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Nontrivial solution of a semilinear Schrödinger equation. (English) Zbl 0864.35036
The authors consider the nonlinear, stationary Schrödinger equation $-\Delta u+V(\cdot)u= f(\cdot,u)$ in $\bbfR^n$. Assume that $V\in C(\bbfR^n)$ and $f\in C^1(\bbfR^n\times \bbfR)$ are 1-periodic in $x_k$, $1\leq k\leq n$, and $D:H^2(\bbfR^n)\to L^2(\bbfR^n)$, $u\mapsto-\Delta u+V(\cdot)u$ is invertible. Moreover, let $f$ satisfy $|f_u(x,u)|\le\text{const}(|u|^{q-2}+|u|^{p-2})$ for $2<q\leq p<2^*$ $(={{2n}\over{n-2}}$ for $n\geq 3$, $=\infty$ otherwise). Then the equation has a nontrivial solution $u\in H^1(\bbfR^n)$, provided that $0<\alpha F(x,u)\leq f(x,u)u$, $u\ne 0$, with some constant $\alpha>2$ and $F(x,u):= \int^u_0 f(x,t)dt$. To obtain this solution, the existence of a Palais-Smale sequence to a suitable level of the functional $${\cal E}(u):= \int_{\bbfR^n} {\textstyle{1\over2}} |\nabla u(x)|^2+ {\textstyle{1\over2}} V(x)u^2(x)- F(x,u(x))dx, \qquad u\in H^1(\bbfR^n),$$ is proved, i.e., there is a $c\in(0,\infty)$ and $\{u_n\}\subset H^1(\bbfR^n)$ such that ${\cal E}(u_n)\to c$ and $\nabla{\cal E}(u_n)\to 0$ in $H^1(\bbfR^n)$. The difficulty here is that ${\cal E}$ satisfies no Palais-Smale condition because of the periodicity. Moreover, the functional is strongly indefinite since 0 lies in a spectral gap of the linear operator $D$.

35J60Nonlinear elliptic equations
49J35Minimax problems (existence)
35J20Second order elliptic equations, variational methods
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