In this paper, we study a class of $p(x)$-biharmonic problems with negative nonlocal terms and weight function. Using the mountain pass theorem and the Ekeland's variational principle, at least three solutions are obtained. We also give some comments on the existence of infinite many solutions for our problem when the nonlinear term is a general function.

In this paper, we study the existence of infinitely many normalized radial solutions for the following quasilinear Schrödinger-Poisson equations:

$$-\mathrm{\Delta }u-\lambda u+(|x{|}^{-1}\ast |u{|}^2)u-\mathrm{\Delta }(u^2)u-|u{|}^{p-2}u=0,x\in {\mathbb{R}}^3,$$

where $p\in (\frac{10}{3},6)$, $\lambda \in \mathbb{R}$. Firstly, the quasilinear equations are transformed into semilinear equations by making a appropriate change of variables, whose associated variational functionals are well defined in $H_r^1({\mathbb{R}}^3)$. Secondly, by constructing auxiliary functional and combining pohožaev identity, we prove that under constraints, the energy functionals related to the equation have bounded Palais-Smale sequences on each level set. Finally, it is obtained that there are infinitely many normalized radial solutions for this kind of quasilinear Schrödinger-Poisson equations.

In this paper, we investigate how to compute the minimum distance between a point and a parametric surface, and then to return the nearest point (foot point) on the surface as well as its corresponding parameter, which is also called the point projection problem of a parametric surface. The geometric strategy algorithm (hereafter GSA) presented consists of two parts as follows. The normal curvature to a given parametric surface is used to find the corresponding foot point firstly, and then the Taylor’s expansion of the parametric surface is employed to compute parameter increments and to get the iteration formula to calculate the orthogonal projection point of test point to the parametric surface. Our geometric strategy algorithm is essentially dependent on the geometric property of the normal curvature, and performs better than existing methods in two ways. Firstly, GSA converges faster than existing methods, such as the method to turn the problem into a root-finding of nonlinear system, subdividing methods, clipping methods, geometric methods (tangent vector and geometric curvature) and hybrid second-order method, etc. Specially, it converges faster than the classical Newton’s iterative method. Secondly, GSA is independent of the initial iterative value, which we prove in Theorem 1. Many numerical examples confirm GSA’s robustness and efficiency.