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Multiple single-peaked solutions of a class of semilinear Neumann problems via the category of the domain boundary. (English) Zbl 0917.35037

The main theorem of the paper establishes existence of multiple positive solutions to the semilinear Neumann problem \[ -\lambda\Delta u+ u=f(u)\quad\text{in }\Omega,\quad \nu\cdot\nabla u|_{\partial\Omega}= 0\tag{1} \] in a (possibly unbounded) domain \(\Omega\) in \(\mathbb{R}^n\), \(n\geq 2\), with smooth compact boundary. In (1), \(\lambda\) is a small positive parameter, \(\nu\) denotes a unit normal vector to \(\partial\Omega\), \(f\in C^1(\mathbb{R},\mathbb{R})\cap C^2(\mathbb{R}\setminus\{0\})\), \(f\) has support \(\mathbb{R}_+\), \(\int^a_0f>0\) for some \(a>0\), and \(f\) has subcritical superlinear growth. As usual, \(\text{Cat}(\partial\Omega)\) denotes the Lyusternik-Schnirel’man category of the boundary.
Main Theorem: Under these hypotheses, (1) has at least \(\text{Cat}(\partial\Omega)\) distinct nonconstant positive solutions \(u\in C^2(\overline\Omega)\) provided \(\lambda\) is sufficiently small. Furthermore, each such solution has a single maximum point \(x_0\) and \(x_0\in \partial\Omega\).
The constrained variational approach employs a restriction of the free functional for (1) to a suitable submanifold of the Sobolev space \(H^1(\Omega)\). The basic idea of the proof is, for small enough \(\lambda>0\), to relate homotopy properties of a sublevel set for the restricted functional to the topology of \(\partial\Omega\). The presentation involves 26 preliminary lemmas or propositions. Related results have been obtained by C.-S. Lin, W.-M. Ni and I. Takagi [J. Differ. Equations 72, No. 1, 1-27 (1988; Zbl 0676.35030)], Z.-Q. Wang [Arch. Ration. Mech. Anal. 120, No. 4, 375-399 (1992; Zbl 0784.35035)], W.-M. Ni and I. Takagi [Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)].

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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