## Sign-changing solutions for an asymptotically linear Schrödinger equation with deepening potential well.(English)Zbl 1221.35114

The existence of sign-changing solutions for the asymptotically linear Schrödinger equation (P): $$- \Delta u+\lambda g(x)u=f(u), x\in\mathbb R^N$$, $$N \geq3$$, deepening potential well $$g$$ and $$\lambda>0$$ is investigated. The following assumptions are made on $$g$$ and $$f$$: the potential $$g$$, an essentially bounded function, $$0 \leq g(x) \leq 1$$, $$\forall x \in\mathbb R^N$$, is identically 0 on a bounded nonempty domain and $$\lim_{|x|\rightarrow \infty} g(x) = 1$$ and the function $$f$$ is continuous, $$tf(t)>0$$, $$\forall t \in\mathbb R$$, $$t \neq0$$, $$\lim_{t \rightarrow 0} {f(t) \over t} = 0$$, it is asymptotically linear (i.e., there exists $$\alpha \in (0,\infty)$$ such that $$\lim_{|t|\rightarrow \infty} {f(t) \over t} = \alpha$$) and $${f(t) \over t\operatorname{sgn}t}$$ is nondecreasing.
The authors prove, by using variational methods, such as the method of invariant sets and the concentration-compactness principle, that if $$\lambda\geq \alpha> \alpha_2(\lambda)$$ and $$\alpha \not\in \sigma_p(L_\lambda)$$ or if $$\alpha > \lambda$$, $$\alpha \not\in \sigma_{p}(L_{\lambda})$$ and there exist $$c, m, R_{0} > 0$$ such that $$1-g(x) > c|x|^{-m}$$, $$\forall|x|\geq r_{0}$$ then the equation (P) has a sign-changing solution. Here $$L_{\lambda}u= -\Delta u + \lambda g(x)u, \forall u \in H^{1}(\mathbb R^N)$$ and $$\alpha_2(\lambda)$$ is the second eigenvalue of $$L_{\lambda}$$.

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35Q55 NLS equations (nonlinear Schrödinger equations) 35J20 Variational methods for second-order elliptic equations 35J60 Nonlinear elliptic equations 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics