## Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity.(English)Zbl 1142.35422

The paper is concerned with the semilinear elliptic equation $$-\Delta u+b(x)u = f(u)$$, $$x\in\mathbb{R}^N$$, where $$b(x)\geq b_0>0$$ for all $$x$$, $$b$$ is periodic in $$x_1,\ldots,x_N$$, $$f(s)/s\to 0$$ as $$s\to 0$$ and $$f(s)/s\to a > \inf\sigma(-\Delta+b)$$ as $$| s| \to\infty$$. The energy functional $$J$$ associated with this equation has the mountain pass geometry in the Sobolev space $$W^{1,2}(\mathbb{R}^N)$$ and critical points of $$J$$ correspond to solutions $$u(x)$$ decaying to 0 as $$| x| \to\infty$$. The main result asserts that under certain additional hypotheses on $$f$$, if $$c$$ denotes the mountain pass level and there are finitely many geometrically distinct solutions below the level $$c+\alpha$$ for some $$\alpha>0$$, then for each $$k>1$$ there are infinitely many geometrically distinct solutions $$u$$ with $$J(u)\in [kc-\alpha,kc+\alpha]$$.
This extends a result by V. Coti Zelati and P. H. Rabinowitz [Commun. Pure Appl. Math. 45, 1217–1269 (1992; Zbl 0785.35029)] who considered a similar problem for superlinear $$f$$. Although the main idea of the proof is rather similar to that in the above-mentioned paper, there are also some major differences. In particular, the study of the Palais-Smale sequences requires new arguments.

### MSC:

 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 47J30 Variational methods involving nonlinear operators

Zbl 0785.35029
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