The effect of the graph topology on the existence of multipeaksolutions for nonlinear Schrödinger equations

Infinitely many positive solutions for the nonlinear Schrödinger equations in $${\mathbb{R}^N}$$

“…− ε 2 u + V (y)u = u p , u > 0, lim |y|→+∞ u(y) = 0, (1.5) or − ε 2 u + u = Q(y)u p , u > 0, lim |y|→+∞ u(y) = 0, (1.6) where ε > 0 is a small parameter, then the number of the critical points of V (y) (or Q(y)) (see for example [1,7,8], [13]- [16], [19,27,29]), the type of the critical points of V (y) (or Q(y)) (see for example [11,21,26], and the topology of the level set of V (y) (or Q(y)) [2,3,12,18], can affect the number of the solutions for (1.5) (or (1.6)). But for the singularly perturbed problems (1.5) and (1.6), the parameter ε will go to zero as the number of the solutions goes to infinity.…”

Bound states to critical quasilinear Schrödinger equations

“…In [15,17], they constructed a family of solutions with several spikes located around any prescribed finite set of local minima of V . For more results of this type, we refer to [6][7][8][9][10][11]13].…”

On the superlinear Lazer–McKenna conjecture

“…Similar phenomenon occurs in other elliptic problems, such as nonlinear Schrödinger equations and nonlinear scalar field equations. See for example [16,17,22,30].…”