1. Introduction

Recently, a great attention has been focussed on the study of inhomogeneous fractional nonlinear Schrödinger equation

(1.1)\begin{equation} \left\{\begin{array}{@{}ll} i\dot u-(-\Delta)^{\alpha} u-m|x|^{\sigma} u+\varepsilon |x|^{\gamma}|u|^{p-1}u=0, & x\in R^{N}, \ t>0\\ u|_{t=0}=u_0\in H^{\alpha}(R^{N}), & N\geq 2, \ \alpha\in (0,1), \ p>1, \end{array}\right. \end{equation}

where $m>0$ and $\varepsilon >0$ are real physical constants, $\sigma$ and $\gamma$ are real constants, $1< p<\frac {N+2\alpha }{(N-2\alpha )_+}$, and this equation comes from various physical contexts in the description of nonlinear waves such as propagation of a laser beam and plasma waves. When $\sigma =2$, the potential term is often called the harmonic potential and occurs in the condensed states. When $\alpha =1$, there is a strong physical background related to the study of (1.1) and one may prefer to the papers [Reference Byeon and Wang3, Reference Smets, Su and Willem34, Reference Zhang37]. When $\sigma =0=\gamma$, to understand the ground state of (1.1) (see proposition 3.1 in [Reference Frank, Lenzmann and Silvestre14] or [Reference Saanouni33]), one may use the well-known fractional Gagliardo–Nirenberg inequality that for some $\theta \in (0,1)$ such that $\frac 1{q}=\frac 12-\frac {\theta \alpha }{N}$, it holds that

\[ \|u\|_{q}\leq C \|u\|_2^{\theta}\|u\|_{\dot{H}^{\alpha}}^{1-\theta} \]

(see proposition A.3 in [Reference Tao35]). When $\sigma =-1$, a new fractional Gagliardo–Nirenberg inequality has been obtained in [Reference Li and Ma20] to prove the existence of the ground state. In the above papers, the Schwartz symmetrization method (so called re-arrangement argument) has been used. However, for $\sigma >0$ and $m>0$, one can not directly use the re-arrangement argument to get the ground state to (1.1). The natural question is if one may get the ground state in broad class of the powers $\sigma,\gamma >0$ without using re-arrangement argument, which is one subject of this paper. In [Reference Ma25], the author has found the ground state in the case when $\sigma =2$ and $\alpha \in \left (\frac 12,\frac {N}{2}\right )$ by restricting the working space to the class of radially functions. In case when $m=0$, the ground states have been obtained [Reference Ma26]. The range of $\alpha$ also plays a crucial role in the existence result of ground states. In this paper, we first set up a new fractional interpolation inequality related to (1.1) and prove that the best constant in the new inequality can be obtained by a radially measurable solution function. We then get the ground states of the fractional Schrödinger equation on the whole space. For $\alpha \in (0,\frac 12]$, the power of the nonlinearity will be restricted into $p\in \left (1,\frac {N}{N-\alpha }\right )$ and the compactness result about radially symmetric functions obtained by Lions [Reference Lions23] will be used. Even in this case, we need to have more conditions about $\gamma$ and $\sigma$. The precise results will be stated (see theorems 1.2 and 1.3 below) after we briefly report some related results obtained by others. In the recent very interesting paper [Reference Ao, Chan, Delatorre, Fontelos, Gonzalez and Wei2], the authors have introduced the ODE approach to nonlinear equations with the fractional Laplacian and their approach may be useful to the problem (1.1).

In [Reference Chen and Guo5], for $\alpha =1$ in (1.1), Chen and Guo have given a criterion for the global existence of solutions of the corresponding Cauchy problem for an inhomogeneous nonlinear Schrödinger equation with harmonic potential:

(1.2)\begin{equation} \left\{\begin{array}{@{}ll} i\psi_t={-}\Delta\psi+|x|^{2} \psi-|x|^{b}|\psi|^{p-1}\psi, & b>0,\\ \psi(x,0)=\psi_0(x), & \end{array}\right. \end{equation}

with radial initial data

\[ \psi_0 \in \widetilde{\Sigma}:=\{u\in H^{1}(R^{N}): u\ is\ radial\ and \ \int_{R^{N}}|x|^{2}|u(x)|^{2}\,{\rm d}\kern0.06em x<\infty\}. \]

