An optimization problem is formulated as
\begin{equation} \min_{\lambda_k,p_k} \sum_{i=1}^K\lambda_k p_k\\ \begin{aligned} s.t:&\sum_{i=1}^K\lambda_k p_k^{2/\alpha}\ge\max(\widehat{T},\widehat{\varepsilon})\\ &0\le p_k\le p_{k,max}\\ &0\le \lambda_k \le \lambda_{k,max} \end{aligned} \end{equation}
where all arguements are constant except $\lambda_k$ and $p_k$. $\alpha$ is greater than 2.
I want to know whether there is an optimal solution? And when exists, how to get the solution. On the contrary, if there is not a solution, what operations are needed to transfer it to a solvable form.
Thanks a lot! : )