Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term

In the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.

Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

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.

GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$\left\{\begin{array}{lll}-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.\end{array}\right.$$ Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.

Multiple bound states of higher topological type for semi-classical Choquard equations

We are concerned with the semi-classical states for the Choquard equation $$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$ where N ⩾ 2, I α is the Riesz potential with order α ∈ (0, N − 1) and 2 ⩽ p < (N + α)/(N − 2). When the potential V is assumed to be bounded and bounded away from zero, we construct a family of localized bound states of higher topological type that concentrate around the local minimum points of the potential V as ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.

Existence of Small-Energy Solutions to Nonlocal Schrödinger-Type Equations for Integrodifferential Operators in ℝN

We are concerned with the following elliptic equations: ( − Δ ) p , K s u + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where ( − Δ ) p , K s is the nonlocal integrodifferential equation with 0 < s < 1 < p < + ∞ , s p < N the potential function V : R N → ( 0 , ∞ ) is continuous, and f : R N × R → R satisfies a Carathéodory condition. The present paper is devoted to the study of the L ∞ -bound of solutions to the above problem by employing De Giorgi’s iteration method and the localization method. Using this, we provide a sequence of infinitely many small-energy solutions whose L ∞ -norms converge to zero. The main tools were the modified functional method and the dual version of the fountain theorem, which is a generalization of the symmetric mountain-pass theorem.

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.

Infinitely Many Solutions for a Superlinear Fractional p-Kirchhoff-Type Problem without the (AR) Condition

In this paper, we investigate the existence of infinitely many solutions to a fractional p-Kirchhoff-type problem satisfying superlinearity with homogeneous Dirichlet boundary conditions as follows: [a+b(∫R2Nux-uypKx-ydxdy)]Lpsu-λ|u|p-2u=gx,u, in Ω, u=0, in RN∖Ω, where Lps is a nonlocal integrodifferential operator with a singular kernel K. We only consider the non-Ambrosetti-Rabinowitz condition to prove our results by using the symmetric mountain pass theorem.

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.