Acharya Ananta

Munoz Victor

Nichols Dustin

Shivaji Ratnasingham

English

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2023, No. 7, 1–10; https://doi.org/10.14232/ejqtde.2023.1.7 www.math.u-szeged.hu/ejqtde/An exact bifurcation diagramfor a p–q Laplacian boundary value problemAnanta Acharya, Victor Munoz, Dustin Nichols andRatnasingham ShivajiBUniversity of North Carolina Greensboro, Greensboro, NC 27412, USAReceived 28 June 2022, appeared 2 March 2023Communicated by John R. GraefAbstract. We study positive solutions to the p–q Laplacian two-point boundary valueproblem: {−µ[(u′)p−1]′ − [(u′)q−1]′ = λu(1 − u) on (0, 1)u(0) = 0 = u(1)when p = 4 and q = 2. Here λ > 0 is a parameter and µ ≥ 0 is a weight parameterinfluencing the higher-order diffusion term. When µ = 0 (the Laplacian case) the exactbifurcation diagram for a positive solution is well-known, namely, when λ ≤ π2 thereare no positive solutions, and for λ > π2 there exists a unique positive solution uλ,µsuch that ∥uλ,µ∥∞ → 0 as λ → π2 and ∥uλ,µ∥∞ → 1 as λ → ∞. Here, we will provethat for all µ > 0 similar bifurcation diagrams preserve, and they all bifurcate from(λ, u) = (π2, 0). Our results are established via the method of sub-super solutions anda quadrature method. We also present computational evaluations of these bifurcationdiagrams for various values of µ and illustrate how they evolve when µ varies.Keywords: positive solutions, p–q Laplacian, Dirichlet boundary conditions, exact bi-furcation diagram.2020 Mathematics Subject Classification: 34B08, 34B18.1 IntroductionWe analyze positive solutions to the boundary value problem:{−µ[(u′)p−1]′ − [(u′)q−1]′ = λ f (u) on (0, 1),u(0) = 0 = u(1)(1.1)when p = 4 and q = 2. Here we will choose f to be a smooth function such that f (0) = 0, andλ > 0, µ ≥ 0 are parameters, with µ influencing the higher-order diffusion term. Study of p–qBCorresponding author. Email: r_shivaj@uncg.edu2 A. Acharya, V. Munoz, D. Nichols and R. ShivajiLaplacian problems have been of interest in the literature (see [1–4, 6]) as they arise as steadystates of reaction-diffusion processes when the diffusion involved is of a certain nonlinearclass. See [6] in particular, where they note that equations of this type arise in biophysics,plasma physics, and chemical reactor design. However, our motivation of this study is purelymathematical. We will begin with the case µ = 0 when the exact bifurcation diagram forpositive solutions is known, and then prove that for all µ > 0 similar bifurcation diagramspreserve, and that they all bifurcate from the branch of trivial solutions at the same pointwhere the bifurcation occurs in the case µ = 0. In particular, in this study we choosef (s) = s(1 − s); s ∈ [0, 1],for which when µ = 0 it is well-known that the bifurcation diagram of positive solutions isexact (see [5, 7]) of the form:Figure 1.1: A prototypical bifurcation diagram of positive solutions for (1.1)when f (s) = s(1 − s) and µ = 0.Namely, for λ ≤ π2 there are no positive solutions, and for λ > π2, there is a uniquepositive