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Published: 10 March 2023

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1

Hsu, Tsing-San, and Huei-Li Lin. "Multiplicity of Positive Solutions for ap-q-Laplacian Type Equation with Critical Nonlinearities." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/829069.

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We study the effect of the coefficientf(x)of the critical nonlinearity on the number of positive solutions for ap-q-Laplacian equation. Under suitable assumptions forf(x)andg(x), we should prove that for sufficiently smallλ>0, there exist at leastkpositive solutions of the followingp-q-Laplacian equation,-Δpu-Δqu=fxu|p*-2u+λgxu|r-2u in Ω,u=0 on ∂Ω,whereΩ⊂RNis a bounded smooth domain,N>p,1<q<N(p-1)/(N-1)<p≤max⁡{p,p^*-q/(p-1)}<r<p^*,p^*=Np/(N-p)is the critical Sobolev exponent, andΔsu=div(|∇u|s-2∇uis thes-Laplacian ofu.

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2

Motreanu, Dumitru. "Quasilinear Dirichlet problems with competing operators and convection." Open Mathematics 18, no. 1 (January 1, 2020): 1510–17. http://dx.doi.org/10.1515/math-2020-0112.

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Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

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3

Motreanu, Dumitru. "Quasilinear Dirichlet problems with competing operators and convection." Open Mathematics 18, no. 1 (December 22, 2020): 1510–17. http://dx.doi.org/10.1515/math-2020-0112.

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Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

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4

Abolarinwa, Abimbola, and Shahroud Azami. "Comparison estimates on the first eigenvalue of a quasilinear elliptic system." Journal of Applied Analysis 26, no. 2 (December 1, 2020): 273–85. http://dx.doi.org/10.1515/jaa-2020-2024.

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AbstractWe study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber–Krahn for the first eigenvalue of a {(p,q)}-Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p-Laplacian, {1<p<\infty}, from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a {(p,q)}-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as {p,q\to 1,1}.

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5

Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Resonant Anisotropic (p,q)-Equations." Mathematics 8, no. 8 (August 10, 2020): 1332. http://dx.doi.org/10.3390/math8081332.

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We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x±∞ can be resonant with respect to the principal eigenvalue of (−Δp(z),W01,p(z)(Ω)). First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.

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6

李, 燕茹. "On a Class of (p(u),q(u))-Laplacian Problem." Pure Mathematics 11, no. 04 (2021): 586–98. http://dx.doi.org/10.12677/pm.2021.114072.

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7

Papageorgiou, Nikolaos S., Dongdong Qin, and Vicenţiu D. Rădulescu. "Nonlinear eigenvalue problems for the (p,q)–Laplacian." Bulletin des Sciences Mathématiques 172 (November 2021): 103039. http://dx.doi.org/10.1016/j.bulsci.2021.103039.

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8

Haghaiegh, Somayeh, and Ghasem Afrouzi. "Sub-super solutions for (p-q) Laplacian systems." Boundary Value Problems 2011, no. 1 (2011): 52. http://dx.doi.org/10.1186/1687-2770-2011-52.

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9

Manouni, Said El, Kanishka Perera, and Ratnasingham Shivaji. "On singular quasi-monotone (p, q)-Laplacian systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 3 (June 2012): 585–94. http://dx.doi.org/10.1017/s0308210510001356.

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10

Humphries, Peter. "Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level." International Mathematics Research Notices 2019, no. 18 (December 8, 2017): 5703–43. http://dx.doi.org/10.1093/imrn/rnx283.

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Abstract We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maaß newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $\Gamma_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Strömberg who proved this for $q = 9$ by different methods.

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11

Onete, Florin-Iulian, Nikolaos S. Papageorgiou, and Calogero Vetro. "A multiplicity theorem for parametric superlinear (p,q)-equations." Opuscula Mathematica 40, no. 1 (2020): 131–49. http://dx.doi.org/10.7494/opmath.2020.40.1.131.

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We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.

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12

PEREZ-LLANOS, MAYTE, and JULIO D. ROSS. "AN ANISOTROPIC INFINITY LAPLACIAN OBTAINED AS THE LIMIT OF THE ANISOTROPIC (p, q)-LAPLACIAN." Communications in Contemporary Mathematics 13, no. 06 (December 2011): 1057–76. http://dx.doi.org/10.1142/s0219199711004543.

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In this work we study the behavior of the solutions to the following Dirichlet problem related to the anisotropic (p, q)-Laplacian operator [Formula: see text] as p, q → ∞. Here Ω ⊂ ℝN× ℝKand [Formula: see text] and [Formula: see text] denote the gradient of u with respect to the first N variables (x variables) and with respect to the last K variables (y variables). We consider a sequence of exponents (pn, qn) that goes to infinity with pn/qn→ R. We prove that un, the solution with p = pn, q = qn, verifies un→ u∞uniformly in [Formula: see text], where u∞is the unique viscosity solution to [Formula: see text] Here [Formula: see text] and [Formula: see text] are the infinity Laplacian in x variables and in y variables, respectively.

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13

Isernia, Teresa. "Fractional p&q-Laplacian problems with potentials vanishing at infinity." Opuscula Mathematica 40, no. 1 (2020): 93–110. http://dx.doi.org/10.7494/opmath.2020.40.1.93.

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In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator.

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14

AFROUZI, G. A., and H. GHORBANI. "POSITIVE SOLUTIONS FOR A CLASS OF p(x)-LAPLACIAN PROBLEMS." Glasgow Mathematical Journal 51, no. 3 (September 2009): 571–78. http://dx.doi.org/10.1017/s0017089509005199.

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AbstractWe consider the system where p(x), q(x) ∈ C1(RN) are radial symmetric functions such that sup|∇ p(x)| < ∞, sup|∇ q(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞, 1 < inf q(x) ≤ sup q(x) < ∞, where −Δp(x)u = −div(|∇u|p(x)−2∇u), −Δq(x)v = −div(|∇v|q(x)−2∇v), respectively are called p(x)-Laplacian and q(x)-Laplacian, λ1, λ2, μ1 and μ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, where R is sufficiently large. We prove the existence of a positive solution when for every M > 0, $\lim_{u \rightarrow +\infty} \frac{h(u)}{u^{p^--1}} = 0$ and $\lim_{u \rightarrow +\infty} \frac{\gamma(u)}{u^{q^--1}} = 0$. In particular, we do not assume any sign conditions on f(0), g(0), h(0) or γ(0).

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15

Shahrokhi-Dehkordi, M. S. "On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain." Communications in Mathematics 25, no. 1 (June 27, 2017): 13–20. http://dx.doi.org/10.1515/cm-2017-0003.

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Abstract Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem-∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2uwhere μ is a positive parameter, 1 < q ≤ p < n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

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16

Tao, Mengfei, and Binlin Zhang. "Solutions for nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities." Advances in Nonlinear Analysis 11, no. 1 (January 1, 2022): 1332–51. http://dx.doi.org/10.1515/anona-2022-0248.

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Abstract In this article, we aimed to study a class of nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in R N {{\mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p, q)-Laplacian systems in the case of N = s p = l q N=sp=lq . It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p, q)-Laplacian systems is the main novelty of this article.

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17

Ercole, Grey. "On the resonant Lane–Emden problem for the p-Laplacian." Communications in Contemporary Mathematics 16, no. 04 (July 14, 2014): 1350033. http://dx.doi.org/10.1142/s0219199713500338.

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We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in [Formula: see text] to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and [Formula: see text]. A consequence of this result is that the best constant of the immersion [Formula: see text] is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space [Formula: see text] and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in [Formula: see text], when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L∞- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq‖∞ which is valid for q ∈ [1, p + ϵ] where [Formula: see text].

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18

Cerda, Patricio, Marco Souto, and Pedro Ubilla. "Existence of solution of some p,q${p,q}$‐Laplacian system under local superlinear conditions." Mathematische Nachrichten 295, no. 1 (January 2022): 44–57. http://dx.doi.org/10.1002/mana.201900424.

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19

Papageorgiou, Nikolaos S., Vicenţiu D. Rădulescu, and Jian Zhang. "Ambrosetti–Prodi problems for the Robin (p,q)-Laplacian." Nonlinear Analysis: Real World Applications 67 (October 2022): 103640. http://dx.doi.org/10.1016/j.nonrwa.2022.103640.

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20

Candito, Pasquale, Salvatore A. Marano, and Kanishka Perera. "On a class of critical (p, q)-Laplacian problems." Nonlinear Differential Equations and Applications NoDEA 22, no. 6 (October 1, 2015): 1959–72. http://dx.doi.org/10.1007/s00030-015-0353-y.

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21

Zerouali, A., B. Karim, O. Chakrone, and A. Boukhsas. "Resonant Steklov eigenvalue problem involving the (p, q)-Laplacian." Afrika Matematika 30, no. 1-2 (September 26, 2018): 171–79. http://dx.doi.org/10.1007/s13370-018-0634-9.

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22

Behboudi, F., and A. Razani. "Twoweak solutions for a singular (p,q)-Laplacian problem." Filomat 33, no. 11 (2019): 3399–407. http://dx.doi.org/10.2298/fil1911399b.

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Here, a singular boundary value problem involving the (p,q)-Laplacian operator in a smooth bounded domain in RN is considered. Using the variational method and critical point theory, the existence of two weak solutions is proved.

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23

Crînganu, Jenică, and Daniel Paşca. "Existence results for Dirichlet problems with (q,p)-Laplacian." Journal of Mathematical Analysis and Applications 387, no. 2 (March 2012): 828–36. http://dx.doi.org/10.1016/j.jmaa.2011.09.042.

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24

Papageorgiou, Nikolaos S., Calogero Vetro, and Francesca Vetro. "On a Robin (p,q)-equation with a logistic reaction." Opuscula Mathematica 39, no. 2 (2019): 227–45. http://dx.doi.org/10.7494/opmath.2019.39.2.227.

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We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter \(\lambda \gt 0\) varies. Also, we show that for every admissible parameter \(\lambda \gt 0\), the problem admits a smallest positive solution.

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25

Beals, Richard, and Nancy K. Stanton. "The Heat Equation for the -Neumann Problem, II." Canadian Journal of Mathematics 40, no. 2 (April 1, 1988): 502–12. http://dx.doi.org/10.4153/cjm-1988-021-8.

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Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

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26

Zerouali, Abdellah Ahmed, and Belhadj Karim. "Existence and non-existence of positive solution for (p, q)-Laplacian with singular weights." Boletim da Sociedade Paranaense de Matemática 34, no. 2 (July 13, 2015): 147–67. http://dx.doi.org/10.5269/bspm.v34i2.25229.

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We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 < q < p < \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ > 0 there exists an interval of eigenvalues for our eigenvalue problem.

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27

Marcos, Aboubacar, and Ambroise Soglo. "Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations." Discrete Dynamics in Nature and Society 2020 (July 25, 2020): 1–23. http://dx.doi.org/10.1155/2020/9756162.

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In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation ∂ρt,x/∂t=divxρt,x∇xG′ρ+Vqx−2∇xG′ρ+V. The potential V is not necessarily smooth but belongs to a Sobolev space W1,∞Ω. Given the initial datum ρ0 as a probability density on Ω, we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding W1,q.Ω↪↪Lq.Ω established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space.

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28

Figueiredo, Giovany, and Calogero Vetro. "The existence of solutions for the modified ( p ( x ) , q ( x ) ) -Kirchhoff equation." Electronic Journal of Qualitative Theory of Differential Equations, no. 39 (2022): 1–16. http://dx.doi.org/10.14232/ejqtde.2022.1.39.

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We consider the Dirichlet problem − Δ p ( x ) K p u ( x ) − Δ q ( x ) K q u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) in Ω , u | ∂ Ω = 0 , driven by the sum of a p ( x ) -Laplacian operator and of a q ( x ) -Laplacian operator, both of them weighted by indefinite (sign-changing) Kirchhoff type terms. We establish the existence of weak solution and strong generalized solution, using topological tools (properties of Galerkin basis and of Nemitsky map). In the particular case of a positive Kirchhoff term, we obtain the existence of weak solution ( = strong generalized solution), using the properties of pseudomonotone operators.

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29

Lu, Guozhen, and Yansheng Shen. "Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth." Advanced Nonlinear Studies 20, no. 3 (August 1, 2020): 579–97. http://dx.doi.org/10.1515/ans-2020-2098.

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AbstractIn this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:\left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right.where {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, {p>1}, {s\in(0,1)}, {ps<N}, {p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in {\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with {p<q\leq p^{\ast}_{s}} and {Q_{u}} and {Q_{v}} are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., {s=1}) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian {(-\Delta)^{s}} (i.e., {p=2} and {0<s<1}) has not been studied in the literature before.

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30

Zerouali, Abdellah, Belhadj Karim, Omar Chakrone, and Abdelmajid Boukhsas. "On a positive solution for $(p,q)$-Laplace equation with Nonlinear." Boletim da Sociedade Paranaense de Matemática 38, no. 4 (March 10, 2019): 219–133. http://dx.doi.org/10.5269/bspm.v38i4.36661.

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In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $1<q<p<\infty$ the results are completely different from those for the usua lSteklov eigenvalue problem involving the $p$-Laplacian with indefinite weight, which is retrieved when $\mu=0$. Precisely, we show that when $\mu>0$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.

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31

Banerjee, Subarsha. "Laplacian spectrum of comaximal graph of the ring ℤ_n." Special Matrices 10, no. 1 (January 1, 2022): 285–98. http://dx.doi.org/10.1515/spma-2022-0163.

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Abstract In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) of the ring Z n {{\mathbb{Z}}}_{n} for n > 2 n\gt 2 . We first determine the structure of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) for various n n . We show that Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) is Laplacian integral for n = p α q β n={p}^{\alpha }{q}^{\beta } , where p , q p,q are primes and α , β \alpha ,\beta are non-negative integers and hence calculate the number of spanning trees of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) for n = p α q β n={p}^{\alpha }{q}^{\beta } . The algebraic and vertex connectivity of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) have been shown to be equal for all n n . An upper bound on the second largest Laplacian eigenvalue of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) . We then investigate some properties and vertex connectivity of an induced subgraph of Γ ( Z n ) \Gamma \left({{\mathbb{Z}}}_{n}) . Some problems have been discussed at the end of this paper for further research.

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32

Motreanu, Dumitru. "Degenerated and Competing Dirichlet Problems with Weights and Convection." Axioms 10, no. 4 (October 22, 2021): 271. http://dx.doi.org/10.3390/axioms10040271.

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This paper focuses on two Dirichlet boundary value problems whose differential operators in the principal part exhibit a lack of ellipticity and contain a convection term (depending on the solution and its gradient). They are driven by a degenerated (p,q)-Laplacian with weights and a competing (p,q)-Laplacian with weights, respectively. The notion of competing (p,q)-Laplacians with weights is considered for the first time. We present existence and approximation results that hold under the same set of hypotheses on the convection term for both problems. The proofs are based on weighted Sobolev spaces, Nemytskij operators, a fixed point argument and finite dimensional approximation. A detailed example illustrates the effective applicability of our results.

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33

Afrouzi, G. a., and M. Bai. "Multiple Solution To (p,q)-laplacian Systems With Concave Nonlinearities." Journal of Mathematics and Computer Science 04, no. 01 (January 18, 2012): 60–70. http://dx.doi.org/10.22436/jmcs.04.01.09.

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Afrouzi, G. A., and M. Mirzapour. "Existence results for a class of (p,q) Laplacian systems." Nonlinear Analysis: Modelling and Control 15, no. 4 (October 25, 2010): 397–403. http://dx.doi.org/10.15388/na.15.4.14311.

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We establish the existence of a nontrivial solution for inhomogeneous quasilinear elliptic systems:−∆pu = λ a(x) u |u|γ−2 + α (α + β)–1 b(x) u |u|α−2 |v|β + f in Ω,−∆qv = µ d(x) v |v|γ−2 + β (α + β)–1 b(x) |u|α v |v|β−2 + g in Ω,(u,v) ∈ W01,p(Ω) × W01,q(Ω).Our result depending on the local minimization.

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Alkhutov, Yu A., and M. D. Surnachev. "On a Harnack inequality for the elliptic (p, q)-Laplacian." Doklady Mathematics 94, no. 2 (September 2016): 569–73. http://dx.doi.org/10.1134/s1064562416050252.

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Tanaka, Mieko. "Generalized eigenvalue problems for(p,q)-Laplacian with indefinite weight." Journal of Mathematical Analysis and Applications 419, no. 2 (November 2014): 1181–92. http://dx.doi.org/10.1016/j.jmaa.2014.05.044.

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Goel, Divya, Deepak Kumar, and K. Sreenadh. "Regularity and multiplicity results for fractional (p,q)-Laplacian equations." Communications in Contemporary Mathematics 22, no. 08 (August 20, 2019): 1950065. http://dx.doi.org/10.1142/s0219199719500652.

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This paper deals with the study of the following nonlinear doubly nonlocal equation: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are parameters. Here [Formula: see text] and [Formula: see text] are sign-changing functions. We prove [Formula: see text] estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case [Formula: see text] Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of [Formula: see text], we show the existence of a solution.

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Oubalhaj, Youness, Belhadj Karim, and Abdellah Zerouali. "Existence results to Steklov system involving the (p, q)-Laplacian." Boletim da Sociedade Paranaense de Matemática 41 (December 23, 2022): 1–9. http://dx.doi.org/10.5269/bspm.51711.

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In this paper, a quasilinear elliptic system involving a pair of (p,q)-Laplacian operators with Steklov boundary value conditions is studied. Using the Mountain Pass Geometry, we prove the existence of at least one weak solution. For the inﬁnitely many weak solutions, we based on Bratsch’s Fountain Theorem.

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Zitouni, Mohamed, Ali Djellit, and Lahcen Ghannam. "Radial positive solutions for (p(x),q(x))-Laplacian systems." Boletim da Sociedade Paranaense de Matemática 41 (December 23, 2022): 1–13. http://dx.doi.org/10.5269/bspm.51625.

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In this paper, we study the existence of radial positive solutions for nonvariational elliptic systems involving the p(x)-Laplacian operator, we show the existence of solutions using Leray-Schauder topological degree theory, sustained by Gidas-Spruck Blow-up technique.

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Yu, Changlong, Jing Li, and Jufang Wang. "Existence and uniqueness criteria for nonlinear quantum difference equations with $ p $-Laplacian." AIMS Mathematics 7, no. 6 (2022): 10439–53. http://dx.doi.org/10.3934/math.2022582.

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<abstract><p>Q-calculus plays an extremely important role in mathematics and physics, especially in quantum physics, spectral analysis and dynamical systems. In recent years, many scholars are committed to the research of nonlinear quantum difference equations. However, there are few works about the nonlinear $ q- $difference equations with $ p $-Laplacian. In this paper, we investigate the solvability for nonlinear second-order quantum difference equation boundary value problem with one-dimensional $ p $-Laplacian via the Leray-Schauder nonlinear alternative and some standard fixed point theorems. The obtained theorems are well illustrated with the aid of two examples.</p></abstract>

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Alves, C. O., G. Ercole, and G. A. Pereira. "Asymptotic behaviour asp→ ∞ of least energy solutions of a (p, q(p))-Laplacian problem." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 6 (January 17, 2019): 1493–522. http://dx.doi.org/10.1017/prm.2018.111.

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AbstractWe study the asymptotic behaviour, asp→ ∞, of the least energy solutions of the problem$$\left\{ {\matrix{ {-(\Delta _p + \Delta _{q(p)})u = \lambda _p \vert u(x_u) \vert ^{p-2}u(x_u)\delta _{x_u}} & {{\rm in}} & \Omega \cr {u = 0} \hfill \hfill \hfill & {{\rm on}} & {\partial \Omega ,} \cr } } \right.$$wherexuis the (unique) maximum point of |u|,$\delta _{x_{u}}$is the Dirac delta distribution supported atxu,$$\mathop {\lim }\limits_{p\to \infty } \displaystyle{{q(p)} \over p} = Q\in \left\{ {\matrix{ {(0,1)} & {{\rm if}} & {N < q(p) < p} \cr {(1,\infty )} & {{\rm if}} & {N < p < q(p)} \cr } } \right.$$and λp > 0 is such that$$\min \left\{ {\displaystyle{{{\rm \Vert }\nabla u{\rm \Vert }_\infty } \over {{\rm \Vert }u{\rm \Vert }_\infty }}:0 \ne u\in W^{1,\infty }(\Omega )\cap C_0(\bar{\Omega })} \right\} \les \mathop {\lim }\limits_{p\to \infty } (\lambda _p)^{1/p} < \infty .$$

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Yu, Chengwei. "Regularity for Quasi-Linear p-Laplacian Type Non-Homogeneous Equations in the Heisenberg Group." Mathematics 10, no. 21 (November 5, 2022): 4129. http://dx.doi.org/10.3390/math10214129.

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When 2−1/Q<p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group Hn, where Q=2n+2 is the homogeneous dimension of Hn.

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PAŞCA, DANIEL. "PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL INCLUSIONS SYSTEMS WITH (q, p)-LAPLACIAN." Analysis and Applications 09, no. 02 (April 2011): 201–23. http://dx.doi.org/10.1142/s0219530511001819.

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Luo, Huxiao, Shengjun Li, and Wenfeng He. "Non-Nehari Manifold Method for Fractional p-Laplacian Equation with a Sign-Changing Nonlinearity." Journal of Function Spaces 2018 (July 18, 2018): 1–5. http://dx.doi.org/10.1155/2018/7935706.

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We consider the following fractional p-Laplacian equation: -Δpαu+V(x)up-2u=f(x,u)-Γ(x)uq-2u, x∈RN, where N≥2, pα⁎>q>p≥2, α∈(0,1), -Δpα is the fractional p-Laplacian, and Γ∈L∞(RN) and Γ(x)≥0 for a.e. x∈RN. f has the subcritical growth but higher than Γ(x)uq-2u; however, the nonlinearity f(x,u)-Γ(x)uq-2u may change sign. If V is coercive, we investigate the existence of ground state solutions for p-Laplacian equation.

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Bonheure, Denis, and Julio D. Rossi. "The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p." Nonlinear Analysis: Theory, Methods & Applications 150 (February 2017): 104–13. http://dx.doi.org/10.1016/j.na.2016.11.001.

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Bai, Yunru, Leszek Gasiński, and Nikolaos S. Papageorgiou. "Nonlinear Eigenvalue Problems for the Dirichlet (p,2)-Laplacian." Axioms 11, no. 2 (January 30, 2022): 58. http://dx.doi.org/10.3390/axioms11020058.

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We consider a nonlinear eigenvalue problem driven by the Dirichlet (p,2)-Laplacian. The parametric reaction is a Carathéodory function which exhibits (p−1)-sublinear growth as x→+∞ and as x→0+. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as λ>0 varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to (p,q)-equations (q≠2).

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BARBU, LUMINIŢA, and GHEORGHE MOROŞANU. ""Eigenvalues of the (p, q, r)-Laplacian with a parametric boundary condition"." Carpathian Journal of Mathematics 38, no. 3 (July 26, 2022): 547–61. http://dx.doi.org/10.37193/cjm.2022.03.03.

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"Consider in a bounded domain $\Omega \subset \mathbb{R}^N$, $N\ge 2$, with smooth boundary $\partial \Omega$ the following nonlinear eigenvalue problem \begin{equation*} \left\{\begin{array}{l} -\sum_{\alpha \in \{ p,q,r\}}\rho_{\alpha}\Delta_{\alpha}u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega,\\[1mm] \big(\sum_{\alpha \in \{p,q,r\}}\rho_{\alpha}\mid \nabla u\mid ^{\alpha-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^{r-2}u ~ \mbox{ on} ~ \partial \Omega, \end{array}\right. \end{equation*} where $p, q, r\in (1, +\infty),~q<p,~r\not\in \{p, q\};$ $\rho_p, \rho_q, \rho_r$ are positive constants; $\Delta_{\alpha}$ is the usual $\alpha$-Laplacian, i.e., $\Delta_\alpha u=\, \mbox{div} \, (|\nabla u|^{\alpha-2}\nabla u)$; $\nu$ is the unit outward normal to $\partial \Omega$; $a\in L^{\infty}(\Omega),$ $b\in L^{\infty}(\partial\Omega)$ {are given nonnegative functions satisfying} $\int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0.$ Such a triple-phase problem is motivated by some models arising in mathematical physics. If $r \not\in (q, p),$ we determine a positive number $\lambda_r$ such that the set of eigenvalues of the above problem is precisely $\{ 0\} \cup (\lambda_r, +\infty )$. On the other hand, in the complementary case $r \in (q, p)$ with $r < q(N-1)/(N-q)$ if $q<N$, we prove that there exist two positive constants $\lambda_*<\lambda^*$ such that any $\lambda\in \{0\}\cup [\lambda^*, \infty)$ is an eigenvalue of the above problem, while the set $(-\infty, 0)\cup (0, \lambda_*)$ contains no eigenvalue $\lambda$ of the problem."

