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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