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Journal articles on the topic 'P(x)-laplacian equation'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Luo, Huxiao, Shengjun Li, and Xianhua Tang. "Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods." Advances in Mathematical Physics 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/5317213.

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We study the existence of nontrivial solution of the following equation without compactness: (-Δ)pαu+up-2u=f(x,u), x∈RN, where N,p≥2, α∈(0,1), (-Δ)pα is the fractional p-Laplacian, and the subcritical p-superlinear term f∈C(RN×R) is 1-periodic in xi for i=1,2,…,N. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional p-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution uλ of perturbation equations. And we prove that uλ→u as λ→0. Finally, by using va

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2

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 equ

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3

ElHammar, Hassnae, Mohamed El Ouaarabi, Chakir Allalou, and Said Melliani. "P(x,・)-Kirchhoff type problem involving the fractional p(x)-Laplacian operator with discontinuous nonlinearities." Filomat 38, no. 6 (2024): 2109–25. http://dx.doi.org/10.2298/fil2406109h.

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The purpose of this paper is mainly to investigate the existence of weak solution of the stationary Kirchhoff type equations driven by the fractional p(x)-Laplacian operator with discontinuous nonlinearities for a class of elliptic Dirichlet boundary value problems. By using the topological degree based on the abstract Hammerstein equation, we conduct our existence analysis. The fractional Sobolev space with variable exponent provides an effective functional framework for these situations.

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4

Zhong, Yansheng. "A Concentration Phenomenon for p-Laplacian Equation." Journal of Applied Mathematics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/148902.

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It is proved that if the bounded function of coefficientQnin the following equation -div ⁡{|∇u|p-2∇u}+V(x)|u|p-2u=Qn(x)|u|q-2u, u(x)=0 as x∈∂Ω. u(x)⟶0 as |x|⟶∞is positive in a region contained in Ω and negative outside the region, the sets{Qn>0}shrink to a pointx0∈Ωasn→∞, and then the sequenceungenerated by the nontrivial solution of the same equation, corresponding toQn, will concentrate atx0with respect toW01,p(Ω)and certainLs(Ω)-norms. In addition, if the sets{Qn>0}shrink to finite points, the corresponding ground states{un}only concentrate at one of these points. These conclusions ex

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5

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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6

Alsaedi, R., K. Ben Ali, and A. Ghanmi. "Existence Results for Singular p(x)-Laplacian Equation." Advances in Pure and Applied Mathematics 13, no. 3 (2022): 62–71. http://dx.doi.org/10.21494/iste.op.2022.0840.

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7

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

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8

GE, BIN. "Sign changing solutions of the p(x)-Laplacian equation." Proceedings - Mathematical Sciences 123, no. 4 (2013): 515–24. http://dx.doi.org/10.1007/s12044-013-0150-7.

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9

Zhan, Hua-shui. "Evolutionary p(x)-Laplacian Equation with a Convection Term." Acta Mathematicae Applicatae Sinica, English Series 35, no. 3 (2019): 655–70. http://dx.doi.org/10.1007/s10255-019-0842-6.

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10

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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11

CHEN, YI, and X. H. TANG. "GROUND STATE SOLUTIONS FOR -SUPERLINEAR -LAPLACIAN EQUATIONS." Journal of the Australian Mathematical Society 97, no. 1 (2014): 48–62. http://dx.doi.org/10.1017/s1446788714000135.

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AbstractIn this paper, we deduce new conditions for the existence of ground state solutions for the $p$-Laplacian equation $$\begin{equation*} \left \{ \begin{array}{@{}ll} -\mathrm {div}(|\nabla u|^{p-2}\nabla u)+V(x)|u|^{p-2}u=f(x, u), \quad x\in {\mathbb {R}}^{N},\\[5pt] u\in W^{1, p}({\mathbb {R}}^{N}), \end{array} \right . \end{equation*}$$ which weaken the Ambrosetti–Rabinowitz type condition and the monotonicity condition for the function $t\mapsto f(x, t)/|t|^{p-1}$. In particular, both $tf(x, t)$ and $tf(x, t)-pF(x, t)$ are allowed to be sign-changing in our assumptions.

