A new augmented singular transform and its partial Newton-correction method for finding more solutions to nonvariational quasilinear elliptic PDEs

Abstract

In this paper, in order to find more solutions to a nonvariational quasilinear PDE, a new augmented singular transform (AST) is developed to form a barrier surrounding previously found solutions so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found solution. Thus a solution found by the algorithm must be new. Mathematical justifications of AST formulation are established. A partial Newton-correction method is designed accordingly to solve the augmented problem and to satisfy a constraint in AST. The new method is applied to numerically investigate bifurcation, symmetry-breaking phenomena to a non-variational quasilinear elliptic equation through finding multiple solutions. Such phenomena are numerically captured and visualized for the first time, and still open for theoretical verification. Since the formulation is general and simple, it opens a door to solve other multiple solution problems.

Introduction

Multiple solutions to nonlinear differential systems exist in many applications. Due to the development of new advanced (synchrotron, laser, etc.) technologies, nowadays, many researchers are interested in finding such multiple solutions [1], [2], [3], [4], [5], [6], [7] and explore new applications.

As a model problem of this paper, we investigate steady-state solutions to a quasi-linear convection–diffusion–reaction equation, which leads to find nontrivial solutions to a nonvariational quasi-linear PDE of the form

$$F\left(u\right)\left(x\right)\equiv -{\Delta }_{p}u\left(x\right)+a\left(x\right)\cdot \nabla u\left(x\right)+\lambda \left(x\right)u\left(x\right)+\kappa f(x,\left|u\left(x\right)\right|)u\left(x\right)=0,u\in U$$

where $U=W_0^{1,p}\left(\Omega \right)$, $\Omega$ is a bounded open domain in ${\mathbb{R}}^n$, ${△}_{p}u\left(x\right)=div({|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right))$ is the nonlinear p-Laplacian differential operator, which has a variety of applications in physical fields, such as in fluid dynamics when the shear stress $\overrightarrow{\tau }$ and the velocity gradient $\nabla u$ of the fluid are related in the manner $\overrightarrow{\tau }\left(x\right)=r\left(x\right){|\nabla u\left(x\right)|}^{p-2}\nabla u\left(x\right)$, where $p=2$, $p<2$, $p>2$ if the fluid is Newtonian, pseudoplastic, dilatant, respectively. The p-Laplacian operator also appears in the study of flow in a porous media ($p=\frac{3}{2}$), nonlinear elasticity ($p>2$), and glaciology ($p\in (1,\frac{4}{3})$). So far, people’s knowledge on solutions to (1.1) is still very limited. We hope to develop some stable and efficient numerical methods for approximating nontrivial solutions and to conduct some numerical investigations on the qualitative behavior of the solutions, such as bifurcation and symmetry-breaking phenomena. The functions $a\left(x\right),\lambda \left(x\right)$ and the physical constant $\kappa$ are given, the nonlinear function $f(x,u)u$ is superlinear in $u$. When the vector function $a\left(x\right)⁄\equiv 0$, the elliptic PDE (1.1) is in general non-variational and quasilinear.

The existence and multiplicity of solutions to nonvariational nonlinear elliptic PDE or systems are studied by many researchers. Most of them focus on the case where the nonvariational nature is caused by two PDEs in a system not by the convection term in a PDE as (1.1). E.g., multiple solutions to nonvariational systems are studied with the topological methods [8], [9], with a combination of methods of sub-supersolution and Leray–Schauder topological degree [10], and fixed point theory [11]. Multiple solutions to a nonvariational semilinear elliptic PDE similar to (1.1) are studied with the reduction method [12].

So far numerical methods for finding multiple solutions to such a nonvariational nonlinear PDE are not yet available in the literature. Without any variational structure, variational methods [13], [14], [15], [16], [17] are not applicable. Thus we have to adopt a more general approach and consider finding multiple solutions (zeros) to the equation

$$F\left(u\right)=0,$$

where $F\in {\mathcal{C}}^1(U,V)$ involves nonlinear differential operator(s), $U$ and $V$ are two Banach spaces. We assume that $F\left(0\right)=0$, but one is interested in finding nontrivial solutions.

Eq. (1.2) is one of the most general and useful mathematical formulations since many problems can be formulated into solving such an equation. Because of the general problem setting, it is quite natural to try using a Newton method due to its independence of any variational structure and fast local convergence. A huge related literature exists. However, most results in the literature assume that (1.2) has a unique solution or is a polynomial system with multiple solutions [18], [19], [20], [21], [22], [23], etc, thus those results are not applicable here. It is known that as multiple solutions are concerned, a well known weakness of a Newton method, its heavy dependence on an initial guess is significantly magnified and severely reduces its effectiveness in finding a new solution. This can be easily understood by using the notions of a continuous Newton flow, its barrier and the local basins of attraction, see Example 1.1. In other words, the space $U$ is divided into multiple local basins separated by barriers where a continuous Newton flow cannot cross, so a Newton flow will be trapped in a local basin of an initial guess. Thus for a Newton method to succeed in finding a new unknown solution, an initial guess must be chosen in the same local basin of the unknown solution. It is too difficult to do so for a highly nonlinear nonvariational differential system.

Example 1.1

$F(x,y)=(2x-x^3,2y+y^2),F^{\prime }(x,y)=\text{diag}(2-3x^2,2+2y)$. $F$ has 6 zeros at $(0,0),(\pm \sqrt{2},0),(0,-2),(\pm \sqrt{2},-2)$. Since $det\left(F^{\prime }(x,y)\right)=(2-3x^2)(2+2y)$, the barriers are given by the lines $x=\pm \sqrt{\frac{2}{3}},y=-1$ surrounding 6 local basins. The curve around $(0,0)$ is the Nehari manifold

$$\mathcal{N}=\{t(x,y):\left|(x,y)\right|=1,F\left(t(x,y)\right)\perp (x,y)\}$$$$=\{(x,y)=(tcos\left(\theta \right),tsin\left(\theta \right)):0\le \theta <2\pi ,{cos}^4\left(\theta \right)t^2-{sin}^3\left(\theta \right)t-2=0\}.$$

To reduce the dependence of a Newton method on an initial guess, several Newton homotopy continuation methods (NHCM) are proposed [24], [25], [26], [27]. However, NHCM requires a continuous (Newton) path [20] from an initial guess to an unknown solution. When a target solution is unknown, the barrier of its local basin is unknown, there is no way to tell anything about such a path. Therefore NHCM can help but not much in finding multiple unknown solutions. Also some results assume that (1.2) is a polynomial system or all solutions are nondegenerate and contained in a compact set [18], [20], [22], thus they are not applicable here (see Fig. 1).

Motivated by using a support $S$ in the design of a local minimax method [28], [29], [30], instead of locating the local basin of an unknown solution, which is too difficult to do so in most cases, the authors in [31] recently proposed to use a singular transform to form a barrier surrounding the support $S$, a subspace, e.g., spanned by previously found solutions. So when an initial guess is selected outside $S$, this is much easier to do so since $S$ is known, an approximation sequence generated by a numerical algorithm cannot pass the barrier and penetrate into $S$ to reach a previously found solution. Consequently a solution found must be new. Since a continuous transform cannot change singularities caused by barriers, to accomplish the goal, the authors in [31] introduced the augmented singular transform (AST). The numerical results are very promising, but theoretical issues are left unsolved. This formulation is significantly improved in [32], where with the new formulation, those theoretical issues left in [31] are resolved. It opens a new door for further research, in particular, it enables us to solve nonvariational quasilinear problems for multiple solutions. Under a very simple assumption $F^{\prime }\left(0\right)\ne 0$, the new formulation can be described as below.

Let $U$ be a Banach space, $V=U^{\ast }$, ${〈\cdot ,\cdot 〉}_{U\times V}$ be their duality relation and $\perp$, the orthogonality, be defined by $〈\cdot ,\cdot 〉=0$. Let $S=[u_1,\dots ,u_k]$ be the subspace spanned by $u_1,\dots ,u_k\in U\cap V$ where $u_1,\dots ,u_k$ are linearly independent. Note that in this case, for $u\in U\cap V$, we may have ${\Vert u\Vert }^2\ne 〈u,u〉$.

