I have a gamma distributed random variable $x$ with pdf $$p_x(x)=\frac{\lambda^r}{\Gamma(r)}x^{r-1}\exp(-\lambda x),$$ where $r$ and $\lambda$ are shape and rate parameters respectively.
If for example I multiply $x$ by a constant value $a$ so $y=ax$, what will be the distribution of $y$? How can we introduce the mean value of $y$ in the pdf expression ?
If $X\sim\text{Gamma}(r,\lambda)$, by which I mean it has the density that you provided, then for $a>0$, $$Y =aX\sim \text{Gamma}(r,\lambda/a).$$ This implies that the $E[Y] = r/(\lambda/a)$.
To prove this, notice that $Y = aX$ is one to one over the support of $X$, and hence we can use the one to one transformation; $X= Y/a$, and so $$f_Y(y) = \frac{f_X(y/a)}{\left|\frac{dy}{dx}\right|_{x = y/a}}.$$
Otherwise, you can use the cdf method, $$P(Y\leq y) = P(aX\leq y) = P(X\leq y/a).$$