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.

Infinitely Many Solutions for Fractional p-Laplacian Schrödinger–Kirchhoff Type Equations with Symmetric Variable-Order

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations a+b∫Ω×Ω|ξ(x)−ξ(y)|p|x−y|N+ps(x,y)dxdyp−1(−Δ)ps(·)ξ+λV(x)|ξ|p−2ξ=f(x,ξ),x∈Ω,ξ=0,x∈∂Ω, where Ω is a bounded Lipschitz domain in RN, 1<p<+∞, a,b>0 are constants, s(·):RN×RN→(0,1) is a continuous and symmetric function with N>s(x,y)p for all (x,y)∈Ω×Ω, λ>0 is a parameter, (−Δ)ps(·) is a fractional p-Laplace operator with variable-order, V(x):Ω→R+ is a potential function, and f(x,ξ):Ω×RN→R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

Infinitely many solutions for a gauged nonlinear Schrödinger equation with a perturbation

In this paper, we use the Fountain theorem under the Cerami condition to study the gauged nonlinear Schrödinger equation with a perturbation in R2. Under some appropriate conditions, we obtain the existence of infinitely many high energy solutions for the equation.

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.

Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

Existence of solutions for a class of $p(x)$-curl systems arising in electromagnetism without (A-R) type conditions

In this paper, we study the existence and multiplicity of solutions for a class of of $p(x)$-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the $p(x)$-curl operator in [4, 15].

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.