A discussion of nonnegative solutions of elliptic equations on symmetric domains∗

نویسندگان

  • P. Poláčik
  • Hiroshi Matano
چکیده

In this note we summarize our recent results on nonnegative solutions of nonlinear elliptic equations on reflectionally symmetric domains. We discuss symmetry properties of such solutions, the structure of their nodal set, and the existence and multiplicity of solutions with a nontrivial nodal set.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains

We consider a semilinear elliptic equation on a smooth bounded domain Ω in R2, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in x for x > 0. Our goal...

متن کامل

Symmetry of nonnegative solutions of elliptic equations via a result of Serrin

We consider the Dirichlet problem for semilinear elliptic equations on a smooth bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction orthogonal to H. Employing Serrin’s result on an overdetermined problem, we show that any nonzero nonnegative solution is necessarily strictly positive. One can thus apply a well-known result of Gidas, Ni and Nirenberg t...

متن کامل

On symmetry of nonnegative solutions of elliptic equations

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction...

متن کامل

A numerical method for solving nonlinear partial differential equations based on Sinc-Galerkin method

In this paper, we consider two dimensional nonlinear elliptic equations of the form $ -{rm div}(a(u,nabla u)) = f $. Then, in order to solve these equations on rectangular domains, we propose a numerical method based on Sinc-Galerkin method. Finally, the presented method is tested on some examples. Numerical results show the accuracy and reliability of the proposed method.

متن کامل

Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form ∆u = ±a2(|x|)u|u|q−1, with q > 0,...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2013