Infinitely many homoclinic solutions for double phase problem on integers

We consider a discrete double phase problem on integers with an unbounded potential and reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our results extend some recent findings expressed in the literature.

Infinitely many solutions for anisotropic elliptic equations with variable exponent

In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak solutions.

Negative Energy Solutions for a New Fractional p x -Kirchhoff Problem without the (AR) Condition

In this paper, we investigate the following Kirchhoff type problem involving the fractional p x -Laplacian operator. a − b ∫ Ω × Ω u x − u y p x , y / p x , y x − y N + s p x , y d x d y L u = λ u q x − 2 u + f x , u x ∈ Ω u = 0 x ∈ ∂ Ω , , where Ω is a bounded domain in ℝ N with Lipschitz boundary, a ≥ b > 0 are constants, p x , y is a function defined on Ω ¯ × Ω ¯ , s ∈ 0 , 1 , and q x > 1 , L u is the fractional p x -Laplacian operator, N > s p x , y , for any x , y ∈ Ω ¯ × Ω ¯ , p x ∗ = p x , x N / N − s p x , x , λ is a given positive parameter, and f is a continuous function. By using Ekeland’s variational principle and dual fountain theorem, we obtain some new existence and multiplicity of negative energy solutions for the above problem without the Ambrosetti-Rabinowitz ((AR) for short) condition.

Multiplicity of Quasilinear Schrödinger Equation

In this paper, we study the quasilinear Schrödinger equation involving concave and convex nonlinearities. When the pair of parameters belongs to a certain subset of ℝ2, we establish the existence of a nontrivial mountain pass-type solution and infinitely many negative energy solutions by using some new techniques and dual fountain theorem. Recent results from the literature are improved and extended.

Existence and Multiplicity of Solutions for a Class of Anisotropic Double Phase Problems

We consider the following double phase problem with variable exponents: −div∇upx−2∇u+ax∇uqx−2∇u=λfx,u in Ω,u=0, on ∂Ω. By using the mountain pass theorem, we get the existence results of weak solutions for the aforementioned problem under some assumptions. Moreover, infinitely many pairs of solutions are provided by applying the Fountain Theorem, Dual Fountain Theorem, and Krasnoselskii’s genus theory.

Multiplicity of Radially Symmetric Small Energy Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the L ∞ -bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose L ∞ -norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools.

Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to 0. These results extend some known results in previous papers.