solution uλ,0 such that ∥uλ,0∥∞ → 0 as λ → π2 and ∥uλ,0∥∞ → 1 as λ → ∞. Here, weextend the study for the case µ > 0. In particular, we prove:Theorem 1.1. Let µ > 0 be fixed. Then for λ ≤ π2, (1.1) has no positive solution, and for λ > π2,(1.1) has a unique positive solution uλ,µ such that ∥uλ,µ∥∞ → 0 as λ → π2 and ∥uλ,µ∥∞ → 1 asλ → ∞. Further, for λ > π2, if µ2 > µ1 then uλ,µ1(x) ≥ uλ,µ2(x) for all x ∈ [0, 1].Remark 1.2. Theorem 1.1 establishes that for each µ > 0, a similar exact bifurcation diagramfor positive solutions to the case when µ = 0 preserves and each bifurcates from (λ, u) =(π2, 0) (see Figure 1.2).Remark 1.3. Our analysis uses the relationship (2.3), which determines the bifurcation dia-gram. The derivation of (2.3) uses p = 4 and q = 2 (see the proof of Lemma 2.2). Establishingsuch a result for any p > q > 1 is an open problem. Further, our analysis is restricted to thespecific f we chose.We prove our results by the method of sub-super solutions (see [4]) and via using thequadrature method discussed in [2] (an extension of the quadrature method first introducedfor the case µ = 0 in [5]). In Section 2 we present preliminaries, in Section 3 we prove Theorem1.1, and in Section 4 we compute the bifurcation diagrams numerically for several values of µand demonstrate their evolution as µ varies.An exact bifurcation diagram for a p–q Laplacian BVP 3Figure 1.2: Prototypical bifurcation diagrams of positive solutions for (1.1) whenµ ≥ 0.2 PreliminariesIn this section, we introduce definitions of a subsolution and a supersolution of (1.1) andstate a sub-supersolution theorem that will be used to prove our existence result for positivesolutions. We also state a result via a quadrature method which we will use in our analysis(combined with an existence result obtained via sub-supersolutions) to establish exact detailson the bifurcation diagram for positive solutions.By a subsolution of (1.1) we mean ψ ∈ C2((0, 1)) ∩ C([0, 1]) that satisfies{−µ[(ψ′)3]′ − ψ′′ ≤ λ f (ψ) on (0, 1),ψ(0) ≤ 0, ψ(1) ≤ 0.(2.1)By a supersolution of (1.1) we mean Z ∈ C2((0, 1)) ∩ C([0, 1]) that satisfies{−µ[(Z′)3]′ − Z′′ ≥ λ f (Z) on (0, 1),Z(0) ≥ 0, Z(1) ≥ 0.(2.2)Then the following result holds:Lemma 2.1. Let ψ and Z be a subsolution and a supersolution of (1.1) respectively such that ψ ≤ Z.Then (1.1) has a solution u ∈ C2((0, 1)) ∩ C([0, 1]) such that u ∈ [ψ, Z].Proof. See [4].Lemma 2.2. Let λ, µ > 0 be fixed and ρ ∈ (0, 1). Then (1.1) has a positive solution with ∥uλ,µ∥∞ = ρif and only if λ and ρ satisfyG(λ, ρ) =∫ ρ0ds√√12µλ[F(ρ)− F(s)] + 1 − 1=12√3µ, (2.3)where F(s) =∫ s0 f (z)dz.4 A. Acharya, V. Munoz, D. Nichols and R. ShivajiFigure 2.1: A prototypical shape of a positive solution to (1.1).Proof. (See also [3].) Suppose uλ,µ is a positive solution of (1.1) with ∥uλ,µ∥∞ = ρ. Since (1.1)is autonomous, uλ,µ must be symmetric about x = 12 , increasing on (0,12 ), and decreasing on( 12 , 1). See Figure 2.1.Multiplying the differential equation in (1.1) by u′λ,µ(x) for x ∈ [0, 12 ], we get−µu′λ,µ(x)[(u′λ,µ(x))3]′ − u′λ,µ(x)[u′λ,µ(x)]′ = u′λ,µ(x)λ f (uλ,µ(x)), (2.4)which can be written as−3µ4[(u′λ,µ(x))4]′ − 12[(u′λ,µ(x))2]′ = λ[F(uλ,µ(x))]′; x ∈[0,12]. (2.5)Integrating (2.5) with respect to x over[0, 12], we obtain3µ[u′λ,µ(x)]4 + 2[u′λ,µ(x)]2 = 4λ[F(ρ)− F(uλ,µ(x))]; x ∈[0,12]. (2.6)Solving (2.6) for [u′λ,µ(x)]2, we obtain[u′λ,µ(x)]2 =√12µλ[F(ρ)− F(uλ,µ(x))] + 1 − 13µ; x ∈[0,12].Since u′λ,µ(x) > 0 for x ∈[0, 12], it follows thatu′λ,µ(x) =√√12µλ[F(ρ)− F(uλ,µ(x))] + 1 − 1√3µ; x ∈[0,12]. (2.7)Integrating (2.7) with respect to x over[0, 12), we obtainx√3µ=∫ uλ,µ(x)0ds√√12µλ[F(ρ)− F(s)] + 1 − 1; x ∈[0,12), (2.8)An exact bifurcation diagram for a p–q Laplacian BVP 5and letting x →( 12)−, we obtain (2.3):G(λ, ρ) =∫ ρ0ds√√12µλ[F(ρ)− F(s)] + 1 − 1=12√3µ.Conversely, suppose λ and ρ ∈ (0, 1) are such that (2.3) is satisfied. Then for each x ∈[0, 12), we can find a unique uλ,µ(x) satisfying (2.8). We can now extend this uλ,µ on [0, 1] suchthat uλ,µ( 12)= ρ and uλ,µ(x) = uλ,µ(1− x) for x ∈( 12 , 1]. With the aid of the Implicit FunctionTheorem, we can show that uλ,µ ∈ C2((0, 1)) ∩ C([0, 1]) and then it is easy to show it satisfies(1.1). Hence, (2.3) determines the bifurcation diagram of positive solutions uλ,µ for (1.1) with∥uλ,µ∥∞ = ρ ∈ (0, 1).Remark 2.3. If µ = 0, (1.1) becomes the boundary value problem:{−u′′ = λ f (u) on (0, 1),u(0) = 0 = u(1)(2.9)and by the quadrature method described in [5], the bifurcation diagram for positive solutionsof (2.9) is determined byλ = 2{∫ ρ0ds√F(ρ)− F(s)}2; ρ ∈ (0, 1). (2.10)3 Proof of Theorem 1.1Claim: Nonexistence of positive solutions for λ ≤ π2.Suppose uλ,µ > 0; (0, 1) is a solution to (1.1) for λ ≤ π2. Multiplying each term of thedifferential equation by sin(πx) and integrating on (0, 1), we have−µ∫ 10[(u′λ,µ(x))3]′ sin(πx)dx −∫ 10u′′λ,µ(x) sin(πx)dx = λ∫ 10uλ,µ(x)[1 − uλ,µ(x)] sin(πx)dx.(3.1)Equivalently, we have−µ∫ 10[(u′λ,µ(x))3]′ sin(πx)dx + λ∫ 10[uλ,µ(x)]2 sin(πx)dx = (λ − π2)∫ 10uλ,µ(x) sin(πx)dx.(3.2)Since λ ≤ π2, we have(λ − π2)∫ 10uλ,µ(x) sin(πx)dx ≤ 0. (3.3)6 A. Acharya, V. Munoz, D. Nichols and R. ShivajiHowever,− µ∫ 10[(u′λ,µ(x))3]′ sin(πx)dx + λ∫ 10[uλ,µ(x)]2 sin(πx)dx= −µsin(πx)[u′λ,µ(x)]3∣∣∣10︸ ︷︷ ︸=0−π∫ 10cos(πx)[u′λ,µ(x)]3dx+ λ ∫ 10 [uλ,µ(x)]2 sin(πx)dx= µπ[ ∫ 1/20cos(πx)[u′λ,µ(x)]3︸ ︷︷ ︸>0dx +∫ 11/2cos(πx)[u′λ,µ(x)]3︸ ︷︷ ︸>0dx]+ λ∫ 10[uλ,µ(x)]2 sin(πx)dx︸ ︷︷ ︸>0> 0.This contradicts (3.3). Hence (1.1) has no positive solution for λ ≤ π2.Claim: Existence of a positive solution uλ,µ for λ > π2.Consider ψ(x) = ε sin(πx) with ε > 0. Then ψ′′(x) = −επ2 sin(πx) and (ψ′(x))3 = ε3π3[cos(πx)]3.Hence− µ[(ψ′(x))3]′ − ψ′′(x)− λψ(x)(1 − ψ(x))= −µ(− 3ε3π4 cos2(πx) sin(πx))−(− επ2 sin(πx))− λε sin(πx)[1 − ε sin(πx)]= ε sin(πx)(3µε2π4 cos2(πx) + π2 − λ + λε sin(πx))< 0; x ∈ (0, 1)for ε ≈ 0 when λ > π2. Clearly the boundary conditions are satisfied by ψ. Thus, ψ is asubsolution of (1.1) for λ > π2. Now Z ≡ 1 is a supersolution of (1.1) and ψ < Z for ε ≈ 0.Hence by Lemma 2.1, (1.1) has a positive solution uλ,µ ∈ [ψ, Z] for all λ > π2.Claim: Existence of a unique positive solution uλ,µ such that ∥uλ,µ∥∞ → 0 as λ → π2 and∥uλ,µ∥∞ → 1 as λ → ∞.Recall G(λ, ρ) from (2.3). Note thatG(λ, ρ) =∫ ρ0ds√√12µλ[F(ρ)− F(s)] + 1 − 1=∫ 10ρ√√12µλ[F(ρ)− F(ρv)] + 1 − 1dv. (3.4)Now, using (3.4) we haveGρ(λ, ρ)=∫ 10N(v)√2λµρ2(2ρ(v3 − 1)− 3v2 + 3) + 1(√2λµρ2(2ρ(v3 − 1)− 3v2 + 3) + 1 − 1)3/2 dv,(3.5)whereN(v) = λµρ2(ρ(v3 − 1)− 3v2 + 3)−√2λµρ2(2ρ(v3 − 1)− 3v2 + 3)+ 1 + 1.An exact bifurcation diagram for a p–q Laplacian BVP 7Clearly the denominator of (3.5) is positive. Further, N(1) = 0. Hence, if we prove thatN′(v) < 0, then N(v) has to be positive on [0, 1). Now,N′(v) = 3λµρ2v(ρv − 2)− 6λµρ2v(ρv − 1)√2λµρ2(2ρ(v3 − 1)− 3v2 + 3)+ 1,so N′(v) < 0 provided that 2 − ρv > 2(1−ρv)√2λµρ2σ(v)+1, where σ(v) =(2ρ(v3 − 1)− 3v2 + 3). Butsince σ′(v) = 6v(ρv − 1) < 0 and σ(1) = 0, we must have σ(v) ≥ 0; v ∈ (0, 1). Hence,N′(v) < 0 provided 2 − ρv > 2(1 − ρv), which is clearly true. So Gρ(λ, ρ) > 0 for λ > 0 andρ ∈ (0, 1). Now combining this with our existence of a positive solution for λ > π2, we seethat there exists a unique ρ ∈ (0, 1) such that G(λ, ρ) = 12√3µ. Further, from (2.3) it is easy tosee that Gλ(λ, ρ) < 0 for λ > 0 and ρ ∈ (0, 1) (See Figure 3.1). Thus, by the Implicit FunctionTheorem, there exists a unique function λ : (0, 1) → (π2, ∞) satisfying G(λ(ρ), ρ) = 12√3µanddλdρ= −Gρ(λ, ρ)Gλ(λ, ρ)> 0. (3.6)Recall that we already established a positive solution for λ > π2. Combining this result with(3.6) we now have a unique positive solution uλ,µ for λ > π2. Further, combining with ournonexistence result for λ ≤ π2, we have the following:limρ→0λ(ρ) = π2(limλ→π2∥uλ,µ∥∞ = 0)limρ→1λ(ρ) = ∞(limλ→∞∥uλ,µ∥∞ = 1).Figure 3.1: Plots of G(λ, ·) for various λ. Observe their intersections with thelevel 12√3µwhen µ = 1.8 A. Acharya, V. Munoz, D. Nichols and R. ShivajiClaim: For µ2 > µ1, uλ,µ1(x) ≥ uλ,µ2(x) for all x ∈ [0, 1].Let µ2 > µ1 and λ > π2 be fixed. Now, let uλ,µ1 be a positive solution to (1.1) with µ = µ1.Then uλ,µ1 satisfies −µ1[(u′λ,µ1(x))3]′ − u′′λ,µ1(x) = λ f (uλ,µ1(x)); x ∈ (0, 1). We proceed byshowing that uλ,µ1 is a supersolution to (1.1) with µ = µ2. Observe that−µ2[(u′λ,µ1(x))3]′ − u′′λ,µ1(x) = −µ2(− 1µ1(λ f (uλ,µ1(x)) + u′′λ,µ1(x)))− u′′λ,µ1(x)=µ2µ1(λ f (uλ,µ1(x)) + u′′λ,µ1(x))− u′′λ,µ1(x)=µ2µ1λ f (uλ,µ1(x)) +(µ2µ1− 1)u′′λ,µ1(x)≥ λ f (uλ,µ1(x)); x ∈ (0, 1)provided that (µ2µ1− 1)(λ f (uλ,µ1(x)) + u′′λ,µ1(x))≥ 0; x ∈ (0, 1). (3.7)Given our assumption that µ2 > µ1, we haveµ2µ1− 1 > 0. By (1.1) with p = 4 and q = 2, it iseasy to see that u′′λ,µ1(x) =−λ f (uλ,µ1 (x))1+3µ1(u′λ,µ1 (x))2 ; x ∈ (0, 1). Henceλ f (uλ,µ1(x)) + u′′λ,µ1(x) = λ f (uλ,µ1(x))(1 − 11 + 3µ1(u′λ,µ1(x))2)≥ 0; x ∈ (0, 1).So (3.7) is satisfied and uλ,µ1 is a supersolution to (1.1) with µ = µ2. Recall that ψ(x) =ε sin(πx) with ε > 0 and ε ≈ 0 is a subsolution to (1.1) for any µ > 0 when λ > π2 and clearlyψ ≤ uλ,µ1 when ε ≈ 0. Thus, the unique positive solution uλ,µ2 to (1.1) with µ2 when λ > π2must be such that uλ,µ2 ∈ [ψ, uλ,µ1 ]. Hence, uλ,µ1(x) ≥ uλ,µ2(x) for all x ∈ [0, 1].4 Computation of bifurcation diagrams as µ variesThe bifurcation diagrams for µ > 0 in Figure 4.1 