# zbMATH — the first resource for mathematics

Inequalities for second-order elliptic equations with applications to unbounded domains. I. (English) Zbl 0860.35004
In recent papers, the authors have studied symmetry and monotonicity properties for positive solutions $$u$$ of elliptic equations of the form $u>0,\;\Delta u+f(u)=0 \text{ in } \Omega, \quad u=0 \text{ on } \partial\Omega \tag{1}$ in several classes of unbounded domains $$\Omega$$ in $$\mathbb{R}^n$$. Here they continue this program by considering another type of domain, $$\Omega= \mathbb{R}^{n-j} \times\omega$$, where $$\omega$$ is a smooth bounded domain in $$\mathbb{R}^j$$.
Denote by $$x=(x_1, \dots, x_{n-j})$$ the coordinates in $$\mathbb{R}^{n-j}$$, and by $$y=(y_1, \dots, y_j)$$ the coordinates in $$\omega$$. The goal is to establish symmetry of solutions of (1) corresponding to symmetries of $$\omega$$. For example, if $$\omega$$ is a ball $$\{|y |<R\}$$, they prove that any solution of (1) depends only on $$|y|$$ and $$x$$, and is decreasing in $$|y|$$. Note that $$u$$ is not assumed to be bounded. Throughout the paper it is assumed that $$f$$ is Lipschitz continuous, with Lipschitz constant $$k$$, on $$\mathbb{R}^+$$ (or on $$[0, \sup u]$$ in the case where $$u$$ is bounded).
Reviewer: V.Mustonen (Oulu)

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
semilinear elliptic equation; symmetry of solutions
Full Text:
##### References:
  C. J. Amick and J. F. Toland, Nonlinear elliptic eigenvalue problems on an infinite strip. Global theory of bifurcation and asymptotic bifurcation , Math. Ann. 262 (1983), no. 3, 313-342. · Zbl 0489.35067  P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints , Ark. Mat. 22 (1984), no. 2, 153-173. · Zbl 0557.35033  H. Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques , J. Funct. Anal. 40 (1981), no. 1, 1-29. · Zbl 0452.35038  H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Uniform estimates for regularization of free boundary problems , Analysis and Partial Differential Equations ed. C. Sadosky, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 567-619. · Zbl 0702.35252  H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Symmetry for elliptic equations in a half space , Boundary Value Problems for Partial Differential Equations and Applications ed. J. L. Lions, et al., RMA Res. Notes Appl. Math., vol. 29, Masson, Paris, 1993, pp. 27-42. · Zbl 0793.35034  H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain , in preparation. · Zbl 0906.35035  H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains. II: Symmetry in infinite strips , · Zbl 0860.35004  H. Berestycki and L. Nirenberg, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains , Analysis, Et Cetera ed. P. Rabinowitz, et al., Academic Press, Boston, 1990, pp. 115-164. · Zbl 0705.35004  H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method , Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1-37. · Zbl 0784.35025  H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains , Comm. Pure Appl. Math. 47 (1994), no. 1, 47-92. · Zbl 0806.35129  J. L. Bona, D. K. Bose, and R. E. L. Turner, Finite-amplitude steady waves in stratified fluids , J. Math. Pures Appl. (9) 62 (1983), no. 4, 389-439 (1984). · Zbl 0491.35049  L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form , Indiana J. Math. 30 (1981), no. 4, 621-640. · Zbl 0512.35038  M. Esteban, Nonlinear elliptic problems in strip-like domains. Symmetry of positive vortex rings , Nonlinear Anal. 7 (1983), no. 4, 365-379. · Zbl 0513.35035  B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle , Comm. Math. Phys. 6 (1981), 883-901. · Zbl 0425.35020  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order , 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001  P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. · Zbl 0704.49004  M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations , Prentice-Hall, Englewood Cliffs, N.J., 1967. · Zbl 0549.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.