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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

Citations:

Zbl 0785.35029
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