are computed using (2.3). In particular, fora sequence of values ρ ∈ (0, 1), we determine the corresponding sequence of λ > 0 suchthat (2.3) is satisfied using the FindRoot function in Mathematica. The bifurcation curves aregenerated using linear interpolation of the points {(λ, ρ)}. Similarly, for the µ = 0 case, weapply (2.10).In Figure 4.2, we generate profiles of positive solutions for λ = 50, µ1 = 5, and µ2 = 30using (2.8) for x ∈ [0, 12 ) and appealing to the symmetry established in Lemma 2.2. This il-lustrates that u50,5(x) ≥ u50,30(x) for all x ∈ [0, 1] as described in Theorem 1.1 for particularchoices of µ1 and µ2. By considering a uniform sequence of x-values lying in [0, 1] and solv-ing (2.8) with corresponding λ, ρ, µ values within a specified tolerance using FindRoot, thenlinearly interpolating the points {(x, uλ,µ(x))}, we obtain the solution profiles.An exact bifurcation diagram for a p–q Laplacian BVP 9Figure 4.1: Evolution of exact bifurcation diagrams of positive solutions to (1.1)as µ ≥ 0 varies.Figure 4.2: Profiles of positive solutions uλ,µ1 and uλ,µ2 to (1.1) for λ = 50,µ1 = 5, and µ2 = 30.AcknowledgementsWe thank the referees for their suggestions which aided us in improving the presentation ofthis paper.References[1] A. Acharya, U. Das, R. Shivaji, Existence and multiplicity results for p–q Lapla-cian boundary value problems, Electron. J. Differential Equations 2021, No. 1, 293–300.MR4494204[2] U. Das, A. Muthunayake, R. Shivaji, Existence results for a class of p–q Laplacian semi-positone boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2020, No. 88, 1–7.https://doi.org/10.14232/ejqtde.2020.1.88; MR4208495; Zbl 1474.35227[3] L. F. O. Faria, O. H. Miyagaki, D. Motreanu, M. Tanaka, Existence results for nonlinearelliptic equations with Leray–Lions operator and dependence on the gradient, Nonlin-ear Anal. 96(2014), 154–166. https://doi.org/10.1016/j.na.2013.11.006; MR3143809;Zbl 1285.3501410 A. Acharya, V. Munoz, D. Nichols and R. Shivaji[4] L. F. O. Faria, O. H. Miyagaki, D. Motreanu, Comparison and positive solutions forproblems with the (p, q)-Laplacian and a convection term, Proc. Edinb. Math. Soc. (2)57(2014), No. 3, 687–698. https://doi.org/10.1186/s13661-016-0665-9; MR3251754;Zbl 1315.35114[5] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem,Indiana Univ. Math. J. 20(1970), No. 1, 1–13. MR0269922[6] G. Li, X. Liang, The existence of nontrivial solutions to nonlinear elliptic equation of p–qLaplacian type on RN , Nonlinear Anal. 71(2009), No. 5–6, 2316–2334. https://doi.org/10.1016/j.na.2009.01.066; MR2524439; Zbl 1171.35375[7] J. Shi, R. Shivaji, Persistence in reaction diffusion models with weak Allee effect,J. Math. Biol. 52(2006), No. 6, 807–829. https://doi.org/10.1007/s00285-006-0373-7;MR2235529; Zbl 1110.92055

An exact bifurcation diagram for a p-q Laplacian boundary value problem

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An exact bifurcation diagram for a p-q Laplacian boundary value problem

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