Infinitely many sign-changing solutions of a critical fractional equation

Abstract

In this paper, we obtain results of nonexistence of nonconstant positive solutions, and also existence of an unbounded sequence of sign-changing solutions for some critical problems involving conformally invariant operators on the unit sphere, in particular to the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together with blow up arguments, a variant of Pohozaev’s type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere.

Appendix

1.1 Sobolev type spaces and conformally invariant operators on the unit sphere

In this subsection, we recall some results for $P_s$ and Bessel potential spaces on spheres which can be found in [8, 42, 45, 48, 50].

The operator $P_s$ has as eigenfunctions, the spherical harmonic functions. Let $Y^{(k)}$ be a spherical harmonic function of degree $k\ge 0$. Then for $s\in (0,n/2)$, we have

$$\begin{aligned} P_s(Y^{(k)})=\frac{\Gamma (k+\frac{n}{2}+s)}{\Gamma (k+\frac{n}{2}-s)} Y^{(k)},~ R_s(Y^{(k)})=\frac{\Gamma (\frac{n}{2}+s)}{\Gamma (\frac{n}{2}-s)} Y^{(k)}, \end{aligned}$$

and, for each $u\in H^s({\mathbb {S}},g_{{\mathbb {S}}^n})$ with $u=\sum _{k}b_kY^{(k)}$, we obtain

$$\begin{aligned} \int _{{\mathbb {S}}^n}(u-{\overline{u}})(P_s-R_s)(u-{\overline{u}}){\mathrm {d}}\vartheta _{g_{{\mathbb {S}}^n}}=\sum _{k=1}^{\infty }\left[ \frac{\Gamma (k+\frac{n}{2}+s)}{\Gamma (k+\frac{n}{2}-s)}-\frac{\Gamma (\frac{n}{2}+s)}{\Gamma (\frac{n}{2}-s)}\right] \int _{{\mathbb {S}}^n}|b_kY^{(k)}|^2{\mathrm {d}}\vartheta _{g_{{\mathbb {S}}^n}} \end{aligned}$$

and therefore $\lambda _{1,s,g_{{\mathbb {S}}^n}}>0$.

If $g\in [{\mathbb {S}}^n]$ is a conformal metric on ${\mathbb {S}}^n$, then $(P^g_s)^{-1}$, acting in $L^2({\mathbb {S}}^n,g)$, is compact, self-adjoint and positive [, Proposition 2.3]. Consequently, 42$P^g_s$ admits an unbounded sequence of eigenvalues depending on g. On the other hand, the existence of eigenvalues for the operator $P^g_s - R^g_s$ was shown in [, Appendix] for the case 1$s\in (0,1)$. We believe that (1.5) holds for $s>1$.

Let $\Delta _{g_{{\mathbb {S}}^n}}$ be the Laplace–Beltrami operator on $({\mathbb {S}}^n,g_{{\mathbb {S}}^n})$. For $s>0$ and $1<p<\infty$, the Bessel potential space $H^s_p({\mathbb {S}}^n)$ is the set consisting of all $u\in L^p({\mathbb {S}}^n)$ such that $(1-\Delta )^{\frac{s}{2}}u\in L^p({\mathbb {S}}^n)$. The norm in $H^s_p({\mathbb {S}}^n)$ is defined by

$$\begin{aligned} \Vert u\Vert _{H^s_p({\mathbb {S}}^n)}:=\Vert (1-\Delta _{g_{{\mathbb {S}}^n}})^{\frac{s}{2}}u\Vert _{L^p({\mathbb {S}}^n)}. \end{aligned}$$

When $p=2$, then $H^s_2({\mathbb {S}}^n)$ coincides with $H^s({\mathbb {S}},g_{{\mathbb {S}}^n})$, and the norms $\Vert \cdot \Vert _{H^s_p({\mathbb {S}}^n)}$ and $\Vert \cdot \Vert _{s,g_{{\mathbb {S}}^n}}$ are equivalent. Recall that $H^s({\mathbb {S}}^n,g)$ denotes the closure of $C^{\infty }({\mathbb {S}}^n)$ under the norm

$$\begin{aligned} \Vert u\Vert ^2_{s,g} := \int _{{\mathbb {S}}^n} uP^g_su~{\mathrm {d}}\vartheta _g. \end{aligned}$$

Theorem 5.1

• (i) If $sp<n$, then the embedding $H^s_p({\mathbb {S}}^n)\hookrightarrow L^{q}({\mathbb {S}}^n)$ is continuous for $1\le q \le np/(n-sp)$ and compact for $q<np/(n-sp)$.

• (ii) If $s\in {\mathbb {N}}$, then $H^s_p({\mathbb {S}}^n)=W^{s,p}({\mathbb {S}}^n, g_{{\mathbb {S}}^n})$.

• (iii) If $s=k+s_0$ with $k\in {\mathbb {N}}$ and $s_0\in (0,1)$, then the embedding $H^s({\mathbb {S}},g) \hookrightarrow W^{k,\frac{2n}{n-2s_0}}({\mathbb {S}}^n,g)$ is continuous.

Proof

The proof of (i) and (ii) can be found in the reference cited at the beginning of this section. For the proof of (iii), we use (1.1) and [5, Theorem 6] to obtain

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {S}}^n}uP^g_s u~{\mathrm {d}}\vartheta _g&\ge C\Vert (1-\Delta _{g_{{\mathbb {S}}^n}})^{\frac{s}{2}}(\varphi u)\Vert _{L^2({\mathbb {S}}^n)}\\&=C\Vert (1-\Delta _{g_{{\mathbb {S}}^n}})^{\frac{s_0}{2}}(1-\Delta _{g_{{\mathbb {S}}^n}})^{\frac{k}{2}}(\varphi u)\Vert _{L^2({\mathbb {S}}^n)}\\&\ge C\Vert (1-\Delta _{g_{{\mathbb {S}}^n}})^{\frac{k}{2}}(\varphi u)\Vert _{L^{\frac{2n}{n-2s_0}}({\mathbb {S}}^n)}\\&\ge C \Vert \varphi u\Vert _{W^{k,\frac{2n}{n-2s_0}}({\mathbb {S}}^n)} \end{aligned} \end{aligned}$$

for all $u\in C^{\infty }({\mathbb {S}}^n)$. $\square$