A New Approach for Numerically Solving Nonlinear Eigensolution Problems

Abstract

By considering a constraint on the energy profile, a new implicit approach is developed to solve nonlinear eigensolution problems. A corresponding minimax method is modified to numerically find eigensolutions in the order of their eigenvalues to a class of semilinear elliptic eigensolution problems from nonlinear optics and other nonlinear dispersive/diffusion systems. It turns out that the constraint is equivalent to a constraint on the wave intensity in L-(p+1) norm. The new approach enables people to establish some interesting new properties, such as wave intensity preserving/control, bifurcation identification, etc., and to explore their applications. Numerical results are presented to illustrate the method.

Appendices

Appendix A: Verification of Weaker PS$_{\mathcal N}$ Condition

For $\lambda (u)$ defined in Sect. 3, we verify its PS condition which is crucial for proving the existence of (infinitely) multiple eigenfunctions and also for the convergence of LMM.

Note that $\lambda (u)$ may have a singular point. On the other hand, various PS conditions are proposed in the literature to prove the existence, but failed to handle such a singularity and are not for computational purpose. According to LMM, all computations are carried out only on the Nehari manifold ${\mathcal N}$, see (2.2), where it enjoys a nice property: $\langle \lambda '(u),u\rangle =0$ for all $u\in {\mathcal N}$ and $\text{ dis }({\mathcal N},0)>t_0>0$ for some $t_0>0$. So we can restrict our analysis only on ${\mathcal N}$ and utilize this property to simplify our analysis. Such an observation motivates us to introduce a new definition.

Definition 2

A ${\mathcal C}^1$-functional $J$ is said to satisfy PS$_{\mathcal N}$ condition, if any sequence $\{ u_k\}\subset {\mathcal N}= \{ u\in H : u\ne 0, \langle J'(u),u\rangle =0\}$ s.t. $\{J(u_k)\}$ is bounded and $J'(u_k)\rightarrow 0$ has a convergent subsequence.

It is clear that PS condition implies PS$_{\mathcal N}$ condition.

Theorem 9

$\lambda (u)$ defined with $C(\lambda )=C$ or $C(\lambda )=C\lambda$ in Sect. 3 satisfies PS$_{\mathcal N}$ condition.

Proof

Let $\{u_k\}\subset {\mathcal N}$ s.t. $\{\lambda (u_k)\}$ is bounded and $\lambda '(u_k)\rightarrow 0$. Since $C(\lambda )=C$ or $C\lambda$, $\{C(\lambda (u_k))\}$ and $\{C'(\lambda (u_k))\}$ are bounded. Note

$$\begin{aligned} H(u)=\frac{1}{2}\Vert \nabla u\Vert ^2_{L^2}-\frac{1}{p+1}\Vert u\Vert ^{p+1}_{L^{p+1}}, \quad I(u)=\frac{1}{2}\Vert u\Vert ^2_{L^2}. \end{aligned}$$

(3.14)

Thus $\;\langle I'(u_k), u_k\rangle =2I(u_k)$ and $\langle H'(u_k), u_k\rangle -2H(u_k)=\frac{1-p}{p+1}\Vert u_k\Vert ^{p+1}_{L^{p+1}}.\;$ Then $u_k\in {\mathcal N}$ implies

$$\begin{aligned} 0=\langle \lambda '(u_k), u_k\rangle =[I(u_k)+C'(\lambda (u_k))]^{-1} [\langle H'(u_k),u_k\rangle -\lambda (u_k)\langle I'(u_k),u_k\rangle ] \end{aligned}$$

or

$$\begin{aligned} 0&= \langle H'(u_k),u_k\rangle -\lambda (u_k)\langle I'(u_k),u_k\rangle =\langle H'(u_k),u_k\rangle -2H(u_k)+2H(u_k)-2\lambda (u_k)I(u_k)\\&= \frac{1-p}{p+1}\Vert u_k\Vert ^{p+1}_{L^{p+1}}+2C(\lambda (u_k)). \end{aligned}$$

When $\{C(\lambda (u_k))\}$ is bounded, so is $\{\Vert u_k\Vert ^{p+1}_{L^{p+1}}\}$. By the Hölder inequality $\{I(u_k)\}$ is bounded. Consequently $\{I(u_k)+C'(\lambda (u_k))\}$ is bounded. Thus

$$\begin{aligned} \lambda '(u_k)\equiv \frac{H'(u_k)-\lambda (u_k) I'(u_k)}{I(u_k)+C'(\lambda (u_k))} \rightarrow 0\Rightarrow H'(u_k)-\lambda (u_k)I'(u_k)\rightarrow 0. \end{aligned}$$

From $\,H(u_k)-\lambda (u_k)I(u_k)=C(\lambda (u_k))$ and (3.14), we see that $\{\Vert \nabla u\Vert ^2_{L^2}\}$ is bounded or $\{u_k\}$ is bounded in $H=H^1$. Next we follow the approach in the proof of Lemma 1.20 in [20]. There is a subsequence, denote by $\{u_k\}$ again, and $u\in H$ s.t. $u_k\rightharpoonup u$ (means weakly) in $H$. By the Rellich theorem, $u_k\rightarrow u$ in $L^2$ and $L^{p+1}$. Then

$$\begin{aligned} \Vert u_k-u\Vert ^2_H&= \int _\Omega [|\nabla u_k(x)-\nabla u(x)|^2+|u_k(x)-u(x)|^2]dx\\&= \langle H'(u_k)-\lambda (u_k)I'(u_k)-H'(u)+\lambda (u)I'(u), u_k-u\rangle \\&+\langle \lambda (u_k)I'(u_k)\!-\!\lambda (u)I'(u), u_k\!-\!u\rangle \!+\!\langle u_k^p\!-\!u^p, u_k-u\rangle +\Vert u_k-u\Vert ^2_{L^2}\!\rightarrow \! 0, \end{aligned}$$

where $\Vert u_k-u\Vert ^2_{L^2}\rightarrow 0$ is clear; the first term

$$\begin{aligned} \langle H'(u_k)-\lambda (u_k)I'(u_k)-H'(u)+\lambda (u)I'(u), u_k-u\rangle \rightarrow 0, \end{aligned}$$

because $H'(u_k)-\lambda (u_k)I'(u_k)\rightarrow 0$ in $H$ and $u_k\rightharpoonup u$; the second term

$$\begin{aligned} |\langle \lambda (u_k)I'(u_k)-\lambda (u)I'(u), u_k-u\rangle |&= |\int _\Omega [\lambda (u_k)u_k(x)- \lambda (u)u(x)][u_k(x)-u(x)]dx|\\&\le \Vert \lambda (u_k)u_k-\lambda (u)u\Vert _{L^2} \Vert u_k-u\Vert _{L^2}\rightarrow 0, \end{aligned}$$

by the Cauchy–Schwarz inequality, the boundedness of $\lambda (u_k)$ and $u_k\rightarrow u$ in $L^2$; and finally

$$\begin{aligned} |\langle u_k^p-u^p, u_k-u\rangle |\le \Vert u_k-u\Vert ^{p+1}_{L^{p+1}}\rightarrow 0 \quad \text{ by }\quad u_k\rightarrow u\in L^{p+1}. \end{aligned}$$

$\square$

So Theorems 2 and 3 still hold when PS condition is replaced by PS$_{\mathcal N}$ condition.

Appendix B: Proof of Theorem 8—Identification of Bifurcation

Proof

We have an expression for the linear operator

$$\begin{aligned} \lambda ''(u)&= \frac{1}{(I(u)+C'(\lambda (u))^2}[(H''(u)-\lambda '(u)I'(u) -\lambda (u)I''(u))(I(u)+C'(\lambda (u))\\&-(I'(u)+C''(\lambda (u))\lambda '(u))(H'(u)-\lambda (u)I'(u))]. \end{aligned}$$

At each $u$ s.t. $\lambda '(u)=0$ or $H'(u)-\lambda (u)I'(u)=0$, we have

$$\begin{aligned} \lambda ''(u)=\frac{H''(u)-\lambda (u)I''(u)}{I(u)+C'(\lambda (u))}. \end{aligned}$$

Taking $H(u)=\int _\Omega \left( \frac{1}{2}|\nabla u(x)|^2-\frac{\beta }{p+1} |u(x)|^{p+1}\right) dx, I(u)=\int _\Omega \frac{1}{2} u^2(x)dx$ into account, we have

$$\begin{aligned} H'(u)=\Delta u-\beta |u|^{p-1}u,\; I'(u)=u,\; H''(u)w=-\Delta w-p\beta |u|^{p-1}w,\;I''(u)w=w \end{aligned}$$

and then

$$\begin{aligned} \lambda '(u_C)=0\Leftrightarrow H'(u_c)-\lambda (u_C)I'(u_C)=0 \Leftrightarrow -\beta u_C^{p}-\lambda (u_C)u_C=0\Leftrightarrow \lambda (u_C) =-\beta u_C^{p-1}. \end{aligned}$$

(3.15)

Note $u_C>0$ can be solved from

$$\begin{aligned} H(u_C)-\lambda (u_C)I(u_C)=C(\lambda (u_C))\;\text{ or }\;-\frac{\beta }{p+1} u_C^{p+1}|\Omega |+ u_C^{p-1}\frac{\beta }{2} u_C^2|\Omega |=C(-\beta u_C^{p-1}). \end{aligned}$$

(3.16)

Let $\eta$ be an eigenvalue of the linear operator $\lambda ''(u_C)$ and $w$ be an associated eigenfunction, i.e.,

$$\begin{aligned} \lambda ''(u_C)w=\frac{-\Delta w-p\beta u_C^{p-1}w+ \beta u_C^{p-1}w}{\frac{1}{2} u_C^2|\Omega |+C'(-u_C^{p-1})}=\eta w. \end{aligned}$$

It leads to

$$\begin{aligned} -\Delta w=\left[ (p-1)\beta u_C^{p-1}+\eta \left( \frac{1}{2} u_C^2|\Omega |+C'(-u_C^{p-1})\right) \right] w \mathop {=\!=}\limits ^{\text{ denote }}\mu w. \end{aligned}$$

Then $\mu =(p-1)\beta u_C^{p-1}+\eta \left( \frac{1}{2} u_C^2|\Omega |+C'(-u_C^{p-1})\right)$ is an eigenvalue of $-\Delta$ and $w$ is its associated eigenfunction. So $\lambda ''(u_C)$ and $-\Delta$ share exactly the same eigenfunctions. We have

$$\begin{aligned} \eta =\frac{\mu -(p-1)\beta u_C^{p-1}}{\frac{1}{2} u_C^2|\Omega |+C'(-u_C^{p-1})}. \end{aligned}$$

It indicates that $\eta$ and $\mu$ have the same multiplicity. Let $\eta _1<\eta _2\le \eta _3\cdots$ be all the eigenvalues of $\lambda ''(u_C)$. We obtain that for $k=1,2,\cdots$, (a) if

$$\begin{aligned} \mu _k<(p-1)\beta u_C^{p-1}<\mu _{k+1}, \end{aligned}$$

(3.17)

then $\eta _i<0, i=1,2,\ldots ,k$ and $\eta _j>0, j=k+1,k+2,\ldots$, thus $u_C$ is nondegenerate with MI$(u_c)=k$; and (b) if

$$\begin{aligned} \mu _k<(p-1)\beta u_C^{p-1}=\mu _{k+1}=\cdots =\mu _{k+r_k}<\mu _{k+1+r_k}, \end{aligned}$$

(3.18)

then $\eta _i<0, i=1,2,\ldots ,k, \eta _i=0, i=k+1,\ldots ,k+r_k, \eta _i>0, i=k+1+r_k, k+2+r_k,\ldots$, thus $u_c$ is degenerate with MI$(u_C)=k$, nullity$(u_C)=r_k\ge 1$ and $u_C$ bifurcates to new solution(s) [7]. Note that by the maximum principle, an one-sign solution either whose value and derivative are equal to zero at an interior point of $\Omega$ or whose value and normal derivative are equal to zero at a boundary point of $\Omega$ must be identically equal to zero. Since a sign-changing solution has nodal line(s) (where values are equal to zero) inside $\Omega$, when a sequence of sign-changing solutions approach to an one-sign solution $u^*$, there are two possibilities: (1) some nodal lines stay inside $\Omega$ thus $u^*$ attains its zero value and zero derivative at an interior point of $\Omega$ or (2) some nodal lines approach to the boundary $\partial \Omega$ thus $u^*$ attains its zero value and zero normal derivative (as a solution) at a boundary point of $\Omega$. In either case, $u^*$ has to be identically equal to zero. When

$$\begin{aligned} \mu _1=0<(p-1)\beta u_C^{p-1}<\mu _2 \end{aligned}$$

$u_C$ is nondegenerate, so no bifurcation takes place. On the other hand, since $u_C>0=\mu _1$ and at each bifurcation point $(p-1)\beta u_C^{p-1}=\mu _{k+1}\ge \mu _2>0, u_C>0$ must satisfy

$$\begin{aligned} u_C\ge \Big [\frac{\mu _2}{(p-1)\beta }\Big ]^{\frac{1}{p-1}}>0 \end{aligned}$$

and can bifurcate only to positive non trivial solutions.

When $C(\lambda )=C$, the equation in (3.16) becomes

$$\begin{aligned} -\frac{\beta }{p+1} u_C^{p+1}|\Omega |+ u_C^{p-1}\frac{\beta }{2} u_C^2|\Omega |=C, \end{aligned}$$

which leads to

$$\begin{aligned} u_C=\Big [\frac{2C(p+1)}{|\Omega |\beta (p-1)}\Big ]^{\frac{1}{p+1}}\quad \text{ and }\quad \lambda (u_C)= -\beta \Big [\frac{2C(p+1)}{|\Omega |\beta (p-1)}\Big ]^{\frac{p-1}{p+1}} \end{aligned}$$

(3.19)

from the last equation in (3.15). It is clear that $u_C$ is monotonically increasing in $C$. When $C$ increases so that the term $(p-1)u_C^{p-1}$ increases and crosses each $\mu _k$, the positive constant solution $u_C$ bifurcates to new positive solution(s).$\square$