Multiple Solutions Of Nonlinear Problems Research Articles

Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian

This paper is devoted to the existence of nodal and multiple solutions of nonlinear problems involving the fractional Laplacian{(−Δ)su=f(x,u)in Ω,u=0on ∂Ω, where Ω⊂Rn (n⩾2) is a bounded smooth domain, s∈(0,1), (−Δ)s stands for the fractional Laplacian. When f is superlinear and subcritical, we prove the existence of a positive solution, a negative solution and a nodal solution. If f(x,u) is odd in u, we obtain an unbounded sequence of nodal solutions. In addition, the number of nodal domains of the nodal solutions are investigated.

Improved spherical continuation algorithm with application to the double-bounded homotopy (DBH)

The homotopy continuation methods are useful tools for finding multiple solutions of nonlinear problems. An important issue of this kind of method is the correct implementation of the path-following techniques used to trace the homotopy trajectory. Therefore, in this work we propose a modification of the spherical algorithm to successfully trace the closed paths of a DBH homotopy. The proposed methodology is depicted with three examples. Finally, a comparison of the results with a standard path-following technique is presented and discussed.

Analysis of search-extension method for finding multiple solutions of nonlinear problem

For numerical computations of multiple solutions of the nonlinear elliptic problem Δu + f(u = 0 in Ω, u = 0 on Γ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u ∈ H 1+α, α > 0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.

Finding multiple solutions of nonlinear problems by means of the homotopy analysis method

Using the boundary layer flows over a permeable plate as an example, we show that the homotopy analysis method (HAM) can be applied to give series solutions of all branches of multiple solutions, even if these multiple solutions are very close and thus rather difficult to distinct even by numerical techniques. A new branch of solutions is found, which has never been reported. This indicates that the homotopy analysis method is a powerful tool for strongly nonlinear problems, especially for those having multiple solutions.

Finding multiple solutions of nonlinear problems by means of the homotopy analysis method

Finding multiple solutions of nonlinear problems by means of the homotopy analysis method

Search-extension method for finding multiple solutions of nonlinear problem

Based on the eigensystem {λj,Oj} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω, u = 0 on ∂Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces. Numerical experiments for f(u) = u 3 in a square and L-shape domain are presented. The results show that there exist at least 3 k -1 distinct nonzero solutions corresponding to each k-ple eigenvalue of -Δ (Conjecture 1).