})) ϕ d x \end{array}$ for all $ϕ \in D^{s, 2} (R^{N})$ . Therefore, $w_{n} : = u_{n} - \hat{u}$ solves $\begin{array}{l} (2.20) & {(- Δ)}^{s} w_{n} = μ_{n} b (x) \frac{g (u_{n}) - g (\hat{u})}{u_{n} - \hat{u}} w_{n} = : \tilde{f} (x, w_{n}) in R^{N} . \end{array}$ Observe that $(G 2)$ and (2.19) yield $\begin{array}{l} (2.21) & | \frac{g (u_{n} (x)) - g (\hat{u} (x))}{u_{n} (x) - \hat{u} (x)} | ⩽ L for all n \in N, \end{array}$ for some $L > 0$ . Recalling that $μ_{n} \in (0, 1]$ , we find $| \tilde{f} (x, t) | ⩽ c (x) | t |$ , where $c (x) : = L | b (x) |$ with $c \in L^{1} (R^{N}) \cap L^{\frac{2_{s}^{*}}{2_{s}^{*} - p}} (R^{N})$ . As before, by Proposition 2.1- $(ii)$ , we have that $u_{n} - \hat{u} \in L^{\infty} (R^{N}) \cap C (R^{N})$ and there exists $C > 0$ such that $\begin{array}{l} ‖ u_{n} - \hat{u} ‖_{L^{\infty} (R^{N})} ⩽ C ‖ u_{n} - \hat{u} ‖_{L^{2_{s}^{*}} (R^{N})} for all n \in N . \end{array}$ As a result, $\begin{array}{l} (2.22) & ‖ u_{n} - \hat{u} ‖_{L^{\infty} (R^{N})} \to 0 . \end{array}$ Set $ε_{n} (x) : = μ_{n} (g (u_{n} (x)) - g (\hat{u} (x)))$ , $n \in N$ . Using (2.21), (2.22) and $μ_{n} \in [0, 1]$ , we see that $\begin{array}{l} (2.23) & ‖ ε_{n} ‖_{L^{\infty} (R^{N})} \to 0 . \end{array}$ Since $w_{n} = u_{n} - \hat{u}$ satisfies ${(- Δ)}^{s} w_{n} = b (x) ε_{n} (x)$ in $R^{N}$ , it follows from the Riesz potential formula that $\begin{array}{l} | w_{n} (x) | ⩽ C_{N, - s} (N - 2 s) \int_{0}^{\infty} \frac{1}{t^{N - 2 s + 1}} ‖ b ε_{n} ‖_{L^{1} (B_{t} (x))} d t . \end{array}$ Arguing as in the proof of Lemma 2.1 and taking (2.23) into account, we can demonstrate that $\begin{matrix} ‖ u_{n} - \hat{u} ‖_{V_{s}} = sup_{x \in R^{N}} (1 + | x |^{N - 2 s}) | u_{n} (x) - \hat{u} (x) | \to 0 . \end{matrix}$ Thus, for n sufficiently large, we arrive at $\begin{matrix} inf_{u \in B_{n}} J (u) = J (u_{n}) = J (\hat{u} + u_{n} - \hat{u}) ⩾ J (\hat{u}) . \end{matrix}$ that is, $\hat{u}$ is a local minimizer of J in $D^{s, 2} (R^{N})$ . The proof of Theorem 1.1 is now complete. □

Remark 2.2.

As in [7], one can show that Theorem 1.1 plays an essential role in proving that (1.1) admits a weak solution whenever (1.1) possesses a weak subsolution and a weak supersolution.

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