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Suppose that $\omega_1,\ldots,\omega_N$ are positive real numbers and $x$ isa complex number with positive real part. The multiple Barnes-Euler zetafunction $\zeta_{E,N}(s,x;\bar\omega)$ with parameter vector$\bar\omega=(\omega_1,\ldots,\omega_N)$ is defined as a deformation of theBarnes multiple zeta function as follows $$\zeta_{E,N}(s,x;\bar\omega)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty\frac{(-1)^{t_1+\cdots+t_N}}{(x+\omega_1t_1+\cdots+\omega_Nt_N)^s}. $$  In this paper, based on the fermionic $p$-adic integral, we define the$p$-adic analogue of multiple Barnes-Euler zeta function$\zeta_{E,N}(s,x;\bar\omega)$ which we denote by$\zeta_{p,E,N}(s,x;\bar\omega).$ We prove several properties of$\zeta_{p,E,N}(s,x; \bar\omega)$, including the convergent Laurent seriesexpansion, the distribution formula, the difference equation, the reflectionfunctional equation and the derivative formula. By computing the values of thiskind of $p$-adic zeta function at nonpositive integers, we show that itinterpolates the higher order Euler polynomials $E_{N,n}(x;\bar\omega)$$p$-adically.  Furthermore, we define the corresponding multiple $p$-adic Diamond-Euler LogGamma function. We also show that the multiple $p$-adic Diamond-Euler Log Gammafunction ${\rm Log}\, \Gamma_{\! D,E,N}(x;\bar\omega)$ has an integralrepresentation by the multiple fermionic $p$-adic integral, and it satisfiesthe distribution formula, the difference equation, the reflection functionalequation, the derivative formula and also the Stirling's series expansions.

On $p$-adic multiple Barnes-Euler zeta functions and the corresponding log gamma functions