A New Augmented Singular Transform and its Partial Newton-Correction Method for Finding More Solutions to p-laplacian Differential Equation

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1 A New Augmented Singular Transform and its Partial Newton-Correction Method for Finding More Solutions to p-laplacian Differential Equation Zhaoiang Li, Zhi-Qiang Wang and Jianin Zhou Abstract In this paper, we derive a new augmented singular transform for finding more solutions. Then a corresponding partial Newton-correction method is designed to solve the augmented problem on the solution set. Mathematical justification of the new formulation, method. It is applied to solve a non-variational nonlinear p-laplacian equation for multiple solutions, which are, for the first time, numericall computed and visualized with their profile and contour plots. Several interesting phenomenons are observed for the first time and open for mathematical verification. Since the new formulation is general and simple, it can be modified to treat other problems, e.g., a quasilinear differential PDE, for multiple solutions. Kewords: Multiple solutions, Augmented Singular Transform, Partial Newton-Correction Method, p-laplacian. Introduction Multiple solutions to nonlinear differential sstems eist in man applications and analsis. Due to the development of new advanced (snchrotron, laser, etc.) technologies, nowadas, man researchers are interested in finding such multiple solutions [9,4,5] for new applications. In this paper, we consider finding multiple solutions (zeros) to the equation F (u) =, (.) Department of Mathematics, Shanghai Normal Universit, Shanghai, China, Supported in part b Innovation Program of Shanghai Municipal Education Commission (No.4YZ78) and NSF of Shanghai (No.5ZR439). Utah State Universit, Logan, UT Department of Mathematics, Teas A&M Universit, College Station, TX 77843, USA. (jzhou@math.tamu.edu). Supported in part b NSF DMS-5384.

2 where F C (U, V ) is nonlinear and involves differential operator(s), U and V are two Banach spaces. We assume that F () =, but one is interested in finding nontrivial solutions. Equation (.) is one of the most general and useful mathematical formulations since man problems can be formulated into solving such an equation. Since the problem setting (.) is general, it is quite natural to use a Newton method due to its fast local convergence and independence of an variational structure. A huge related literature eists. However, most results in the literature assume that (.) has a unique solution or is a polnomial sstem with multiple solutions [7,,,2,23,etc], thus those results are not applicable here. It is known that as multiple solutions are concerned, a wellknown weakness of a Newton method, i.e., its heav dependence on an initial guess is significantl magnified and severel reduces its effectiveness in finding a new solution. This can be understood b using the notions of a continuous Newton flow, its barrier and the local basins of attraction, see [2]. In other words, the space U is divided into multiple local basins separated b 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 be successful in finding a new unknown solution, an initial guess must be chosen in the same local basin of the unknown (target) solution. It is too difficult to do so for a highl nonlinear differential sstem. Several Newton homotop continuation methods (NHCM) [,2] have been proposed to reduce the dependence of a Newton method on its initial guess. NHCM requires a continuous (Newton) path [] 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 wa to tell anthing about such a path. Thus NHCM will help but not much in finding multiple unknown solutions. Also some results assume that (.) is a polnomial sstem or all solutions are nondegenerate and contained in a compact set [,7,23], and thus are not applicable here. Motivated b using a support S in the design of a local minima method [2,3,25], instead of locating the local basin of an unknown solution, which is too difficult to do so in most cases, the authors in [2] recentl proposed to use a singular transform to form a barrier surrounding the support S, a subspace spanned, e.g., b previousl found solutions. So when an initial guess is selected outside S, this is much easier to do so since S is known, an approimation sequence generated b a numerical algorithm cannot pass the barrier and penetrate into S to reach a previousl found solution. Consequentl a solution found must be new. Since a continuous transform cannot change singularities, to accomplish the goal, the authors in [2] introduced the following augmented singular transform (AST), where F (). Let U be a Banach space, V = U,, be their dualit relation and be defined b, =. Also let S = [u,, u k ] be the subspace spanned b u,, u k U V where u,..., u k are linearl independent. Denote S = {v U : v, u =, u S} and S = {u S : u = }. For each u S and t, define an augmented singular 2

