Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions

Abstract

In this paper, in order to solve an elliptic partial differential equation with a nonlinear boundary condition for multiple solutions, the authors combine a minimax approach with a boundary integral-boundary element method, and identify a subspace and its special expression so that all numerical computation and analysis can be carried out more efficiently based on information of functions only on the boundary. Some mathematical justification of the new approach is established. An efficient and reliable local minimax-boundary element method is developed to numerically search for solutions. Details on implementation of the algorithm are also addressed. The existence and multiplicity of solutions to the problem are established under certain regular assumptions. Some conditions related to convergence of the algorithm and instability of solutions found by the algorithm are verified. To illustrated the method, numerical multiple solutions to some examples on domains with different geometry are displayed with their profile and contour plots.

Normalized Goldstein-type local minimax method for finding multiple unstable solutions of semilinear elliptic PDEs

“…Nowadays, with the development of new experimental techniques, it has been possible to observe local unstable equilibria or transient excited states in numerous physical/chemical/biological systems. Consequently, their theoretical and numerical studies have attracted increasing attentions [2,4,15,21,27,30]. However, these local unstable equilibria or transient excited states are related to critical points that are not local extrema, and then called saddle points.…”

Existence of Positive Solution to Schrödinger-type Semipositone Problems with Mixed Nonlinear Boundary Conditions

“…Nonlinear Schrödinger equations have been intensively studied to demonstrate the existence of solutions that act according to V on the whole space R N (see [5,24,32] or references therein) or on bounded domains with linear boundary conditions under the assumption f (0) ≥ 0 [19]. A numerical solution was treated in [28] for linear Schrödinger equations with a nonlinear boundary condition.…”

“…(ii) The condition on g is restrictive. For the more general case for g, we refer to [28], where it is treated by a linear Schrödinger equation.…”

Convergence for a planar elliptic problem with large exponent Neumann data

“…It was first proved in [19] for more general nonlinearities, that there exists at least one solution which changes sign. If the nonlinearity is odd in u, as in our case, it is mentioned in [19] that there exist infinitely many sign-changing solutions by a standard argument (see the reference therein), so it makes sense to study the properties of both positive and sign-changing solutions.…”

“…where p > 1, Ω ⊂ R 2 is a smooth bounded domain and ∂/∂ν denotes the derivative with respect to the outward normal to ∂Ω. Elliptic problem with nonlinear boundary condition has been widely studied in the past by many authors and it is still an area of intensive research, see for instance [2,5,6,7,8,17,19,26]. Problem (1.1) has a variational structure.…”