The paper concerns with positive solutions of problems of the type \(-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}\) in \(\Omega \subseteq \mathbb {R}^N\), \(N\ge 3\), \(2^*={2N\over N-2}\), \(2<p<2^*\). Here \(\Omega \) can be an exterior domain, i.e. \(\mathbb {R}^N{\setminus }\Omega \) is bounded, or the whole of \(\mathbb {R}^N\). The potential \(a\in L^{N/2}_{\mathop {\mathrm{loc}}\nolimits }(\mathbb {R}^N)\) is assumed to be strictly positive and such that there exists \(\lim _{|x|\rightarrow \infty }a(x):=a_\infty >0\). First, some existence results of ground state solutions are proved. Then the case \(a(x)\ge a_\infty \) is considered, with \(a(x)\not \equiv a_\infty \) or \(\Omega \ne \mathbb {R}^N\). In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small \({\varepsilon }\).

纸张关切的类型的问题正解\（ - \德尔塔U + A（X）\，为u = u ^ {P-1} + \ varepsilonÚ^ {2 ^ * - 1} \）在\（ \ Omega \ subseteq \ mathbb {R} ^ N \），\（N \ ge 3 \），\（2 ^ * = {2N \ over N-2} \），\（2 <p <2 ^ * \ ）。在这里\（\ Omega \）可以是一个外部域，即\（\ mathbb {R} ^ N {\ setminus} \ Omega \）有界，或者是整个\（\ mathbb {R} ^ N \）。假定势\（a \ in L ^ {N / 2} _ {\ mathop {\ mathrm {loc}} \ nolimits}（\ mathbb {R} ^ N）\）严格为正，因此存在\（\ lim _ {| x | \ rightarrow \ infty} a（x）：= a_ \ infty> 0 \）。首先，证明了基态解的一些存在性结果。然后考虑\（a（x）\ ge a_ \ infty \）的情况，其中\（a（x）\ not \ equiv a_ \ infty \）或\（\ Omega \ ne \ mathbb {R} ^ N \ ）。在这种情况下，对于小\（{\ varepsilon} \），不存在基态解，并且证明了束缚态解的存在。