3 transform A(u, t, t,, t k ) = t F (t(u + It is clear that A has a singularit at t = and an point u t i u i )). (.2) i= / S can be epressed as u = t(u + t u + + t k u k ) for some u S and constants t, t,..., t k. We have u (U \ S) when t and u = when t =. Since F () =, we can solve t F (tu) =, u = tu, but when F () =. F (tu) F () lim t t This method fails, F (tu) = o(t). We want to find p, such that = F ()u =, for an u U, F (tu) F () lim, t t p F (tu) = O(t p ), we solve t p F (tu) =, for u =, get u = tu, t. So we define a new augmented singular transform in the following section. 2 A New Augmented Singular Transform and its Mathematical Validation We consider finding multiple solutions to F (u) =. Let u,..., u k be previousl found solutions and S = [u,..., u k ]. Assume there is no solution u S ecept ±u,..., ±u k. Assume F () = but lim t F (tu) for all u. Define an augmented singular t p transform A(u, t, t,..., t k ) = t F p (t(u + t i u i )) (2.) for each u S, u = where is in the dualit sense. For u U and v U, we sa u v u, v U U =. We assume that u,..., u k are nice so that u,..., u k U U. Thus S U U and U = S S. Assume (u, t, t,..., t k ) (ū, t, t,..., t k ) and A(ū, t, t,..., t k ) =. Then t. In other words, w = t(ū + k i= t i u i ) S is a new solution. Let U and V be a Banach spaces and U be the dual space of U with dualit,. Let F : U V be C with F (u) = N(u) + G(u) where N(u) = u =, G(u) = u =, N(tu) = t p N(u), G(tu) = t l G(u) with p where N = N(, u, u,...) with N(, tu, t u,...) = t p N(, u, u,...); There are three cases: a) l > p, b) l < p, c) l = p. i= 3

4 Since = A(ū, t, t,..., t k ) = t F ( t(ū + t p i u i )) = N(ū + t = implies that a) for l > p, N(ū + b) for l < p, t l p = implies i= i= i= t i u i ) + t l p G(ū + t i u i ), i= t i u i ) = ū + t i u i = ; i= G(ū + t i u i ) = ū + i= t i u i =. i= c) for l = p, we assume N(u) + G(u) = u =. It leads to ū + k t i= i u i =. Then in each case, we obtain ū = k t i= i u i S S = {}, which contradicts to ū =. Net we consider the case where G(u) = G (u) + + G n (u) with G i (tu) = t l i all l i s are distrinct. The term with l i = p will be absorbed into N with the condition N(u) = u =, which means this part does not form an eigenvalue problem. So onl have to discuss the cases with l i > p and l i < p. Since all l i s are distinct, those terms will not be cancelled each other. The result follows. G(tu) Finall we consider the cases where for each u =, we have I. lim t t p G(tu) lim t =. t p When t =, we have = A(ū, t, t,... t k ) = N(ū + G(t(ū t + k i u i ) + lim t i= i u i )). t t p i= It leads to a contradiction in either case I or case II. We ma consider the case where G(u) is a power series. and = ; II. Splitting Indirect Newton method in implementation. Partial (subspace) Newton method will be used to update u and orthogonal conditions A(u, t, t,..., t k ) u, u u,..., u k u are used to determine t, t,..., t k. Partial Newton method is invariant in S. have Since we assume (u, t, t,..., t k ) (ū, t, t,..., t k ) and A(ū, t, t,..., t k ) =, if t =, we = A(ū G(t(ū, t, t,..., t + k k ) = lim t i= i (u i u ))) G(), t t p t 4

5 which leads to In particular, G(t(ū + t i u i )) = o(t p ). i= G(t(ū + k lim t i= i (u i u ))) G() t t = G (), ū + t i (u i u ) =, i= or ū + k i= t i (u i u ) ker(g ()) = ker(f (u )) = K. Thus if ū + k i= t i u i K, then t. In particular, if u is nondegenerate, i.e., ker(f (u )) = K = {}, then t. This result holds true if other solutions are degenerate. Small neighborhood of decides the basin structure b using spectral decomposition U with respect to F () in a Hilbert space. Banach space? Change the basin structure b augmented method, i.e., add -d corridor each time. Motivated b the eigensolution problem with λ replaced b t l p? So F (ū + t i u i ) t l p G(ū + t i u i ) = F (ū + t i u i ) λg(ū + t i u i ) = i= i= i= i= with λ = t l p. Morse inde and basin structure in Hilbert space. One ma also dicuss that t. Partial Newton method is no longer invariant in S. When orthogonal conditions fail to satisf, we can rela them b a minimization process min { A(u, λ, t,..., t k ), u 2 + λ,t,...,t k A(u, λ, t,..., t k ), u i 2 }. It is a subspace Newton method. We partition the entire space i= U = S S, where S constains previousl found linear independent eigenfunctions u,..., u k. We write an eigensolution in the form t(u + k i= t iu i ) where u S with u =. It is clear that an solution u can be written in this form. We use a Newton method onl in the subspace S to update u S and use k + orthogonal conditions to update k + parameters (t, t,..., t k ) in an eigensolution of problem. This will take care of the remaining k d space S and to keep the partial Newton method invariant in S. As a result, if the Newton direction is zero in the partial Newton method, it ields a solution. The solution t(u + k i= t iu i ) will be a new one if t. Net we assume that F () but F (u ) = and ker(f (u )) = K. Denote G(u) = F (u u ). 5

6 We have G() =, ker(g ()) = K. Denote S = [K, u u,..., u k u ] where u, u,..., u k are previousl found solutions and we know ker(f (u )) = K. Use this method together with certain decreasing strateg, e.g., a -d minimum search along the Newton direction using the original minimizing objective functional, (tr to keep awa from saddle points) one ma find all minima. This minimization along the Newton direction ν, i.e. min τ f(t(u τν + k i= t iu i )) will not penetrate into S since ν S and u S unless ν = τu.u. We ma use the semilinear elliptic PDE on dumbbell shaped domain to find its three local minima. Find and save a new critical point b APNM and determine if a critical point is a local minimum b small perturbation and descent method. Will ν = τu, τ? The standard convergence ma fail. Tr once b ν = F ( w) [u, S] to sta awa from the previous point. The algorithm ma continue. Note = F ( w)u, u implies that F is degenerate at w in the direction of u. When F = f, this implies that the function g(τ) = f(t(u + τu + k i= t iu i )) has an inflection point at τ =. 3 A Partial Newton-Correction Method Note that the above results are independent of an numerical method applied to solve A(u, t, t,, t k ) =. One ma think of appling a usual Newton method w n+ = w n + ν n where w n = (u n, t n, t n,, t n k ), un S and the Newton direction ν n is solved from A (w n )ν n = A(w n ) (3.) with A (w n ) = (A u(w n ), A t(w n ), A t (w n ),, A t k (w n )), ν n = (ν n, dt n, dt n,, dt n k ). The difficult is that we cannot guarantee the constraint ν n S to be satisfied. Consequentl we cannot guarantee ū n+ = u n + ν n S or u n+ = ūn+ ū n+ S, a condition in our AST (2.) formulation. Thus the usual Newton method is not well-defined to work with AST (2.) due to the constraint on the u-variable. Motivated b the idea from [2, 25], we design a new partial Newton-correction method (PNCM). To eplain the idea, we define a solution set M A = {p(u) = t(u + t u + + t k u k ) : t, u S, A(u, t, t,, t k ) [u, S]} = M F = {p(u) = t(u + t u + + t k u k ) : t, u S, F (p(u)) [u, S]}. (3.2) It is clear that an solution w / S satisfies w M A. When S = {}, we have M A = M F = {tu : u H, u =, F (tu) u} = N, (3.3) where N is called the Nehari manifold in the literature [7]. The solution set M A generalizes the notion of the Nehari manifold. Note that M A N for an finite-dimensional subspace S U. For S = [u,..., u k ], denote the set-valued S- mapping P : S 2 H b P (u) = {w = t(u + t u + + t k u k ) : F (w) u, u,..., u k }, u S. 6

