Saddle-shaped Solutions of Bistable Elliptic Equations Involving the Half-laplacian
نویسنده
چکیده
We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (−∆)u = f(u) in all the space R, where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions 2m = 4 and 2m = 6. These results are relevant in connection with the analog for fractional equations of a conjecture of De Giorgi on the 1-D symmetry of certain solutions. Saddleshaped solutions are the simplest candidates, besides 1-D solutions, to be global minimizers in high dimensions, a property not yet established.
منابع مشابه
Qualitative Properties of Saddle-shaped Solutions to Bistable Diffusion Equations
We consider the elliptic equation −∆u = f(u) in the whole R , where f is of bistable type. It is known that there exists a saddleshaped solution in R. This is a solution which changes sign in R and vanishes only on the Simons cone C = {(x, x) ∈ R×R : |x| = |x|}. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when 2m = 6 every saddle-s...
متن کاملThe Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent
In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.
متن کاملSaddle-shaped Solutions of Bistable Diffusion Equations in All of R
We study the existence and instability properties of saddleshaped solutions of the semilinear elliptic equation −∆u = f(u) in the whole R, where f is of bistable type. It is known that in dimension 2m = 2 there exists a saddle-shaped solution. This is a solution which changes sign in R and vanishes only on {|x1| = |x2|}. It is also known that this solution is unstable. In this article we prove ...
متن کاملExistence of three solutions for a class of quasilinear elliptic systems involving the $p(x)$-Laplace operator
The aim of this paper is to obtain three weak solutions for the Dirichlet quasilinear elliptic systems on a bonded domain. Our technical approach is based on the general three critical points theorem obtained by Ricceri.
متن کاملEnergy Estimates and 1-d Symmetry for Nonlinear Equations Involving the Half-laplacian
We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation (−∆)u = f(u) in R. Our energy estimates hold for every nonlinearity f and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimen...
متن کامل