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)].

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 |