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.
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Supported in part by NSF DMS-0713872/0820327/1115384.
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
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
or
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
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
where \(\Vert u_k-u\Vert ^2_{L^2}\rightarrow 0\) is clear; the first term
because \(H'(u_k)-\lambda (u_k)I'(u_k)\rightarrow 0\) in \(H\) and \(u_k\rightharpoonup u\); the second term
by the Cauchy–Schwarz inequality, the boundedness of \(\lambda (u_k)\) and \(u_k\rightarrow u\) in \(L^2\); and finally
\(\square \)
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
At each \(u\) s.t. \(\lambda '(u)=0\) or \(H'(u)-\lambda (u)I'(u)=0\), we have
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
and then
Note \(u_C>0\) can be solved from
Let \(\eta \) be an eigenvalue of the linear operator \(\lambda ''(u_C)\) and \(w\) be an associated eigenfunction, i.e.,
It leads to
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
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
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
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
\(u_C\) 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
and can bifurcate only to positive non trivial solutions.
When \(C(\lambda )=C\), the equation in (3.16) becomes
which leads to
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).\(\square \)
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Wang, C., Zhou, J. A New Approach for Numerically Solving Nonlinear Eigensolution Problems. J Sci Comput 64, 109–129 (2015). https://doi.org/10.1007/s10915-014-9925-y
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DOI: https://doi.org/10.1007/s10915-014-9925-y
Keywords
- Semilinear elliptic eigensolution problem
- Implicit approach
- Minimax method
- Order in eigenvalues
- Morse index
- Bifurcation