Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

Abstract

We consider the standing-wave problem for a nonlinear Schrodinger equation, corresponding to the semilinear elliptic problem $$\begin{aligned} -\Delta u+V(x)u=|u|^{p-1}u, u\in H^1(\mathbb {R}^2), \end{aligned}$$ where $$V(x)$$ is a uniformly positive potential and $$p>1$$ . Assuming that $$\begin{aligned} V(x)=V_\infty +\frac{a}{|x|^m}+O\left( \frac{1}{|x|^{m+\sigma }}\right) \text {as} |x|\rightarrow +\infty , \end{aligned}$$ for instance if $$p>2$$ , $$m>2$$ and $$\sigma >1$$ we prove the existence of infinitely many positive solutions. If $$V(x)$$ is radially symmetric, this result was proved in [43]. The proof without symmetries is much more difficult, and for that we develop a new intermediate Lyapunov–Schmidt reduction method, which is a compromise between the finite and infinite dimensional versions of it.