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12

Xie, Jin. "Weak Comparison Principle for Weighted Fractional p -Laplacian Equation." Journal of Function Spaces 2020 (December 19, 2020): 1–7. http://dx.doi.org/10.1155/2020/6675031.

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The aim of this paper is to establish a weak comparison principle for a class fractional p -Laplacian equation with weight. The nonlinear term f x , s > 0 is a Carathéodory function which is possibly unbounded both at the origin and at infinity and such that f x , s s 1 − p decreases with respect to s for a.e. x ∈ Ω .

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13

Ahmedatt, Taghi, Ahmed Aberqi, Abedlfettah Touzani, and Chihab Yazough. "On some nonlinear hyperbolic p(x,t)-Laplacian equations." Journal of Applied Analysis 24, no. 1 (2018): 55–69. http://dx.doi.org/10.1515/jaa-2018-0006.

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Abstract This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation u_{tt}=Lu+f(x,t)\quad\text{in }\Omega_{T}=\Omega\times(0,T), where L is a nonlinear operator and {\phi(x,t,\cdot\,)} , {f(x,t)} and the exponents of the nonlinearities {p(x,t)} and {\mu(x,t)} are given functions.

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14

Afrouzi, G. A., S. Mahdavi, and Z. Naghizadeh. "The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition." Nonlinear Analysis: Modelling and Control 12, no. 2 (2007): 143–55. http://dx.doi.org/10.15388/na.2007.12.2.14705.

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The Nehari manifold for the equation −∆pu(x) = λu(x)|u(x)|p−2 + b(x)|u(x)|γ−2u(x) for x ∈ Ω together with Dirichlet boundary condition is investigated in the case where 0 < γ < p. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form of t → J(tu) where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as λ changes, and show how existence results for positive solutions of the equation are linked to the properties of Nehari manifold.

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15

Sugie, Jitsuro, and Masakazu Onitsuka. "A non-oscillation theorem for nonlinear differential equations with p-Laplacian." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 3 (2006): 633–47. http://dx.doi.org/10.1017/s0308210500005096.

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The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

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16

Zhang, Chao, Shulin Zhou, and Bin Ge. "Gradient estimates for the p(x)-Laplacian equation in RN." Annales Polonici Mathematici 114, no. 1 (2015): 45–65. http://dx.doi.org/10.4064/ap114-1-4.

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17

Zhan, Huashui, and Zhaosheng Feng. "Definite Condition of the Evolutionary $\vec{p}(x)$−Laplacian Equation." Analysis in Theory and Applications 38, no. 3 (2022): 297–321. http://dx.doi.org/10.4208/ata.oa-2021-0029.

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18

Yin, Jingxue, Jinkai Li, and Yuanyuan Ke. "Existence of positive solutions for the $p(x)$-Laplacian equation." Rocky Mountain Journal of Mathematics 42, no. 5 (2012): 1675–758. http://dx.doi.org/10.1216/rmj-2012-42-5-1675.

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19

Mei, Yu, Fu Yongqiang, and Li Wang. "Existence of Solutions for thep(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent." Abstract and Applied Analysis 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/894925.

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This paper deals with thep(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness inW01,p(x)(Ω)space is established, then by applying it we obtain the existence of solutions for the followingp(x)-Laplacian problem:-div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u), x∈Ω, u=0, x∈∂Ω,whereΩ⊂ℝNis a bounded domain,0∈Ω,1<p-≤p(x)≤p+<N, andf(x,u)satisfiesp(x)-growth conditions.

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20

Hamid, El Ouardi. "Existence, Uniqueness and Blow-up Result of Solutions for an Evolution p(x) - laplacian Equation." British Journal of Applied Science & Technology 21, no. 3 (2017): 1–13. https://doi.org/10.9734/BJAST/2017/32806.

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In this paper we are investigate in the evolution equation p(x)- laplacian with the initial boundary value question. We translate the parabolic equation into the elliptic equation by using a finite difference method, and then the existence and uniqueness solution are obtained. The blow-up property is shown, by using the energy method. We perform, using Matlab (Ode45 subroutine), some numerical experiments just to illustrate our general results.