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Repovš, Dušan D., and Calogero Vetro. "The behavior of solutions of a parametric weighted $ (p, q) $-Laplacian equation." AIMS Mathematics 7, no. 1 (2021): 499–517. http://dx.doi.org/10.3934/math.2022032.

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<abstract><p>We study the behavior of solutions for the parametric equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0, $\end{document} </tex-math></disp-formula></p> <p>under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $, $ a_1, a_2 \in L^\infty(\Omega) $ with $ a_1(z), a_2(z) > 0 $ for a.a. $ z \in \Omega $, $ p, q \in (1, \infty) $ and $ \Delta_{p}^{a_1}, \Delta_{q}^{a_2} $ are weighted versions of $ p $-Laplacian and $ q $-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $ f(z, x) $ asymptotically as $ x \to \pm \infty $ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When $ \lambda $ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.</p></abstract>

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BARBU, LUMINIŢA, and GHEORGHE MOROŞANU. "On a Steklov eigenvalue problem associated with the (p,q)-Laplacian." Carpathian Journal of Mathematics 37, no. 2 (June 9, 2021): 161–71. http://dx.doi.org/10.37193/cjm.2021.02.02.

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"Consider in a bounded domain \Omega \subset \mathbb{R}^N, N\ge 2, with smooth boundary \partial \Omega, the following eigenvalue problem (1) \begin{eqnarray*} &~&\mathcal{A} u:=-\Delta_p u-\Delta_q u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega, \nonumber \\ &~&\big(\mid \nabla u\mid ^{p-2}+\mid \nabla u\mid ^{q-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^ {r-2}u ~ \mbox{ on} ~ \partial \Omega, \nonumber \end{eqnarray*} where 1<r<q<p<\infty or 1<q<p<r<\infty; r\in \Big(1, \frac{p(N-1)}{N-p}\Big) if p<N and r\in (1, \infty) if p\ge N; a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega) are given nonnegative functions satisfying \[ \int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0. \] Under these assumptions we prove that the set of all eigenvalues of the above problem is the interval [0, \infty). Our result complements those previously obtained by Abreu, J. and Madeira, G., [Generalized eigenvalues of the (p, 2)-Laplacian under a parametric boundary condition, Proc. Edinburgh Math. Soc., 63 (2020), No. 1, 287–303], Barbu, L. and Moroşanu, G., [Full description of the eigenvalue set of the (p,q)-Laplacian with a Steklov-like boundary condition, J. Differential Equations, in press], Barbu, L. and Moroşanu, G., [Eigenvalues of the negative (p,q)– Laplacian under a Steklov-like boundary condition, Complex Var. Elliptic Equations, 64 (2019), No. 4, 685–700], Fărcăşeanu, M., Mihăilescu, M. and Stancu-Dumitru, D., [On the set of eigen-values of some PDEs with homogeneous Neumann boundary condition, Nonlinear Anal. Theory Methods Appl., 116 (2015), 19–25], Mihăilescu, M., [An eigenvalue problem possesing a continuous family of eigenvalues plus an isolated eigenvale, Commun. Pure Appl. Anal., 10 (2011), 701–708], Mihăilescu, M. and Moroşanu, G., [Eigenvalues of -\triangle_p-\triangle_q under Neumann boundary condition, Canadian Math. Bull., 59 (2016), No. 3, 606–616]."

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Bu, Weichun, Tianqing An, Deliang Qian, and Yingjie Li. "(p(x), q(x))-Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term ." Fractal and Fractional 6, no. 5 (May 6, 2022): 255. http://dx.doi.org/10.3390/fractalfract6050255.

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In the present article, we study a class of Kirchhoff-type equations driven by the (p(x), q(x))-Laplacian. Due to the lack of a variational structure, ellipticity, and monotonicity, the well-known variational methods are not applicable. With the help of the Galerkin method and Brezis theorem, we obtain the existence of finite-dimensional approximate solutions and weak solutions. One of the main difficulties and innovations of the present article is that we consider competing (p(x), q(x))-Laplacian, convective terms, and logarithmic nonlinearity with variable exponents, another one is the weaker assumptions on nonlocal term Mv(x) and nonlinear term g.

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