Their result may be stated as the following. Let $b>0, N>1+\frac {b}{2},1+\frac {4+2b}{N}< p<\frac {N+2}{N-2}+\frac {2b}{N-1}$ $(1+\frac {4+2b}{N}< p<\infty, if N\leq 2), A=\frac {N(p-1)-2b}{2},B=\frac {2(p-1)-(N(p-1)-2b)}{2}$ and let $V(\lambda )=\left (\frac {A-2}{p+1-A}\right )^\frac{A-2}{2B}||Q||^\frac{p-1}{B}_{L^{2}}\lambda ^{-\frac {A-2}{2B}}$, where $Q$ is the ground state solution to the equation

\[ -\Delta Q+Q-|x|^{b}|Q|^{p-1}Q=0. \]

Define $\mathscr {S}=\{\phi \in \widetilde {\Sigma }:||\phi ||_{L^{2}}< V(||\nabla \phi ||^{2}_{L^{2}}+||x\phi ||^{2}_{L^{2}})\}$. If $\varphi _0\in \mathscr {S}$, then the corresponding solution to (1.2) is global in time and uniformly bounded in $\widetilde {\Sigma }$. Furthermore, $\mathscr {S}$ is unbounded in $\widetilde {\Sigma }$. The proof relies on a Gagliardo–Nirenberg inequality with best constant and an energy method. Moreover, a similar result holds true for the critical power $p=1+\frac {4+2b}{N}$, where the assumption $\varphi _0\in \mathscr {S}$ is replaced with $\varphi _0\in \widetilde {\Sigma }$ and $||\varphi _0||_{L^{2}}<||Q||_{L^{2}}$. In addition, this criterion is sharp in the following sense. For any $\lambda >0$ and $c\in \mathbb {C}$ with $|c|\geq 1$, if $\varphi _0(x)= c\lambda ^{\frac {N}{2}}Q(\lambda x)$, then $\varphi _0\in \widetilde {\Sigma }, ||\varphi _0||_{L^{2}}\geq ||Q||_{L^{2}}$ and the corresponding solution to (1.2) blows up in finite time. See also [Reference Li, Wang and Zeng21, Reference Smets, Su and Willem34] for related results. For $\sigma =-s$ and $\gamma =-t$ with $0\leq t<2s$, via a detailed study of a fractional version Hardy–Sobolev–Maz'ya inequality, with particular attention to the optimality of the constants involved, Mallick [Reference Mallick28] has studied the existence of positive solutions of the nonlinear equation involving fractional powers of the Laplacian with cylindrical potentials. Moreover, he has also considered the symmetric and asymptotic properties of the positive solution. The study of the problem (1.1) is also closely related to weighted Hardy type inequalities [Reference Frank and Seiringer13, Reference Li and Ma20]. See also [Reference Chhetri and Girg7, Reference Cho, Hwang, Kwon and Lee9, Reference Hajaiej, Molica Bisci and Vilasi15–Reference Le Coz19, Reference Luo and Zhang24, Reference Moroz and Van Schaftingen29, Reference Paul and Santra32] for related results of nonlocal problems.

We use the following notations: $A\lesssim B$ denotes an estimate of the form $A\leqslant CB$ for some absolute constant $C>0$; for $q\geq 1$, $L^{q}:=L^{q}({R}^{N})$ is the Lebesgue space endowed with the usual norm $\|\,\cdot \,\|_q:=\|\,\cdot \,\|_{L^{q}}$; $\|\,\cdot \,\|:=\|\,\cdot \,\|_2$, and $\|(-\Delta )^\frac{\alpha }{2}\cdot \|=\|\cdot \|_{\dot H^{\alpha }}$. We also use the brief notation that $\int \cdot {\rm d}\kern0.06em x:=\int _{R^{N}} \cdot {\rm d}\kern0.06em x$.

Recall that for $s\in (0,1)$, the fractional Hilbert space $H^{s}=H^{s}(R^{N})$ is defined by [Reference Adams1, Reference Chen, Li and Ma6, Reference Di Nezza, Palatucci and Valdinoci11, Reference Dyda, Ihnatsyeva and Vähäkangas12, Reference Musina and Nazarov30]

\[ H^{s}(R^{N})=\{u\in L^{2}(R^{N}); (1+|\xi|^{2})^{s/2}\mathcal{F}(u(\xi))\in L^{2}(R^{N})\} \]

where $\mathcal {F}(u)$ denotes the Fourier transformation of $u$, with norm

\[ \|u\|_{H^{s}}=\sqrt{\int_{R^{N}}(1+|\xi|^{2})^{s}|\mathcal{F}(u(\xi))|^{2}\,{\rm d}\xi}. \]

We also denote by

\[ \|(-\Delta)^\frac{s}{2} u\|^{2}=\int_{R^{N}}|\xi|^{2s}|\mathcal{F}(u(\xi))|^{2}\,{\rm d}\xi, \quad u\in C_0^{\infty}(R^{N}). \]

