Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I. (English) Zbl 1072.35019

Ambrosetti, Antonio; Malchiodi, Andrea; Ni, Wei-Ming

Commun. Math. Phys. 235, No. 3, 427-466 (2003).

Summary: We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like \(-\varepsilon^2 \Delta u + V(| x|)u = u^p\), \(u\in H^1(\mathbb R^n)\). Under suitable assumptions on the auxiliary potential \(M(r) = r^{n-1} V^\theta(r)\), \(\theta(p+1)/(p-1)-1/2\), we provide necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions.
For Part II, see Indiana Univ. Math. J. 53, No. 2, 297–329 (2004; Zbl 1081.35008).

MSC:

35B25	Singular perturbations in context of PDEs
35B32	Bifurcations in context of PDEs
35J60	Nonlinear elliptic equations
35Q55	NLS equations (nonlinear Schrödinger equations)