Denote $S^{\perp }=\{v\in U:〈v,u〉=0,\forall u\in S\}$ and $S_1^{\perp }=\{u\in S^{\perp }:\Vert u\Vert =1\}$. Then we have $U=S\oplus S^{\perp }$. To see this, first since $S\subset U\cap V$ is finite-dimensional, by the Gram–Schmidt orthogonalization process, ${\bar{v}}_j=u_j-{\sum }_{i=1}^{j-1}\frac{〈u_j,v_i〉}{〈v_i,v_i〉}{v}_i,v_j=\frac{{\bar{v}}_j}{\Vert \overline{v_j}\Vert },j=1,\dots ,k$, we obtain an orthonormal basis $\{v_1,\dots ,v_k\}$ for $S$. Then for each $u\in U$, $u^{\perp }=u-{\sum }_{i=1}^k\frac{〈u,v_i〉}{〈v_i,v_i〉}{v}_i\perp v_j,j=1,\dots ,k$.

For each $u\in S_1^{\perp }$ and $t\ne 0$, define an augmented singular transform (AST) [32]

$$A(u,t,t_1,\dots ,t_k)=\frac{1}{t}F\left(t(u+\sum _{i=1}^k{t}_i{u}_i)\right).$$

It is clear that $A$ has a singularity at $t=0$ and any point $u^{\ast }\notin S$ can be expressed as $u^{\ast }=t(u+t_1{u}_1+\cdots +t_k{u}_k)$ for some $u\in S_1^{\perp }$ and constants $t,t_1,\dots ,t_k$. We have $u^{\ast }\in (U\setminus S)$ when $t\ne 0$, i.e., it forms a barrier surrounding the subspace $S$. Thus instead of solving $F\left(u\right)=0$ for $u^{\ast }$, we can solve $A(u,t,t_1,\dots ,t_k)=0$ for a NEW solution not in $S$. We assume a standard convergence

$$w=(u,t,t_1,\dots ,t_k)\to \bar{w}=(\bar{u},t^{\ast },t_1^{\ast },\dots ,t_k^{\ast })$$

is obtained from solving $A(u,t,t_1,\dots ,t_k)=0$ by a numerical method. Such a standard convergence assumption is necessary to avoid the case where $u$ does not converge but $tu$ converges ($t\to 0$). Thus when $t\to t^{\ast }=0$, we have

$$0=\underset{t\to 0}{lim}A(u,t,t_1,\dots ,t_k)=\underset{t\to 0}{lim}\frac{F\left(t\bar{u}\right)-F\left(0\right)}{t}=F^{\prime }\left(0\right)\bar{u},$$

where $\bar{u}=u+{\sum }_{i=1}^k{t}_i{u}_i$. Thus if $\bar{u}⁄\in ker\left(F^{\prime }\left(0\right)\right)$, e.g., $ker\left(F^{\prime }\left(0\right)\right)=\left\{0\right\}$, then $t^{\ast }\ne 0$ and $\bar{w}=t^{\ast }\bar{u}\in U\setminus S$ is a NEW solution. However, if $F^{\prime }\left(0\right)=0$ or $ker\left(F^{\prime }\left(0\right)\right)=U$, we have

$$0=\underset{t\to 0}{lim}\frac{F\left(tu\right)-F\left(0\right)}{t}=F^{\prime }\left(0\right)u,\forall u\in U,$$

or, $F\left(tu\right)=o\left(t\right)$, this method fails. Fortunately, the improvement of the formulation from [31] to (1.4) enables us to generalize the formulation as to consider

$$\underset{t\to 0}{lim}\frac{F\left(tu\right)-F\left(0\right)}{t^p}\ne 0,$$

or, $F\left(tu\right)=O\left(t^p\right)$ for some $p\ge 1$. Thus we can solve $\frac{1}{t^p}F\left(tu\right)=0$ for a NEW solution not in $S$.

The paper is organized as follows. In Section 2, we give a new augmented singular transform(AST) and its mathematical validation. In Section 3, we describe a partial Newton-Correction method(PNCM) and its flow chart. In Section 4, we apply AST and PNCM developed in the previous sections to numerically compute multiple solutions and explore possible bifurcation phenomena to the nonvariational quasilinear PDE (1.1). Some numerical results are presented for the multiple positive solutions of (1.1). The bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of (1.1). The final section is for some concluding discussions.