We introduce, for $\sigma \in (0,2)$, $\alpha \in (0,1)$, a fractional Sobolev space $D^{\alpha }$ with the norm

\[ \|u\|_{D^{\alpha}}:=\left(\||x|^{\frac\sigma 2}u\|^{2}+\|(-\Delta)^\frac\alpha 2 u\|^{2}\right)^\frac 12, \]

such that under this norm $D^{\alpha }$ is the completion of $C_0^{\infty }(R^{N})$. We also let $\dot H^{\alpha }$ be the completion of $C_0^{\infty }(R^{N})$ under the usual norm $\|(-\Delta )^\frac\alpha 2 u\|$. For $\alpha >1$, we may use Fourier transform to define $H^{\alpha }$ and we refer to [Reference Cho and Ozawa8]. Then we may define $D^{\alpha }$ as above.

For $Z$ being any space of functions on $R^{N}$, we denote by $Z_{rd}$ the space of radial functions in $Z$. Here and hereafter, we denote, for $\phi \in D_{rd}^{\alpha },$ the mass and energy functionals for (1.1):

\[ M(\phi)=\int |\phi|^{2}\,{\rm d}\kern0.06em x:= \|\phi\|^{2}, \]

and

\[ E(\phi)=\frac{1}{2}\|(-\Delta)^\frac\alpha 2 \phi\|^{2} +\frac m2\int |x|^{\sigma}|\phi|^{2}\,{\rm d}\kern0.06em x -\frac{1}{p+1}\int |x|^{\gamma}|\phi|^{p+1}\,{\rm d}\kern0.06em x, \]

where the integration is over the whole space $R^{N}$. We remark that the ground state to (1.1) may be considered as the mountain pass critical point of the energy functional $E(\cdot )$ on the working space $D_{rd}^{\alpha }$ [Reference Costa10].

We define the functional

\[ J(u):=\frac{\|(-\Delta)^\frac\alpha 2u\|^{B}\||x|^{\frac\sigma 2}u\|^{A}}{\int|x|^{\gamma}|u|^{1+p}\,{\rm d}\kern0.06em x} \]

on the space $D_{rd}^{\alpha }$, where

\begin{align*} & A:=(p+1)\theta,\\ & B:=(p+1)(1-\theta), \end{align*}

where $\theta =\frac {N+\sigma -\frac {2N+2\gamma }{p+1}}{2\alpha +\sigma }$.

We now consider the minimization problem

(1.3)\begin{equation} d:=\inf_{0\neq \phi \in D^{\alpha}_{rd}}\left\{ J(u); u\in D^{\alpha}_{rd}, u\neq 0 \right\}. \end{equation}

There is no previous result about this problem with $\sigma >0$. We now set up the interpolation inequality as below.

For $\alpha \in (0,1)$, a more general result than proposition 1.1 is proven in theorem 1.1 in [Reference Nguyen and Squassina31]. Since we have different purpose, we shall present a simpler and direct proof of proposition 1.1 for completeness. We now explain why $\theta$ can be chosen in this way. So we do the dimension analysis. Note that for $p>1$, $\gamma '\geq 0$, $a>0,b>0$, we let the scaling $\psi = u_{a,b}:= au\left (\frac {x}{b}\right )$ with $\frac {x}{b}=z$:

\begin{align*} \||\cdot|^{\gamma'}u_{a,b}\|^{p+1}_{L^{p+1}}& =\int|x|^{\gamma'(p+1) }u^{p+1}_{a,b}(x)\,{\rm d}\kern0.06em x\\ & =a^{p+1}\int{|x|^{\gamma'(p+1)}u^{p+1}\left(\frac{x}{b}\right){\rm d}\kern0.06em x}\\ & =a^{p+1} b^{N+\gamma'(p+1) }\int{|x|^{\gamma'(p+1) }u^{p+1}(z)\,{\rm d}z}\\ & =a^{p+1} b^{N+\gamma'(p+1)}\||\cdot|^{\gamma'}u\|^{p+1}_{L^{p+1}}. \end{align*}

Note that for $p=1$ and $\gamma '=\frac \sigma 2$, we have

\[ \||\cdot|^{\frac{\sigma}{2}}u_{a,b}\|^{2}_{L^{2}}=a^{2} b^{N+\sigma}\||\cdot|^{\frac{\sigma}{2}}u\|^{2}_{L^{2}}. \]

We also have

\[ \|(-\Delta)^{\frac{\alpha}{2}}u_{a,b}\|^{2}_{L^{2}}=a^{2} b^{N-2\alpha}\|(-\Delta)^{\frac{\alpha}{2}}u\|^{2}_{L^{2}}. \]

