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Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains. (English) Zbl 0705.35004
Analysis, et cetera, Res. Pap. in Honor of J. Moser’s 60th Birthd., 115-164 (1990).
[For the entire collection see Zbl 0688.00009.]
Let $$S=\{x=(x_ 1,...,x_ n)$$; $$y=(x_ 2,...,x_ n)\in \omega \}$$ be a cylindrical domain in $${\mathbb{R}}^ n$$, where $$\omega$$ is a bounded domain in $${\mathbb{R}}^{n-1}=\{y=(x_ 2,...,x_ n)\}$$ with $$C^ 2$$- boundary and let $$\nu$$ denote the exterior unit normal to S at any boundary point. The authors consider equations of the form (here $$u_ 1=u_{x_ 1})$$ $(1)\quad \Delta u-\beta (u)u_ 1+f(y,u)=0\text{ in } S$ (and some more general ones) under Neumann condition (2) $$u_{\nu}=0$$ on $$\partial S$$ or Dirichlet (3) $$u=0$$ on $$\partial S$$. The solutions are supposed to belong to $$C^ 2(\bar S)$$ and, to satisfy for some constant k, (4) $$u>k$$ in S, (5) $$\lim_{x_ 1\to -\infty}u(x_ 1,y)=k$$ uniformly for $$y\in {\bar \omega}$$. In many of the results, conditions on u as $$x_ 1\to +\infty$$ are also imposed. The function f is supposed to be continuous where defined, and in many cases, to be differentiable in u; $$\beta$$ (y) is assumed to be continuous.
In the present paper the authors take up questions of the following type: Is u monotonous in $$x_ 1?$$ Is it symmetric in $$x_ 1$$ about some value? In case $$\beta =0$$, and f is odd in u, is u antisymmetric in $$x_ 1$$, about some value? If the condition $$u(x_ 1,y)\to K>k$$ as $$x_ 1\to +\infty$$ is required, is the solution unique - up to $$x_ 1$$- translation?
In the present paper the authors use: the method of moving planes of B. Gidas, W. M. Ni and L. Nirenberg [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)], and “the method of sliding domains”: shifting a solution u along the $$x_ 1$$ axis and then comparing the shifted u with another solution, or with the original u. Both methods were used in their previous paper [J. Geom. Phys. 5, No.2, 237-275 (1988; Zbl 0698.35031)]. However a new ingredient is needed to carry out these procedures: some fairly precise knowledge of the asymptotic behaviour of the solution near $$x_ 1=\pm \infty$$. The authors rely on some results of Agmon, Nirenberg and of Pasy, which are described in Section 2. These results involve “exponential solutions” of the form $$v=e^{\lambda x_ 1}\phi (y)$$ of linearized equations $(6)\quad (\Delta -\beta (y)\partial_ 1-a(y))v=0$ under boundary condition (2) or (3), where $$a(y)=-f_ u(y,k)$$. This means that $$\phi$$ (y)$$\not\equiv 0$$ satisfies $(7)\quad (-\Delta_ y+a(y))\phi =(\lambda^ 2-\lambda \beta (v))\phi$ and $$\phi$$ satisfies $$\phi_{\nu}=0$$ or $$\phi =0$$ on $$\partial \omega.$$
Section 3 is devoted to the spectral analysis of equations (7). In Section 4 the results of Sections 2 and 3 are applied to obtain asymptotic behaviour near $$(x_ 1=)+\infty$$ of solutions (1) under condition (2) or (3).
In Section 5 the authors study travelling front solutions in S satisfying (2) and (4), (5) with $$k=0$$. These investigations are related to several models in biology, chemical kinetics and combustion (see D. G. Aronson and H. F. Weinberger [Lect. Notes Math. 446, 5-49 (1975; Zbl 0325.35050)] and P. C. Fife [Lect. Notes Biomath. 28 (1979; Zbl 0403.92004)]).
Section 6 is concerned with solitary wave solutions $$u>0$$ in S, $$u(x_ 1,y)\to 0$$ as $$| x_ 1| \to \infty$$, of $$\Delta u+f(y,u)=0$$ under condition (2) or (3). In Section 7 the authors study solutions of equations $(8)\quad u-c\cdot \alpha (y)u_ 1+f(y,u)=0\text{ in } S$ and $(9)\quad u-(c+\alpha (y))u_ 1+f(y,u)=0\text{ in } S$ under the condition $(10)\quad u_{\nu}=0\text{ on } \partial S.$ In (8) $$\alpha$$ (y)$$\geq 0$$ in $$\omega$$ and in (9) $$\alpha$$ (y) is a given function and the constant c is to be determined. More precisely the authors study solutions of (8), (9) under (10) satisfying the assumptions: $$k<u<K$$; $$u(x_ 1,y)\to K$$ as $$x_ 1\to +\infty$$. These investigations have connections with the work of the first author and B. Larrouturou [J. Reine Angew. Math. 396, 14-40 (1989; Zbl 0658.35036)].
Reviewer: I.J.Bakelman

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations