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We consider the singularly perturbed nonlinear elliptic problem ε² - Δ V (x)v + f(v) = 0, v > 0, lim v(x) = 0. |x|→ ∞ Under almost optimal conditions for the potential V and the nonlinearity f, we establish the existence of single-peak solutions whose peak points converge to local minimum points of V as ε → 0. Moreover, we exhibit a threshold on the condition of V at infinity between existence and nonexistence of solutions. [ABSTRACT FROM AUTHOR]

For N = 1,2, we consider singularly perturbed elliptic equations ε2Δ u - V(x) u + f(u)= 0, u(x)> 0 on RN, lim|x|→∞u(x)= 0. For small ε > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007). [ABSTRACT FROM AUTHOR]

For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti-Malchiodi-Ni [3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3]. [ABSTRACT FROM AUTHOR]