We present some results concerning the problem $-\Delta u=\lambda \frac{u}{{\left|x\right|}^2}+u^q$, $u>0$ in $\Omega$, ${u|}_{\partial \Omega }=0$, where $0<q<(N+2)/\left(N-2\right)$, $q\ne 1$, $\lambda \ge 0$ and $\Omega$ is a smooth bounded domain containing the origin. In particular, bifurcation and uniqueness results are discussed.

We give a sufficient condition for [μ-M, μ+M] × {0} to be a bifurcation interval of the equation u = L(λu + F(u)), where L is a linear symmetric operator in a Hilbert space, μ ∈ r(L) is of odd multiplicity, and F is a nonlinear operator. This abstract result provides an elementary proof of the existence of bifurcation intervals for some eigenvalue problems with nondifferentiable nonlinearities. All the results obtained may be easily transferred to the case of bifurcation from infinity.

The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms: $$\left\{\begin{array}{cc}{(-\Delta )}^{s}u+V\left(x\right)u+\phi u-\frac{1}{2}u{(-\Delta )}^s{u}^2=f(x,u),\hfill & x\in {ℝ}^3,\hfill \\ {(-\Delta )}^t\phi =u^2,\hfill & x\in {ℝ}^3,\hfill \end{array}\right.$$ where ${(-\Delta )}^{\alpha }$ is the fractional Laplacian for $\alpha =s$, $t\in (0,1]$ with $s<t$ and $2t+4s>3$. Under assumptions on $V$ and $f$, we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem.