On scaling algorithm for finding positive solution of elliptic equation

Abstract

Using a numerical method, we will show the existence of positive solution for the elliptic equation $-\mathrm{\Delta }u(x)=-\lambda u(x)+u(x)|u(x){|}^{p-2}$ with Dirichlet boundary condition for varying λ.

Introduction

Because of wide applications of numerical solutions of PDEs in many branches of science and engineering, many authors are studying this arguments by different tools and approaches. Therefore many algorithms have been found effective [4].

DIA, the direct iteration algorithm; MIA, monotone iteration algorithm; MPA, mountain pass algorithm and SIA, the scaling iteration algorithm and etc. Each algorithm has its advantages and disadvantages and should be chosen according to certain a priori knowledge of the solution manifold structure, if any such knowledge is available. For example, DIA and MIA are useful in capturing the so-called stable solutions while MPA and SIA are useful in capturing the so-called mountain pass solutions which are unstable.

In this paper we want to find numerical solution of semilinear elliptic boundary value problems

$$\left\{\begin{array}{cc}-\mathrm{\Delta }u(x)=-\lambda u(x)+u(x)|u(x){|}^{p-2}\hfill & x\in \Omega \hfill \\ u(x)=0\hfill & x\in \mathrm{\partial }\Omega \hfill \end{array}\right.\text{,}$$

where Ω is a bounded region with smooth boundary in R^N, λ is a real parameter, and Δ is the Laplacian operator. We also assume that p satisfies condition $2<p<2^{\ast }=\frac{2N}{N-2}$.

The study of (1) is motivated by the fact that the equation has wide applications to physical models (see [5] and references therein).

It is well known that the steady-state solutions of

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

correspond to critical points of the functional $J:H_0^{1\text{,}2}(\Omega )\to R$

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

where $F(x\text{,}u(x))={\int }_0^{u}f(x\text{,}s)\mathrm{d}s$. For convenience we will denote $H_0^{1\text{,}2}(\Omega )$ by H and will use the standard norm (see, e.g., [1] or [6])

$$\Vert u{\Vert }_H^2={\int }_{\Omega }|\nabla u(x){|}^2\mathrm{d}x\text{.}$$

The corresponding Euler functional for Eq. (1) is given by:

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

First we claim that J(u) has neither a global minimum nor a global maximum. In fact, we can choose a sequence {u_n} satisfying ${\int }_{\Omega }|u(x){|}^p\mathrm{d}x=1$ and ${\int }_{\Omega }|\nabla u(x){|}^2\mathrm{d}x\to \infty$, so that $J(u)\to \infty$ and $n\to \infty$, i.e., J(u) is not bounded from above. On the other hand, for fixed u ≢ 0, we have

$$J(\mathit{tu})=\frac{1}{2}{t}^2{\int }_{\Omega }|\nabla u(x){|}^2\mathrm{d}x+\frac{\lambda }{2}{t}^2{\int }_{\Omega }|u(x){|}^2\mathrm{d}x-\frac{1}{p}|t{|}^p{\int }_{\Omega }|u(x){|}^p\mathrm{d}x=t^2\left[\frac{1}{2}{\int }_{\Omega }|\nabla u(x){|}^2\mathrm{d}x+\frac{\lambda }{2}{\int }_{\Omega }|u(x){|}^2\mathrm{d}x-\frac{1}{p}|t{|}^{p-2}{\int }_{\Omega }|u(x){|}^p\mathrm{d}x\right]\to -\infty \text{,}$$

as $t\to \infty$. Hence J(u) is not bounded from below, so we have proved.

Theorem 1

J(u) is neither bounded from above nor bounded from below.

Mountain Pass Lemma [3] can be applied and we have

Theorem 2

For every $\lambda >0$, the problem (1) has a positive solution.

The above theorem proved in [2]. We now provided sketch of proof:

Proof

For the existence of solution to (1) it is sufficient to check conditions of Mountain Pass Lemma are satisfied.

By using Sobolev imbedding inequality we have

$$J(u)⩾\frac{1}{2}{\int }_{\Omega }|\nabla u(x){|}^2\mathrm{d}x-\frac{1}{p}{\int }_{\Omega }|u(x){|}^p\mathrm{d}x⩾\frac{1}{2}{\int }_{\Omega }|\nabla u(x){|}^2\mathrm{d}x+\frac{1}{p}{c}^p{\left({\int }_{\Omega }|\nabla u(x){|}^2\right)}^{\frac{p}{2}}\mathrm{d}x=\left[\frac{1}{2}\frac{c^p}{p}\Vert u{\Vert }^{p-2}\right]\Vert u{\Vert }^2$$

So by choosing $\rho =(\frac{p}{4c^p}{)}^{\frac{1}{p-2}}$ and $\alpha =\frac{1}{4}(\frac{p}{4c^p}{)}^{\frac{2}{p-2}}$ we will $J(u)>0$ for $u\in \{u\in H:\Vert u\Vert <\rho \}-\{0\}$, $J(u)⩾\alpha$ for $\Vert u\Vert =\rho$ so J(u) satisfies the first condition.

For given fixed u ≢ 0 in H, let us consider the map $t\to J(\mathit{tu})$. Then for $t>0$ we have $J(\mathit{tu})$ is negative if t is large enough and is positive for sufficiently small t. Thus by continuity there exists $t_0>0$ such that $J(t_{0}u)=0$ and so J(u) satisfies the second condition.

Also, J satisfies third condition. Therefore by the Mountain Pass Lemma J has a nontrivial critical point. Since $J(u)=J(|u|)$, without loss of generality, we may assume that u is a nonnegative solution of (1). □

We want to obtain this solution numerically by using scaling iteration algorithm.