7 Denote an S- selection p : S H b Then the solution set can be epressed b p(u) P (u) u S. (3.4) M A = {P (u) : u S }. So the problem becomes to find u S b a Newton method such that F (p(u)) =. For this composite function, if we appl a Newton method directl, we need p to be differentiable. Instead, we design a partial Newton-correction method and assume a weaker condition in its local convergence analsis. In other words, we will develop a Newton method without assuming that F (p(u)) is differentiable in u. M F In [25], when F (u) = f(u) for some functional f, a negative gradient at a point w on is used to search for multiple critical points of f. When the negative gradient search leads to a point off M F, a correction formula is designed to pull the point back to M F. The method is quite successful in finding multiple solutions to variational PDEs and sstems [2,2,26]. Here in this paper, in order for us to solve a non-variational problem for multiple solutions and to speed up numerical computation, we tr to use a faster Newton method instead. We need to modif the idea. First we cannot directl appl a Newton method to F (w) = due to its wellknown barrier difficult. This difficult is overcome b the AST (.2) and its corresponding partial Newton method in [2]. We modif (.2) b a new AST (2.) and also devise a new partial Newton-correction method (PNCM). The idea is described in the following two steps. Choose an initial guess u S and set n = ; Step : (Correction) find w n = (u n, t n, t n,, t n k ) such that wn = t n (u n + k i= tn i u i ) M A. Thus A(w n ) [u n, S]; Step 2: (Partial Newton Iteration) update the variable u n b u n+ = the Newton direction ν n S is solved from a partial differential equation un +ν n u n +ν n S where A u(w n )ν n = A(w n ) or F ( w n )ν n = F (w n ). (3.5) Since the new point t n (u n+ + t n u + + t n k u k) will be off the solution set M A, a correction process is required. So we set n = n + and return to Step and the algorithm iteration can continue. Eample 3.. To justif the assumption F ( w)u, u, let us check F (u)() = L(u)()+ N(, u()) where L is linear and N is nonlinear satisfing N(, tu()) = t p N(, u()) for some p >. For S = {}, when w = t u u M F for some u =, t u, we have = F ( w), u = t u L(u), u + t p u N(u), u, F ( w)u, u = d dτ τ= F ( w + τu), u = d dτ τ=[(t u + τ) L(u), u + (t u + τ) p N(u), u ] = L(u), u + pt p u N(u), u. 7

8 3. A Flow Chart of a Partial Newton-Correction Method Step :(Initialization) Given S = [u,, u k ] and an initial guess w S. Denote w = t (ū + k t i u i ) for some ū S and t. Set m =. Using ( t, t,, t k ) as an initial i= guess to compute t, t,, t k from A(u, t, t,, t k ) u, u,, u k. Step 2:(Partial Newton Search) Find the Newton direction ν m S b solving the linear sstem A u(u m, t m, t m,, t m k )ν = A(u m, t m, t m,, t m k ) S. Step 3:(Correction) Set u m+ = (u m + ν m )/( u m + ν m ) S. Using t m, t m,, t m k as an initial guess to solve tm+, t m+,, t m+ k from A(u m+, t, t,, t k ) u m+, u,, u k. Step 4:(Terminate Condition) If A(u m+, t m+, t m+,, t m+ k ) L ε or ν m U δ then output u k+ = t m+ (u m+ + k t m+ i u i ) and stop; else go to Step 2 with u m+, t m+, t m+,, t m+ i= k as the initial guesses and update m = m +. 4 Numerical Eperiments For eample, as our model problem, the investigation of stead-state solutions to semilinear convection-diffusion-reaction equation leads to numericall approimate nontrivial solutions to a nonvariational PDE of the form F (u)() p u() + a() u() + λ()u() + κf(, u() )u() =, u U (5.) where p u() = div( u() p 2 u()) is the nonlinear p-laplacian differential operator, which has a variet of applications in phsical fields, such as in fluid dnamics when the shear stress τ and the velocit gradient u of the fluid are related in the manner τ() = r() u() p 2 u(), where p = 2, p < 2, p > 2 if the fluid is Newtonian, pseudoplastic, dilatant, respectivel. The p-laplacian operator also appears in the stud of flow in a porous media (p = 3 ), nonlinear elasticit (p > 2), and glaciolog (p (, 4 )). So far, 2 3 knowledge about solutions to 5. is still ver limited. We hope to eamine the qualitative behavior of solutions and find new phenomena through numerical investigation.the functions a(), λ() and constant κ are given and the nonlinear function f(, u)u is superlinear in u, U = W,2 (Ω) and Ω is a bounded open domain Ω R n. It is known that solutions u,..., u k to the nonvariational PDE (5.) satisf the condition S = [u,..., u k ] U V where U = W,2 (Ω), V = W,2 (Ω) and Ω R n is a bounded open domain. When a(), (5.) is a non-variational semilinear elliptic PDE [3]. Numerical methods for finding multiple solutions to such a nonvariational PDE are not et available in the literature. 8