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21

Ait Hammou, M., and E. H. Rami. "Existence of weak solutions for a $p(x)$-Laplacian equation via topological degree." Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta 59 (May 2022): 15–24. http://dx.doi.org/10.35634/2226-3594-2022-59-02.

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We consider the $p(x)$-Laplacian equation with a Dirichlet boundary value condition $$ \begin{cases} -\Delta_{p(x)}(u)+|u|^{p(x)-2}u= g(x,u,\nabla u), &amp;x\in\Omega,\\ u=0, &amp;x\in\partial\Omega. \end{cases} $$ Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions, the existence of weak solutions for this equation.

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22

Khosravi Rashti, Mahnaz, Mohsen Alimohammady, and Sirous Ghobadi. "Multiple Solutions for a p (x)-Laplacian Problem: A Variational Approach." Journal of Mathematics 2023 (March 23, 2023): 1–11. http://dx.doi.org/10.1155/2023/4261555.

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We focus on an anisotropic p (x)-Laplacian equation defined on a bounded domain with smooth boundary. Moreover, using variational methods to find the existence of at least three solutions due to Ricceri is discussed.

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23

Garain, Prashanta. "Properties of solutions to some weighted p-Laplacian equation." Opuscula Mathematica 40, no. 4 (2020): 483–94. http://dx.doi.org/10.7494/opmath.2020.40.4.483.

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In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,\] on smooth domain and for varying nonlinearity \(f\).

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24

Seshadev, Padhi Jaffar Ali and John R. Graef. "Positive Solutions to a Derivative Dependent p-Laplacian Equation with Riemann-Stieltjes Integral Boundary Conditions." Annals of Communications in Mathematics 3, no. 1 (2020): 7–25. https://doi.org/10.5281/zenodo.10043218.

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This paper is concerned with the existence of two nontrivial positive solutions to a class of boundary value problems involving a p-Laplacian of the form (Φp(x 0 ))0 + g(t)f(t, x, x0 ) = 0, t ∈ (0, 1), x(0) − ax 0 (0) = α[x], x(1) + bx 0 (1) = β[x], where Φp(x) = |x| p−2x is a one dimensional p-Laplacian operator with p > 1, a and b are real constants, and α and β are given by the Riemann-Stieltjes integrals α[x] = Z1 0 x(t)dA(t), β[x] = Z1 0 x(t)dB(t), with A and B functions of bounded variation. The approach used is based on fixed point index theory. The results obtained in this paper are

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25

Kamel, Tahri, Brahimi Noureddine, and Abd eljalil Keboucha. "An existence and uniqueness of solution for p--Laplacian Kirchhoff type equation with singular term." Asia Mathematika 5, no. 3 (2021): 1——13. https://doi.org/10.5281/zenodo.5808899.

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This work is devoted to study the existence of positive solution for a class of $p$--Laplacian Kirchhoff type equation with singular nonlinearity:   \( \left\{ \begin{array}{ll} L_{p}(u)=f(x)|u|^{-\gamma}-\lambda|u|^{p^{*}-2} u & \text { in } \Omega, \\ u=0 & \text { on } \partial \Omega, \end{array} \right. \) where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^{n}(n \geq 3), \lambda>0\) is a real parameter. Here \(\gamma \in(0,1)\) is a constant, \(a, b \geq 0\) such that \(a+b>0\) are parameters, the weight function \(f: \Omega \right

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26

Poulou, Marilena N., та Nikolaos M. Stavrakakis. "Eigenvalue problems for a quasilinear elliptic equation onℝN". International Journal of Mathematics and Mathematical Sciences 2005, № 18 (2005): 2871–82. http://dx.doi.org/10.1155/ijmms.2005.2871.

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We prove the existence of a simple, isolated, positive principal eigenvalue for the quasilinear elliptic equation−Δpu=λg(x)|u|p−2u,x∈ℝN,lim|x|→+∞u(x)=0, whereΔpu=div(|∇u|p−2∇u)is thep-Laplacian operator and the weight functiong(x), being bounded, changes sign and is negative and away from zero at infinity.

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27

Antontsev, S. N., I. V. Kuznetsov, and S. A. Sazhenkov. "A SHOCK LAYER ARISING AS THE SOURCE TERM COLLAPSES IN THE P(X)-LAPLACIAN EQUATION." Issues of Analysis 27, no. 3 (2020): 31–53. http://dx.doi.org/10.15393/j3.art.2020.8990.

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28

Wang, Pengyan. "Monotonicity of solutions for fractional p-equations with a gradient term." Open Mathematics 20, no. 1 (2022): 465–77. http://dx.doi.org/10.1515/math-2022-0035.

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Abstract In this paper, we consider the following fractional p p -equation with a gradient term: ( − Δ ) p s u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) . {\left(-\Delta )}_{p}^{s}u\left(x)=f\left(x,u\left(x),\nabla u\left(x)). We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional p p -laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, w

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29

Yang, Dandan, Zhenyu Bai, and Chuanzhi Bai. "Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs." Axioms 13, no. 11 (2024): 762. http://dx.doi.org/10.3390/axioms13110762.

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In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of so

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30

El Bahja, Hamid, Abderrahmane El Hachimi, and Ali Alami Idrissi. "Semidiscretization for a Doubly Nonlinear Parabolic Equation Related to the p(x)-Laplacian." International Journal of Differential Equations 2019 (April 11, 2019): 1–8. http://dx.doi.org/10.1155/2019/6107841.

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This paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor.

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31

Zang, Aibin. "p(x)-Laplacian equations satisfying Cerami condition." Journal of Mathematical Analysis and Applications 337, no. 1 (2008): 547–55. http://dx.doi.org/10.1016/j.jmaa.2007.04.007.

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32

Liang, Yuan, Xianbin Wu, Qihu Zhang, and Chunshan Zhao. "MULTIPLE SOLUTIONS OF A $p(x)$-LAPLACIAN EQUATION INVOLVING CRITICAL NONLINEARITIES." Taiwanese Journal of Mathematics 17, no. 6 (2013): 2055–82. http://dx.doi.org/10.11650/tjm.17.2013.3074.

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33

Lourêdo, Aldo Trajano, Manuel Milla Miranda, and Marcondes Rodrigues Clark. "Variable exponent perturbation of a parabolic equation with p ( x ) -Laplacian." Electronic Journal of Qualitative Theory of Differential Equations, no. 60 (2019): 1–14. http://dx.doi.org/10.14232/ejqtde.2019.1.60.

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Lang, Honglei, Changchun Liu, and Zhenbang Li. "Some properties of solutions for an evolution p(x)-Laplacian equation." Lobachevskii Journal of Mathematics 32, no. 1 (2011): 48–60. http://dx.doi.org/10.1134/s1995080211010082.

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35

Karami, F., K. Sadik, and L. Ziad. "A variable exponent nonlocal p(x)-Laplacian equation for image restoration." Computers & Mathematics with Applications 75, no. 2 (2018): 534–46. http://dx.doi.org/10.1016/j.camwa.2017.09.034.

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36

FILIPPUCCI, ROBERTA, PATRIZIA PUCCI, and MARCO RIGOLI. "NONLINEAR WEIGHTED p-LAPLACIAN ELLIPTIC INEQUALITIES WITH GRADIENT TERMS." Communications in Contemporary Mathematics 12, no. 03 (2010): 501–35. http://dx.doi.org/10.1142/s0219199710003841.

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In this paper, we give sufficient conditions for the existence and nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div {g(|x|)|Du|p-2Du} ≥ h(|x|)f(u)ℓ(|Du|). We achieve our conclusions by using a generalized version of the well-known Keller–Ossermann condition, first introduced in [2] for the generalized mean curvature case, and in [11, Sec. 4] for the nonweighted p-Laplacian equation. Several existence results are also proved in Secs. 2 and 3, from which we deduce simple criteria

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37

Borsuk, Mikhail. "Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary." Ukrainian Mathematical Bulletin 17, no. 4 (2020): 455–83. http://dx.doi.org/10.37069/1810-3200-2020-17-4-1.

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This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

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38

Cuadro, Johnny, and Gabriel López. "Strong Unique Continuation for Solutions of ap(x)-Laplacian Problem." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/108671.

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We study the strong unique continuation property for solutions to the quasilinear elliptic equation-div(|∇u|p(x)-2∇u)+V(x)|u|p(x)-2u=0 in ΩwhereV(x)∈LN/p(x)(Ω),Ωis a smooth bounded domain inℝN, and1<p(x)<NforxinΩ.

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39

ZHANG, CHAO, and SHULIN ZHOU. "ENTROPY AND RENORMALIZED SOLUTIONS FOR THE p(x)-LAPLACIAN EQUATION WITH MEASURE DATA." Bulletin of the Australian Mathematical Society 82, no. 3 (2010): 459–79. http://dx.doi.org/10.1017/s0004972710000432.

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AbstractIn this paper we prove the existence and uniqueness of both entropy solutions and renormalized solutions for the p(x)-Laplacian equation with variable exponents and a signed measure in L1(Ω)+W−1,p′(⋅)(Ω). Moreover, we obtain the equivalence of entropy solutions and renormalized solutions.

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40

Cui, Na, and Hong-Rui Sun. "Existence of solutions for critical fractional p-Laplacian equations with indefinite weights." Electronic Journal of Differential Equations 2021, no. 01-104 (2021): 11. http://dx.doi.org/10.58997/ejde.2021.11.

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This article concerns the critical fractional p-Laplacian equation with indefinite weights $$ (-\Delta_p)^su=\lambda g(x)|u|^{p-2}u+h(x)|u|^{p_s^*-2}u \quad \text{in }\mathbb{R}^N, $$ where \(0<s<1<p<\infty\), \(N>sp\) and \(p_s^*=Np/(N-sp)\), the weight functions \(g\) may be indefinite, and \(h\) changes sign. Specifically, based on the results of asymptotic estimates for an extremal in the fractional Sobolev inequality and the discrete spectrum of fractional p-Laplacian operator, we establish an existence criterion for a nontrivial solution to this problem. For more inf

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Torres Ledesma, César E. "Multiplicity result for non-homogeneous fractional Schrodinger--Kirchhoff-type equations in ℝn". Advances in Nonlinear Analysis 7, № 3 (2018): 247–57. http://dx.doi.org/10.1515/anona-2015-0096.

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AbstractIn this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff typeM\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x)in {\mathbb{R}^{n}}, where (-Δ)_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡\infty, ps¡n, f : \mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R} is a continuous function, V : \mathbb{R}^{n}\to\mathbb{R}^{+} is a potential function and g : \mathbb{R}^{n}\to\mathbb{R} is a perturbatio

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42

Merah, Ahlem, and Fatiha Mesloub. "On a viscoelastic plate equation with a polynomial source term and p(x,t)-Laplacian operator in the presence of delay term." Journal of Innovative Applied Mathematics and Computational Sciences 2, no. 1 (2022): 92–107. http://dx.doi.org/10.58205/jiamcs.v2i1.30.

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In this paper, the blow-up of solutions for a Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p(x,t)-Laplacian operator in the presence of time delay is obtained. Under suitable conditions on g and the variable exponent of the p(x,t)-Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.

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43

Ma, To Fu, та Maurício Luciano Pelicer. "Perturbations near resonance for thep-Laplacian inℝN". Abstract and Applied Analysis 7, № 6 (2002): 323–34. http://dx.doi.org/10.1155/s1085337502203073.

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We study a multiplicity result for the perturbedp-Laplacian equation−Δpu−λg(x)|u|p−2u=f(x,u)+h(x) in ℝN, where1<p<Nandλis nearλ 1, the principal eigenvalue of the weighted eigenvalue problem−Δpu=λg(x)|u|p−2uinℝN. Depending on which sideλis fromλ 1, we prove the existence of one or three solutions. This kind of result was firstly obtained by Mawhin and Schmitt (1990) for a semilinear two-point boundary value problem.

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44

Qing, Yi, and Zhao Junning. "Singular limit of solutions of the p-Laplacian equation." Nonlinear Analysis: Theory, Methods & Applications 43, no. 6 (2001): 733–41. http://dx.doi.org/10.1016/s0362-546x(99)00231-x.

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45

Yao, Fengping. "Weighted Lorentz estimates for subquadratic quasilinear elliptic equations with measure data." Electronic Journal of Qualitative Theory of Differential Equations, no. 7 (2024): 1–25. http://dx.doi.org/10.14232/ejqtde.2024.1.7.

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In this work we mainly prove the following interior gradient estimates in weighted Lorentz spaces g − 1 [ M 1 ( μ ) ] ∈ L w , l o c q , r ( Ω ) ⟹ | D u | ∈ L w , l o c q , r ( Ω ) , where g ( t ) = t a ( t ) for t ≥ 0 and M 1 ( μ ) ( x ) is the first-order fractional maximal function M 1 ( μ ) ( x ) := sup r > 0 r | μ | ( B r ( x ) ) | B r ( x ) | , for a class of non-homogeneous divergence quasilinear elliptic equations with measure data in the subquadratic case − div ⁡ [ a ( ( A D u ⋅ D u ) 1 2 ) A D u ] = μ in Ω , whose model cases are the classical elliptic p -Laplacian equation with me

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46

Serag, H. M., and S. A. Khafagy. "Existence of Weak Solutions for Nonlinear Systems Involving Degenerated p-Laplacian Operators." Sarajevo Journal of Mathematics 5, no. 1 (2024): 41–54. http://dx.doi.org/10.5644/sjm.05.1.04.

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We study the existence of weak solutions for the nonlinear system\begin{equation*}\left.\begin{aligned}-\Delta _{P,_{p}}u&=a(x)|u|^{p-2}u-b(x)|u|^{\alpha }|v|^{\beta }v+f, \\-\Delta _{Q,q}v&=-c(x)|u|^{\alpha }|v|^{\beta }u+d(x)|v|^{q-2}v+g,\end{aligned}\right\}\end{equation*}where, the degenerated p-Laplacian is defined as $\Delta _{P,p}u=div [P(x)$ $|\nabla u|^{p-2}\nabla u].$ We prove the existence of weak solutions for this system defined on bounded domains using the theory of monotone operators. We also consider the case of an unbounded domain. 2000 Mathematics Subject Classificati

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47

Zhan, Huashui, and Zhen Zhou. "The Evolutionary p(x)-Laplacian Equation with a Partial Boundary Value Condition." Discrete Dynamics in Nature and Society 2018 (2018): 1–7. http://dx.doi.org/10.1155/2018/1237289.

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Consider a diffusion convection equation coming from the electrorheological fluids. If the diffusion coefficient of the equation is degenerate on the boundary, generally, we can only impose a partial boundary value condition to ensure the well-posedness of the solutions. Since the equation is nonlinear, the partial boundary value condition cannot be depicted by Fichera function. In this paper, when α<p--1, an explicit formula of the partial boundary on which we should impose the boundary value is firstly depicted. The stability of the solutions, dependent on this partial boundary value cond

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Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Resonant Anisotropic (p,q)-Equations." Mathematics 8, no. 8 (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 t

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Chung, Soon-Yeong, and Min-Jun Choi. "Blow-Up Solutions and Global Solutions to Discretep-Laplacian Parabolic Equations." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/351675.

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We discuss the conditions under which blow-up occurs for the solutions of discretep-Laplacian parabolic equations on networksSwith boundary∂Sas follows:ut(x,t)=Δp,ωu(x,t)+λ|u(x,t)|q-1u(x,t),(x,t)∈S×(0,+∞);u(x,t)=0,(x,t)∈∂S×(0,+∞);u(x,0)=u0≥0,x∈S¯, wherep>1,q>0,λ>0, and the initial datau0is nontrivial onS. The main theorem states that the solutionuto the above equation satisfies the following: (i) if0<p-1<qandq>1, then the solution blows up in a finite time, providedu¯0>ω0/λ1/q-p+1, whereω0:=maxx∈S⁡∑y∈S¯‍ω(x,y)andu¯0:=maxx∈S u0(x); (ii) if0<q≤1, then the nonnegative solu

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50

Liao, Menglan, Qiang Liu, and Hailong Ye. "Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations." Advances in Nonlinear Analysis 9, no. 1 (2020): 1569–91. http://dx.doi.org/10.1515/anona-2020-0066.

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Abstract In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p-Laplacian with $\begin{array}{} p \gt \max\{\frac{2N}{N+2s},1\} \end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial ener

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