Ackermann, Nils Multiple single-peaked solutions of a class of semilinear Neumann problems via the category of the domain boundary. (English) Zbl 0917.35037 Calc. Var. Partial Differ. Equ. 7, No. 3, 263-292 (1998). 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)]. Reviewer: Charles A.Swanson (Vancouver) Cited in 1 Document 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 Keywords:homotopy of sublevel set; multiple positive solutions; Lyusternik-Schnirel’man category; constrained variational approach Citations:Zbl 0676.35030; Zbl 0784.35035; Zbl 0796.35056 × Cite Format Result Cite Review PDF Full Text: DOI