Existence and Multiplicity of Solutions for a Biharmonic Equation with p(x)-Kirchhoff Type

Based on the basic theory and critical point theory of variable exponential Lebesgue Sobolev space, this paper investigates the existence and multiplicity of solutions for a class of nonlocal elliptic equations with Navier boundary value conditions when (AR) condition does not hold and improves or generalizes the original conclusions.

On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ -Laplacian

In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ -Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.

Four Solutions for Fractional p-Laplacian Equations with Asymmetric Reactions

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.

Multiplicity of solutions for the discrete boundary value problem involving the p-Laplacian

PurposeThe purpose of this paper is the study of existence and multiplicity of solutions for a nonlinear discrete boundary value problems involving the p-laplacian.Design/methodology/approachThe approach is based on variational methods and critical point theory.FindingsTheorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4.Originality/valueThe paper is original and the authors think the results are new.

Existence of Nontrivial Solutions for Sixth-Order Differential Equations

We show the existence of at least one nontrivial solution for a nonlinear sixth-order ordinary differential equation is investigated. Our approach is based on critical point theory.

Existence of infinitely many solutions for fractional p-Laplacian Schrödinger–Kirchhof-Type equations with general potentials

In this paper, we use the genus properties in critical point theory to prove the existence of infinitely many solutions for fractional [Formula: see text]-Laplacian equations of Schrödinger-Kirchhoff type.

Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator

In this paper, we consider a perturbed partial discrete Dirichlet problem with the (p,q)-Laplacian operator. Using critical point theory, we study the existence of infinitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term f, we obtain the sufficient conditions for the existence of infinitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exemplified by an example.

An eigenvalue of anisotropic discrete problem with three variable exponents

UDC 517.5We study the existence of a continuous spectrum of an anisotropic discrete problem, involving variable exponent.The proposed technical approach is based on the variational methods and critical point theory.

Infinitely many solutions to fractional differential equations with instantaneous and non-instantaneous impulses

The goal of this paper is to study fractional differential equations involving instantaneous and non-instantaneous impulses with Sturm-Liouville boundary conditions. By using critical point theory and variational approach, infinitely many solutions are obtained. The interesting point is that the potential has an oscillating asymptotic behavior. Also one example is presented to illustrate the main result.