As is already classical, linear partial differential operators \(P(D)\) with constant coefficients are not necessarily surjective on the space \({\mathcal A}(\Omega)\) of real analytic functions even if \(\Omega\) is convex. The same difficulty was found for the space of Roumieu type ultradifferentiable functions, which was caused by the similarity of the topological linear structure. The introduction of this article gives a concise but overall description to the history of this theme. Then it presents a new criterion for the surjectivity of \(P(D)\) on the space of Roumieu type non-quasianalytic ultradifferentiable functions \({\mathfrak E}_\omega(\Omega)\) for \(\Omega\) not necessarily convex: \(\Omega\) is \(P\)-convex, and there exists an exhausting sequence of compact subsets \(K_n\) of \(\Omega\) and a strictly increasing sequence \(\gamma_n>1\) such that for any \(G\in D_\omega(\Omega)'\) with \(\text{supp } G\subset\Omega\backslash K_{m+1}\) we can find \(H\in D_\omega(\Omega)'\) which is regular near \(K_m\) with moderate norm and satisfies \(P(D)H=G\). This is also equivalent even if we limit \(G\) to delta functions. Thus the result can be considered as an analogue of the characterization of surjectivity in the space of analytic functions by means of the existence of good fundamental solutions as was initiated by Kawai. This result has also a strong analogy with the existence of continuous linear right inverse for \(P(D)\) where a condition with singular support replaced by support was employed. In the case of semielliptic operators in the Gevrey space \(\Gamma^d(\Omega)\), a complete characterization is possible, which shows a gap phenomenon for the Gevrey index as was first found by Braun.