9 Since PNCM does not assume a variational structure, in this section, it is applied to numericall find multiple solutions to (5.) for the focusing and defocusing with a(), and the non-variational cases. However numerical solutions found are no longer in a variational (Morse inde) order. The first two variational cases are used to check PNCM and the last nonvariational case is used b PNCM to do new numerical investigation. For 5., we define an augmented singular transform G(u, t, t,..., t k ) = t p F (t(u + t i u i )) (5.2) for each u S, u = where is in the dualit sense. At a point w = t(u+ k i= t iu i ) M G, the Newton direction ν S is solved from (3.5), a linear elliptic equation in U. G u(u, t, t,, t k )ν = G(u, t, t,, t k ), Ω, ν() =, Ω. (5.3) It can be done b man numerical solvers using a finite difference- method (FDM), a finiteelement method (FEM), etc. In our numerical computation, a FDM is applied b calling the Matlab subroutines NUMGRID and DELSQ for a(). Approimate solutions to the p-laplacian equation b W,2 method with the weak form. For J(u) = [ p + u() p+ f(, u())]d. For each basis function v i, we have d + λd, v i = J (u), v i = Thus d W,q can be used as a search direction. Ω Ω i= [ u() p 2 u() v i () f (, u())v i ()]d. The k+ scalars t, t,, t k in the correction process are solved from the k+ orthogonal conditions (3.2) b calling the Matlab subroutine FSOLVE with initial coefficients alwas set, unless specificall indicated otherwise, as ( t, t,, t k ) = (,,, ) for an initial ū S, and as indicated in Step 3 in PNCM for other u S. To monitor the convergence of our numerical computation in solving G(u, t, t,, t k ) =,, we check two error terms: the residual error G( ) and the norm of the Newton direction ν U at an approimation solution w M G are both less than ε =. In order to show the profile and contours of a solution u k clearl in one figure, a vertical translation u k.5 ma(abs(u k )) is used in the figures below. In the numerical eperiments, when FDM is used, we take h =.. in order to clearl see the mesh grids in figures, a coarse mesh is used to redraw the profiles and contours. One ma also find more information on a solution, its numerical error and convergence b zooming-in at the top portion of its figure. 9

10 In our numerical eperiments, an initial guess u is selected accordingl b solving { u () = c(), Ω, if u is concave down at, where c() =, if u u () =, Ω is concave up at, (5.4), otherwise, and then a normalization. We set c(, 2 ) = if (, 2 ) (, 2 ). and c(, 2 ) = otherwise. Thus the peak-location(s) (, 2 ) of an initial guess can be convenientl selected. Since an positive function can be at most near but not eactl orthogonal to a positive solution, in order to find multiple positive solutions and also to see how a fied initial guess u leads to different solutions after S changes, we will use an initial guess of the form w = t(ū + t ū + + t k ū k ) S instead of u S where ū S, ū,, ū k are determined accordingl b two Gram-Schmidt orthogonalization processes (GSO): () Initial guess last Gram-Schmidt orthogonalization (LGSO) [u, S] = S [ū ] ū = u u, v i = u i i u i, ū j ū j, j= ū i = v i, 2 i k, v v i k+ = u k (5.5) u, ū j ū j, ū = v k+. v k+ Thus S = [ū,, ū k ] = [u,, u k ] is not changed but the initial guess u is changed. With ū S, PNCM finds a new solution not in S. (2) Initial guess first Gram-Schmidt orthogonalization (FGSO) [u, S] = [u ] S i ū = u, v i = u i u i, ū ū u i, ū j ū j, ū i = v i v i, i k. (5.6) j= Thus ū = u remains the same but S = [u,, u k ] is changed to S = [ū,, ū k ], denoted b S again. With u S, PNCM finds different solutions with the same u but different S. Thus PNCM ma also find a previousl found solution not contained in the changed S. Consider the following nonvariational equation with U = H (Ω) and V = W,2 (Ω): j= F (u)() = p u() + u() r u() 3 = in Ω = (, ) 2. (5.7) Then we have a() = (, 2 ). We have ker(f ()) = {}. PNCM is applicable.

11 For k =,,, we set S = [u, u 2,, u k ]. Thus the AST is G(u, t, t,, t k )() (5.8) = { [ (u() + t i u i ())] 2 + [ (u() + t i u i ())] 2 p 2 2 (u() + t i u i ())} 2 { [ (u() t [ p 2 (u() + i= t i u i ())] 2 + [ (u() + 2 i= t i u i ()) + 2 (u() + 2 i= t p 4 ( ) r 2 (u() + t i u i ()) 3. i= i= i= t i u i ())] 2 p 2 2 t i u i ())] i= 2 (u() + i= t i u i ())} At a given point w = u+ k i= t iu i M G, the Newton direction ν S is solved from (3.5), a linear elliptic equation of the form A 2 ν() 2 Where B 2 ν() C 2 ν() D ν() E ν() F ν() = G(u, t, t 2 2,, t k )(); in Ω. 2 2 (5.9) A = H p (p 2)H p 4 2 ( w ) 2, B = 2(p 2)H p 4 2 w w, 2 C = H p (p 2)H p 4 2 ( w 2 ) 2, D = (p 2)( 2 w 2 E = (p 2)( 2 w 2 F = 3 t p 4 ( ) r 2 w 2, H = ( w ) 2 + ( w 2 ) 2, + 2 w )H p w )H p M = ( w ) 2 2 w + 2 w w 2 2 N = 2( w 2 w + w 2 2 Q = 2( w 2 w + w w 2 ), 2 w 2 ), w + (p 2)(p 4)H p 6 w 2 M + (p 2)H p 4 2 N t, p 2 w + (p 2)(p 4)H p 6 w 2 M + (p 2)H p 4 2 Q 2 2 t 2, p 2 2 w + ( w 2 ) 2 2 w, 2 The following 4 eamples without a variational structure show how PNCM with LGSO finds new solutions. In general, if we choose an initial guess u and the initial coefficients 2 2 i=

12 (t, t,, tk ) more fleibl or more accordingl to some known information of solutions, we can find more (desirable) solutions. Case : (p =.75, r = 6 in (5.7). Using different u and LGSO): B choosing an initial guess u with different peak-locations (, 2 ) and taking LGSO strateg as (5.5), 9 solutions u,, u9 are found b PNCM. Their profiles and the peak locations of u are shown in Figures, 2 and 3. u =7.2 at (.9,.9), p=.75, r=6 u2 =6.992 at (.9,.9), (.9,.9), p=.75, r=6 u3 =7.56 at (.9,.9), (.9,.9), p=.75, r=6 G =9.3964e 7, v H=.27e 6 G =4.4895e 7, v H=.42e G =.422e 7, v H=.7e (u ) (u2 ).5 (u3 ) Figure : NV-Case. Solutions u with (, 2 ) at (.6,.6). u2 with (, 2 ) at (.6,.6) and (.6,.6), u3 with (, 2 ) at (.6,.6) and (.6,.6) u =6.967 at (.9,.9),(.9,.9), (.9,.9) u =6.99 at (.9,.9), (.9,.9), (.9,.9), (.9,.9) u =7.425 at (.9,.9),(.9,.9), (.9,.9), (.9,.9) G =.4e 9, v H=3.498e 8, p=.75, r=6 G =6.846e 7, v H=.34e 6, p=.75, r=6 G =5.883e 7, v H=.2728e 6, p=.75, r= (u4 ) (u5 ) (u6 ) Figure 2: NV-Case. Solutions u4 with (, 2 ) at (.8,.8),(.8,.8) and (.5,.5), u5 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), u6 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,,,, correspondingl Case 2 (p = 2.25, r = 6 in (5.7). Using different u and LGSO): We choose different peak-locations (, 2 ) for u and take LGSO strateg as (5.5). 9 solutions u,, u9 are found b PNCM. Their profiles and the peak locations of u are shown in Figures 4, 5 and 6, the are ver nice smmetric. 2

