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In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential,\[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \]where $n \geq 1$
, $0< s<1$
, $\omega >-\lambda _{1,s}$
, $2< p< {2n}/{(n-2s)^+}$
, $\lambda _{1,s}>0$
is the lowest eigenvalue of $(-\Delta )^s + |x|^2$
. The fractional Laplacian $(-\Delta )^s$
is characterized as $\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$
for $\xi \in \mathbb {R}^n$
, where $\mathcal {F}$
denotes the Fourier transform. This solves an open question in [M. Stanislavova and A. G. Stefanov. J. Evol. Equ. 21 (2021), 671–697.] concerning the uniqueness of ground states.
We prove the existence of nontrivial ground state solutions of the critical quasilinear Hénon equation 
$\displaystyle -\Delta _p u=|x|^{\alpha _1}|u|^{p^{*}(\alpha _1)-2}u-|x|^{\alpha _2}|u|^{p^{*}(\alpha _2)-2}u\ \ {\rm in}\ \mathbb {R}^{N}.$ It is a new problem in the sense that the signs of the coefficients of critical terms are opposite.
In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$
and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$
. We also prove an existence result of standing waves and prove their orbital stability.
$(p,q)$-LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS
We study the existence of positive ground state solutions for the following class of 
$(p,q)$-Laplacian coupled systems where 
$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$
$1<p\leq q<N$. Here the coefficient 
$\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition 
$|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where 
$\unicode[STIX]{x1D6FF}\in (0,1)$ and 
$\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.
For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,
where 
$$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$
$N\ges 3$, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and 
$I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided 
$N \ges 4$ or in dimension N = 3 if in addition 3/2 < α < 3 and 
$\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.
In this paper, we study a time-independent fractional Schrödinger equation of the form (−Δ)su + V(x)u = g(u) in ℝN, where N ≥, s ∈ (0,1) and (−Δ)s is the fractional Laplacian. By variational methods, we prove the existence of ground state solutions when V is unbounded and the nonlinearity g is subcritical and satisfies the following geometry condition:
