Two numerical methods for finding multiple solutions of a superlinear Dirichlet problem

Abstract

Using numerical methods based on variational method, we will obtain positive, negative, and exactly-once sign-changing solutions to the boundary value problems of −Δu = f(u) for x ∈ Ω with Dirichlet boundary condition.

Introduction

In this work, we seek numerical solutions to the boundary value problem

$$\left\{\begin{array}{cc}-\mathrm{\Delta }u(x)+f(u)=0\text{,}\hfill & x\in \Omega \text{,}\hfill \\ u(x)=0\text{,}\hfill & x\in \mathrm{\partial }\Omega \text{,}\hfill \end{array}\right.$$

where Ω is a bounded domain in R^N with smooth boundary, Δ is the standard Laplacian operator, and f ∈ C¹(R, R) such that f(0) = 0.

We assume that there exist constants A > 0 and $p\in \left(1\text{,}\frac{N+2}{N-2}\right)$ such that ∣f′(u)∣ ⩽ A(∣u∣^p−1 + 1) for all u ∈ R. It follows that f is subcritical, i.e., there exists m ∈ (0, 1) such that

$$\frac{m}{2}f(u)u⩾F(u)\text{,}$$

where $F(u)={\int }_0^{u}f(s)\mathrm{d}s$, for all u ∈ R. A vital assumption that we make is that f is superlinear, i.e.,

$$\underset{|u|\to \infty }{\lim }\frac{f(u)}{u}=\infty \text{.}$$

Finally, we make the assumption that f satisfies

$$f^{\prime }(u)>\frac{f(u)}{u}\text{for}u\ne 0\text{.}$$

Let H be Sobolev space $H_0^{1\text{,}2}(\Omega )$, in which case the zero Dirichlet conditions allow the inner product $〈u\text{,}v〉={\int }_{\Omega }\nabla u·\nabla v\mathrm{d}x$ (see [1]). We define the action functional $J:H\to \mathcal{R}$ by

$$J(u)={\int }_{\Omega }\left\{\frac{1}{2}|\nabla u{|}^2-F(u)\right\}\mathrm{d}x\text{.}$$

One easily sees that critical points of J are weak solution to (1). In fact by regularity theory for elliptic boundary value problems u is a solution (classical) to (1) if and only if u is a critical point of the functional J.

When J is bounded below on H, J has a minimizer on H which is a critical point of J. In many cases such as (1) J is not bounded below on H, but is bounded below on an appropriate subset of H and a minimizer on this set (if it exists) may give rise to solution of the corresponding differential equation. A good candidate for an appropriate subset of H are

$$\begin{array}{cc}\hfill & S=\{u\in H-\{0\}:〈J^{\prime }(u)\text{,}u〉=0\}\text{,}\hfill \\ \hfill & S_1=\{u\in S:u_{+}\ne 0\text{,}{u}_{-}\ne 0\text{,}〈J^{\prime }(u_{+})\text{,}{u}_{+}〉=0\}\text{,}\hfill \end{array}$$

where we note that nontrivial solutions to (1) are in S (a closed subset of H) and sign-changing solutions are in S₁ (a closed subset of S).

Let 0 < λ₁ < λ₂ ⩽ λ₃ ⩽ ⋯ be the eigenvalues of −Δ with zero Dirichlet boundary condition in Ω. The following result is proved in [2].

Theorem 1

If f′(0) < λ₁, then (1) has at least three nontrivial solutions, w₁ > 0 in Ω, w₂ < 0 in Ω, and w₃. The function w₃ changes sign exactly once in Ω, i.e., (w₃)⁻¹(R − 0) has exactly two connected components.

Theorem 1 has been proved in [2] via a variational argument.

We note that if f′(0) > λ₁, then by multiplying (1) by an eigenfunction corresponding to λ₁ and integrating by parts, it is easily seen that (1) does not have one signed solutions. But for existence of sign-changing solution for f′(0) > λ₁ we have:

Theorem 2 [5]

If f′(0) ∈ [λ₁, λ₂) then (1) has a solution which change sign exactly once. Sign-changing exactly-once solutions exist provided that f′(0) < λ₂. We want to investigate all above results numerically.

Example

In this paper’s numerical investigation we let N = 2 and Ω = (0, 1) × (0, 1). For example we let

$$f(u)=\lambda u+u^5\text{,}$$

where f satisfy the assumption for N = 2.

We know from [2], [4], a classical solution to the PDE (1) are saddle type critical point of the action functional J, so we can apply mountain pass lemma to obtain one signed solutions and modified mountain pass algorithm [6] to obtain sign-changing solution.