13 u = at (.9,.9), (.9,.9), (.9,.9), (.9,.9) u =6.992 at (.9,.9), (.9,.9), p=.75, r=6 u =7.29 at (.9,.9), (.9,.9), p=.75, r=6 G =5.2296e 2, v H=3.33e 8, p=.75, r=6 G =6.66e 7, v H=7.32e 6 G =4.895e 7, v H=8.363e (u7 ) (u8 ).5 (u9 ) Figure 3: NV-Case. Solutions u7 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,,,, correspondingl, u8 with (, 2 ) at (.6,.6) and (.6,.6), c(, 2 ) =, correspondingl, u9 with (, 2 ) at (.6,.6) and (.6,.6), c(, 2 ) =,, correspondingl u =3.282 at (.72,.72), p=2.25, r=6 u =3.49 at (.7,.7), (.7,.7), p=2.25, r=6 u =3.727 at (.72,.72),(.72,.72), p=2.25, r=6 G =4.2475e, v H=4.3568e 6 G =.36e 6, v H=3.628e 7 G =.77e 4, v H=3.795e (u ) (u2 ).5 (u3 ) Figure 4: NV-Case 2. Solutions u with (, 2 ) at (.6,.6). u2 with (, 2 ) at (.6,.6) and (.6,.6), u3 with (, 2 ) at (.6,.6) and (.6,.6) u4 = at (.72,.7), (.7,.7), (.7,.7) u5 = at (.68,.68),(.68,.68),(.68,.68),(.68,.68), u6 =3.729 at (.74,.74), (.74,.74), (.74,.74), G =9.358e 8, v H=9.94e 7, p=2.25, r=6 G =5.2647e 8, v H=8.3743e 7, p=2.25, r=6 (.74,.74), G =2.58e, v H=9.3996e 6, p=2.25, r= (u4 ) 5: NV-Case 2. Solutions u4 (u Figure with, 2 ) at (.8,.8),(.8,(u.8) 5 ) ( 6 ) and (.5,.5), u5 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), u6 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,,,, correspondingl Case 3 (p = 2.25, r = 6 in (5.7). Using a fied asmmetric u and LGSO): B using a fied asmmetric initial guess u obtained in (5.4) with peak-location (, 2 ) at (-., -.9), c(, 2 ) = and taking LGSO strateg as (5.5), PNCM finds 4 different solutions u,, u4, where u in Figure 4 and u2, u3, u4 are listed in Figure 7, u3 is asmmetric, 3

