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