If proposition 1.1 is true, then for any constant $c$, we have

\[ ab^{\frac{N}{p+1}+\gamma'}\leq c(ab^{\frac{N+\sigma}{2}})^{1-\theta}(ab^{\frac{N-2\alpha}{2}})^{\theta}. \]

By this we have

\[ b^{\frac{N}{p+1}+\gamma'}\leq cb^{\left(\frac{N+\sigma}{2}\right)(1-\theta)+\left(\frac{N}{2}-\alpha\right)\theta}, \]

which implies that

\[ 1\leq cb^{\left(\frac{N+\sigma}{2}\right)(1-\theta)+\left(\frac{N}{2}-\alpha\right)\theta-\frac{N}{p+1}-\gamma'}. \]

Then we get

\[ \left(\frac{N+\sigma}{2}\right)(1-\theta)+\left(\frac{N}{2}-\alpha\right)\theta-\frac{N}{p+1}-\gamma'=0, \]

which implies that

\[ \theta=\frac{N+\sigma-\frac{2N}{p+1}-2\gamma'}{2\alpha+\sigma}. \]

As the consequences, we may obtain some of our main results (and the very new part is when $\alpha \in \left (0,\frac 12\right ]$, which has not been treated before).

Using this proposition, we shall prove that there is a ground state of (1.9) below.

Theorem 1.3 Take $\epsilon =1$, $\gamma \geq 0$, $\frac 12<\alpha <\frac N2$, and $\frac {2\gamma +2\alpha -N-4\sigma }{N-2\alpha }\leq p\leq \frac {2(N+\gamma )}{N-2\alpha }-1$. Then, there is a ground state solution to (1.1) in the following meaning

(1.9)\begin{equation} (-\Delta)^{\alpha} \phi +|x|^{\sigma}\phi -|x|^{\gamma}\phi|\phi|^{p-1}=0 ,\quad 0\neq \phi \in D_{rd}^{\alpha}. \end{equation}

To prove the result above, we need to set up a compactness result, which is parallel to the classical Sobolev injection (see [Reference Chen and Guo5, Reference Lions23]). We state here such result in case $\frac 12<\alpha <\frac N2$.

Lemma 1.4 Let $\gamma \geq 0$, $\frac 12<\alpha <\frac N2$, and $\frac {2\gamma +2\alpha -N-4\sigma }{N-2\alpha }\leq p\leq \frac {2(N+\gamma )}{N-2\alpha }-1$. Then, the following injection

(1.10)\begin{equation} D^{\alpha}_{rd}(R^{N})\hookrightarrow\hookrightarrow\Sigma:=\{u\in L^{1}_{loc}(R^{n}); \int_{R^{n}} |u|^{p+1}|x|^{\gamma}\,{\rm d}\kern0.06em x<\infty\} \end{equation}

is compact, and we may simply write $\Sigma :=L^{p+1}(|x|^{\gamma }\,{\rm d}\kern0.06em x)$.

We may also show the existence of other standing states to (1.1) in the following way. We now define on $D^{\alpha }_{rd}$ the action for (1.1) by

\[ S(u)=E(u)+\frac 12 M(u) \]

and let $Q(u)=< S'(u),u>$. Set

\[ M=\{u\in D^{\alpha}_{rd}; u\not=0, Q(u)=0 \} \]

and

\[ m_0=\inf\{S(u),u\in {M}\}. \]

It is clear that when restricted on $M$, the action $S$ is simplified to

\[ S(u)=a\int |x|^{\gamma} |u|^{p+1}=a\int |(-\Delta)^\frac\alpha 2 u|^{2} +m|x|^{\sigma}|u|^{2}+|u|^{2}\,{\rm d}\kern0.06em x \]

where $a=\left (\frac 12-\frac {1}{p+1}\right )$. Hence $m_0\geq 0$.

We shall show in next section that

The proof of lemma 1.5 replies on theorem 1.2 and this result will play an important role in the argument of theorem 1.6 below.

Then, we can show the following result about the existence of standing waves and their orbital stability for (1.1).

As usual, in the orbitally stable part, we always suppose that the global existence of solutions to (1.1) with initial datum near to the standing waves.

The rest of the paper is organized as follows. In §2, we prove lemmas 1.4 and 1.5. We recall some useful inequalities and some tools like compactness result needed in the sequel. In §3, we prove proposition 1.1. In §4, we prove theorem 1.2. We prove theorem 1.3 in § 5. We establish the existence of standing states of (1.1) and the orbital stability in §6. Thus, we prove theorem 1.6 in the last section.