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Generalized linking theorem with an application to a semilinear Schrödinger equation. (English) Zbl 0947.35061
Summary: Consider the semilinear Schrödinger equation $-\Delta u+ V(x)u= f(x,u),\quad u\in H^1(\mathbb{R}^N).\tag{$$*$$}$ It is shown that if $$f$$, $$V$$ are periodic in the $$x$$-variables, $$f$$ is superlinear at $$u= 0$$ and $$\pm\infty$$ and 0 lies in a spectral gap of $$-\Delta+V$$, then $$(*)$$ has at least one nontrivial solution. If in addition $$f$$ is odd in $$u$$, then $$(*)$$ has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.

##### MSC:
 35J60 Nonlinear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35J10 Schrödinger operator, Schrödinger equation 47H11 Degree theory for nonlinear operators
##### Keywords:
subcritical; periodic; superlinear; spectral gap; degree-theory