The (p,q)-Laplacian systems on locally finite graphs

Non-variational Solutions for Anisotropic (p, q)-Laplacian Systems with Nonhomogeneous Growth

This paper applies a general framework to explore the existence of multiple positive solutions for the fractional integral boundary value problem of high-order Caputo and Hadamard fractional coupled Laplacian systems with delayed or advanced arguments. We first focus on a generalized fractional homomorphic coupled boundary value problem with Hilfer fractional derivatives. Then we present the Green’s function corresponding to this Hilfer fractional system and its important properties. On this basis, by constructing a positive cone and applying a generalized cone fixed point theorem, we have established some novel criteria to ensure that the generalized fractional system has at least three positive solutions. As applications, we also obtain the multiplicity of the positive solutions of the Caputo and Hadamard fractional-order coupled Laplacian systems under two special Hilfer derivatives, respectively. Finally, we provide several examples to inspect the applicability of the main results.

Abstract In this paper, we study the T-periodic solutions of the parameter-dependent ϕ-Laplacian equation \begin{equation*} (\phi(x'))'=F(\lambda,t,x,x'). \end{equation*} Based on the topological degree theory, we present some atypical bifurcation results in the sense of Prodi–Ambrosetti, i.e., bifurcation of T-periodic solutions from λ = 0. Finally, we propose some applications to Liénard-type equations. Dedicated to Professor Maria Patrizia Pera on the occasion of her 70th birthday

This article concerns a novel coupled implicit differential system under φ–Riemann–Liouville (RL) fractional derivatives with p-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains [c,∞). The explicit iterative solution’s existence and uniqueness (EaU) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias (UHR) stabilities are investigated. Ultimately, we provide a numerical application of a coupled φ-RL fractional turbulent flow model to illustrate and test the effectiveness of our outcomes.

Abstract In this paper, we focus on studying the existence of positive weak solutions for a class of fractional Laplacian systems with respect to another function ψ ⁢ ( ⋅ ) {\psi(\,\cdot\,)} with multiple parameters. These systems arise in various applications, such as Lévy flights. We establish new conditions for the existence of a positive solution for the considered system using the method of sub-super solutions, which extends and complements previously known results in the literature.

In recent years, significant attention has been focused on issues related to the 1 -Laplacian operator, Δ 1 . Motivated by these advancements and driven by the exploration of ( p , q ) -Laplacian systems, we delve into the phenomenon of symmetry breaking in a Hénon-type ( p , 1 ) -Laplacian system. Our primary finding establishes the existence of non-negative and non-radial solutions under specific parameter conditions. Initially, we examine the associated problem concerning the ( p , q ) -Laplacian system, which is of independent interest. This is followed by an analysis of the sequence of solutions as q approaches 1 + asymptotically.

The complex-shifted Laplacian systems arising in a wide range of applications. In this work, we propose an absolute-value based preconditioner for solving the complex-shifted Laplacian system. In our approach, the complex-shifted Laplacian system is equivalently rewritten as a 2×2 block real linear system. With the Toeplitz structure of uniform-grid discretization of the constant-coefficient Laplacian operator, the absolute value of the block real matrix is fast invertible by means of fast sine transforms. For more general coefficient function, we then average the coefficient function and take the absolute value of the averaged matrix as our preconditioner. With assumptions on the complex shift, we theoretically prove that the eigenvalues of the preconditioned matrix in absolute value are upper and lower bounded by constants independent on the matrix size, indicating a matrix-size independent linear convergence rate of the MINRES solver. Interestingly, numerical results show that the proposed preconditioner is still efficient even if the assumptions on the complex shift are not met. The fast invertibility of the proposed preconditioner and the robust convergence rate of the preconditioned MINRES solver lead to a linearithmic (nearly optimal) complexity of the proposed solver. The proposed preconditioner is compared with several state-of-the-art preconditioners via several numerical examples to demonstrate the efficiency of the proposed preconditioner.

Existence of positive weak solutions for a class of fractional Laplacian systems with sign-changing weight functions

We investigate the existence of ground state solutions for a ( p , q ) -Laplacian system with p , q > 1 and potential wells on a weighted locally finite graph G = ( V , E ) . By making use of the method of Nehari manifold and the Lagrange multiplier rule, we prove that if the nonlinear term F takes on the super- ( p , q ) -linear growth and the potential functions a ( x ) and b ( x ) satisfy some suitable conditions, then for any fixed parameter λ ≥ 1 , the system is provided with a ground state solution ( u λ , v λ ) . Additionally, we set up the convergence property of the solutions set { ( u λ , v λ ) } when λ → + ∞ .

Analysis of positive solutions for classes of Laplacian systems with sign change weight functions and nonlinear boundary conditions

In this present paper, we concern to investigate the existence and multiplicity of solutions for a new class of systems of differential equations over the (p, q)-Laplacian via ?-Hilfer fractional operators in the space ?-fractional H?,?,?;? (p,q) (?). To investigate such results, we used variational methods.

Sub/Super Solutions Methods for a Class of Fractional Laplacian Systems

Abstract We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a ( p , q ) \left(p,q) -Laplacian system with a parameter on locally finite graphs. The main tool is an abstract critical point theorem in [G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1–10]. A key point in this study is that we overcome the difficulty to prove that the Gâteaux derivative of the variational functional for poly-Laplacian operator admits a continuous inverse, which is caused by the special definition of the poly-Laplacian operator on graph and mutual coupling of two variables in system.

Existence, uniqueness, and decay results for singular $$\Phi $$-Laplacian systems in $${\mathbb {R}}^N$$

Analysis of solutions for a class of $$(p_1(x),\ldots ,p_n(x))$$-Laplacian systems with Hardy potentials

The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for differential equations and have wide applications. Therefore, this article considers a class of nonlinear Hadamard fractional coupling (p1,p2)-Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam–Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results.

Existence and multiplicity of solutions to perturbed $$\Delta ^{\alpha ,\beta }_{\alpha _1,\beta _1}$$-Laplacian system in $$\mathbb R^N$$ involving critical nonlinearity

We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a (p, q)-Laplacian coupled system with perturbations and two parameters λ1 and λ2 on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-(p, q) conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-(p, q) conditions which is weaker than the well-known Ambrosetti-Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters λ1 and λ2.

In this paper, we consider a class of non-homogeneous fractional (p (.), q (.))−Laplacian systems. By using variational methods and combining with the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we prove the existence of solutions to the system.