14 u =3.954 at (.74,.74), u 7 =3.45 at (.72,.72) u =3.944 at (.74,.74), u 8 7 ma (.72,.72),(.72,.72), G =2.58e 6, v H=9.3996e 7, =3.434 at (.72,.72), u =3.554 at (.74,.74), (.74,.74), p=2.25, r=6 9 8 ma (.72,.72), G =6.88e 7, v H=6.676e 7,p=2.25, r=6 G =7.543e 8, v H=5.7569e 7 p=2.25, r= (u7 ) 6: NV-Case 2. Solutions u7 with (u8 ) ( Figure, 2 ) at (.6,.6), (.6,(u.6), 9 ) (.6,.6) and (.6,.6), c(, 2 ) =,,,, correspondingl, u8 with (, 2 ) at (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,, correspondingl, u9 with (, 2 ) at (.6,.6) and (.6,.6), c(, 2 ) =,, correspondingl u, u2, u4 have ver nice smmetr. u2 =3.49 at (.7,.7), (.7,.7), p=2.25, r=6 u3 = at (.7,.46), u3 min= at (.86,.86), u4 =3.727 at (.72,.72), (.72,.72), p=2.25, r=6 G =6.338e 2, v H=3.4334e 9 G =5.3433e, v H=3.7994e 8,p=2.25, r=6 G =3.67e 4, v H=3.742e (u2 ) (u4 )5.7 Figure 7: NV-Case 3. (u Solutions u2, u3 and u4 to equation 3) Case 4: (p = 2.5, r = 6 in (5.7). Using different u and LGSO): We choose different peak-locations (, 2 ) for u and take LGSO strateg as (5.5). 9 solutions u,, u9 are found b PNCM. Their profiles and the peak locations of u are shown in Figures 8, 9 and, the have ver nice smmetr. u = at (.68,.68), p=2.5, r=6 u2 = at (.66,.66), (.66,.66), p=2.5, r=6 u3 =24.84 at (.66,.66),(.66,.66), p=2.5, r=6 G =4.58e, v =3.989e 6 G =6.2338e, v =8.2583e 6 G =4.687e 6, v =.383e 6 H H H (u ) (u2 ) (u3 ) Figure 8: NV-Case 4. Solutions u with (, 2 ) at (.6,.6), u2 with (, 2 ) at (.6,.6) and (.6,.6), u3 with (, 2 ) at (.6,.6) and (.6,.6) 4

15 u = at (.64,.64),(.64,.64), (.64,.64) u = at (.64,.64),(.64,.64),(.64,.64),(.64,.64), u = at (.7,.7), (.7,.7), (.7,.7), (.7,.7) G =3.863e, v H=.6246e 8, p=2.5, r=6 G =2.6724e 6, v H=5.982e 7, p=2.5, r=6 G =2.7283e 7, v H=2.624e 6, p=2.5, r= (u4 ) (u5 ).5 (u6 ) Figure 9: NV-Case 4. Solutions u4 with (, 2 ) at (.8,.8),(.8,.8) and (.5,.5), u5 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), u6 with (, 2 ) at (.6,.6), (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,,,, correspondingl u =26.7 at (.7,.7), u 7 = at (.68,.68), u =26.7 at (.7,.7) u = at (.68,.68), u = at (.7,.7), (.7,.7), p=2.5, r=6 (.68,.68), G =4.459e 7, v H=4.6999e 7, p=2.5, r=6 G =3.53e 6, v H=5.638e ma (.68,.68), G =3.2933e, v H=3.425e 6, p=2.5, r= (u7 ) (u8 ) (u9 ) Figure : NV-Case 4. Solutions u7 with (, 2 ) at (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,, correspondingl, u8 with (, 2 ) at (.6,.6), (.6,.6) and (.6,.6), c(, 2 ) =,, correspondingl, u9 with (, 2 ) at (.6,.6) and (.6,.6), c(, 2 ) =,, correspondingl Conclusion Remark: B using the information obtained from previousl found solutions, a new augmented singular transform (AST) is derived in this paper to find more solutions. Its formulation is mathematicall validated. An applicabilit result. The results significantl improved those established in [2] and resolved several theoretical issues left unsolved in [2]. Then a partial Newton-correction method (PNCM) is developed to carr out numerical computation with eas selection of initial guesses. Its local convergence is also established under a correction condition. In numerical eamples, PNCM is first applied to solve a non-variational PDE for multiple solutions. Such solutions are for the first time to be numerical computed and visualized. Since our formulation is ver general and simple, it has a great common interest in science and engineering communit. Also our approach is quite eas to follow, we epect that the results presented in this paper will stimulate people to further modif or generalize our approach to solve various multiple solution problems. We are working on modifing the formulation so that it can solve non-variational quasilinear 5

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