We verify the existence of radial positive solutions for the semi-linear equation −∆u = up − V(y)uq, u >0, in RN where N ≥ 3, p is close to p∗ := (N + 2)/(N−2), and V is a radial smooth potential. If q is super-critical, namely q > p∗, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with diﬀerent rates of concentration, provided V(0) < 0. On the other hand, if N/(N −2) < q < p∗, we prove that this Problem has a radial solution behaving like a super-position of ﬂat bubbles with diﬀerent rates of concentration, provided limr→∞V(r) < 0.
Musso, M., & Pimentel, J. (2020). A semilinear elliptic equation with competing powers and a radial potential. Journal d'Analyse Mathematique, 140(1), 283-298. https://doi.org/10.1007/s11854-020-0089-4