In decision-making process, decision-makers may make different decisions because of their different experiences and knowledge. The abnormal preference value given by the biased decision-maker (the value that is too large or too small in the original data) may affect the decision result. To make the decision fair and objective, this paper combines the advantages of the power average (PA) operator and the Bonferroni mean (BM) operator to define the generalized fuzzy soft power Bonferroni mean (GFSPBM) operator and the generalized fuzzy soft weighted power Bonferroni mean (GFSWPBM) operator. The new operator not only considers the overall balance between data and information but also considers the possible interrelationships between attributes. The excellent properties and special cases of these ensemble operators are studied. On this basis, the idea of the bidirectional projection method based on the GFSWPBM operator is introduced, and a multi-attribute decision-making method, with a correlation between attributes, is proposed. The decision method proposed in this paper is applied to a software selection problem and compared to the existing methods to verify the effectiveness and feasibility of the proposed method.

In this paper, we study the following Schrödinger–Kirchhoff type equation: $\begin{array}{rl}& (a+b{\iint }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+ps}}\mathrm{d}x\mathrm{d}y){(-\mathrm{\Delta })}_p^{s}u+\lambda V(x)|u{|}^{p-2}u\\ & =h(x)|u{|}^{r-2}u+g(x)|u{|}^{q-2}u,\end{array}$ $x\in {\mathbb{R}}^N$, where $s\in (0,1)$, $2\le p<\mathrm{\infty }$, N>ps, $1<q<p<r<p_s^{\ast }$ and $a,\lambda >0$, $b\ge 0$ are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution $u_{b,\lambda }^{+}$ by a truncated functional and discuss their asymptotical behavior for $\lambda \to +\mathrm{\infty }$ when p<r0, we obtain a positive energy solution $u_{\lambda }^{+}$ and discuss their asymptotical behavior for $\lambda \to +\mathrm{\infty }$ when $2p\le r<p_s^{\ast }$, where $p_s^{\ast }=\frac{Np}{N-sp}$. Moreover, we obtain other asymptotical behavior of positive energy solution when $p<r<p_s^{\ast }$. Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation $ M\left(\,\,\displaystyle{\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\text{d}x\text{d}y}\right)(-\Delta)_{p}^{s} u+V(x)|u|^{p-2}u = \lambda f(x,u),\,\, \text{in}\,\,\mathbb{R}^N, $ where $0 < s < 1 < p < \infty$, $sp < N$, $\lambda > 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.