Symmetric Mountain Pass Theorem Research Articles

Infinitely Many High Energy Solutions for a Fourth-Order Equations of Kirchhoff Type in ℝN

In this paper we study the following fourth-order elliptic equations of Kirchhoff type $$\Delta^2u - (a+b\int_{\mathbb{R}^N} | \triangledown u|^2dx)\Delta u + V(x)u=f(x, u), \;\;x\in\mathbb{R}^N,$$ where Δ2 := Δ(Δ) is the biharmonic operator, a, b > 0 are constants, V ∈ C(ℝN, ℝ) and f ∈ C(ℝN × ℝ, ℝ). Under some appropriate assumptions on V(x) and f(x, u), new results on the existence of infinitely many high energy solutions for the above equation are obtained via Symmetric Mountain Pass Theorem.

Infinitely many fast homoclinic solutions for a class of superquadratic damped vibration systems

Consider the following damped vibration system $$\begin{aligned} \ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in \mathbb {R} \qquad \qquad (1) \end{aligned}$$ where $$q\in C(\mathbb {R},\mathbb {R})$$ , $$L\in C(\mathbb {R},\mathbb {R}^{N^{2}})$$ and $$W\in C(\mathbb {R}\times \mathbb {R}^{N},\ \mathbb {R})$$ . Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many fast homoclinic solutions for (1) when L is not required to be either uniformly positive definite or coercive and W satisfies some general super-quadratic conditions at infinity in the second variable but does not satisfy the classical superquadratic growth conditions at infinity.

Schrödinger-Poisson system without growth and the Ambrosetti-Rabinowitz conditions

We consider the following Schrodinger-Poisson system $ \left\{ \begin{array}{l} {\rm{ - }}\Delta u + V\left(x \right)u + \phi u = \lambda f\left(u \right)\; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}, \\ - \Delta \phi = {u^2}, \mathop {\lim }\limits_{|x| \to + \infty } \phi = 0, \; \; \; \; \; \; \; \; \; \; \; \; {\rm{in}}\; {\mathbb{R}^3}. \end{array} \right. $ Unlike most other papers on this problem, the Schrodinger-Poisson system without any growth and Ambrosetti-Rabinowitz condition is considered in this paper. Firstly, by Jeanjean's monotonicity trick and the mountain pass theorem, we prove that the problem possesses a positive solution for large value of $\lambda$. Secondly, we establish the multiplicity of solutions via the symmetric mountain pass theorem.

Existence of Infinitely Many High Energy Solutions for a Fourth-Order Kirchhoff Type Elliptic Equation in R<sup>3</sup>

In this paper, we consider the following fourth-order equation of Kirchhoff type where a, b > 0 are constants, 3 p V ∈ C (R3, R); Δ2: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on V (x). We make some assumptions on the potential V (x) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature.

Infinitely many solutions for a quasilinear Schrödinger equation with Hardy potentials

In this article, we study the following quasilinear Schrodinger equation −∆u − µ u |x| 2 + V(x)u − (∆(u 2 ))u = f(x, u), x ∈ R N, where V(x) is a given positive potential and the nonlinearity f(x, u) is allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions by a change of variable and Symmetric Mountain Pass Theorem.

Infinitely many solutions for a class of biharmonic equations with indefinite potentials

In this paper, we consider the following sublinear biharmonic equations $ \begin{equation*} \Delta^2 u + V\left( x \right)u = K(x)|u|^{p-1}u, x\in \mathbb{R}^N, \end{equation*} $ where $N\geq5, ~0 \lt p \lt 1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.

INFINITELY MANY SOLUTIONS FOR CRITICAL FRACTIONAL EQUATION WITH SIGN-CHANGING WEIGHT FUNCTION

In this work, we consider the fractional Schrödinger type equations with critical exponent, concave nonlinearity and sign-changing weight function on $ \mathbb{R}^N $. With the aid of the symmetric Mountain Pass Theorem, we prove this problem has infinitely many small energy solutions.

Parametric and nonparametric A-Laplace problems: Existence of solutions and asymptotic analysis

We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a suitable parameter goes to 0 + . In this case, we point out the existence of a blow-up phenomenon. The analysis developed in this paper extends and complements various qualitative and asymptotic properties for some cases described by homogeneous differential operators.

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.

Infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents

ABSTRACT In this paper, we show the existence of infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. More precisely, we consider where is a continuous function, Ω is a bounded domain in with for all is the variable-order fractional Laplace operator, and are two continuous functions, are two parameters and In addition to using the new version of Clark's theorem due to Liu and Wang to prove the existence of infinitely many solutions for the above problem, we also apply the symmetric mountain pass theorem, fountain theorem and dual fountain theorem to obtain the same conclusion. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero.

Multiplicity Results for Kirchhoff Equations with Hardy-Littlewood-Sobolev Critical Nonlinearity

In this paper, we deal with the multiplicity results for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity in $\mathbb {R}^{N}$. By using the second concentration-compactness principle and concentration-compactness principle at infinity to prove that the (PS)c condition holds locally and together with the new version of symmetric mountain pass theorem of Kajikiya, we prove that the problem admits infinitely many solutions under suitable conditions.

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.

Existence and multiplicity of solutions for Klein\u2013Gordon\u2013Maxwell systems with sign-changing potentials

In this paper, we study the following nonlinear Klein–Gordon–Maxwell system: \t\t\t{−Δu+V(x)u−(2ω+ϕ)ϕu=f(x,u)+λh(x)|u|q−2u,x∈R3,Δϕ=(ω+ϕ)u2,x∈R3,(Pλ)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta u+ V(x)u-(2\\omega +\\phi )\\phi u = f(x,u)+\\lambda h(x) \\vert u \\vert ^{q-2}u, & x\\in \\mathbb{R}^{3},\\\\ \\Delta \\phi = (\\omega +\\phi )u^{2}, & x\\in \\mathbb{R}^{3}, \\end{cases}\\displaystyle \\quad (\\mathrm{P_{\\lambda }}) $$\\end{document} where ω and λ are positive constants, V is a continuous function with negative infimum, qin (1,2), hin L^{frac{2}{2-q}}(mathbb{R}^{3}) is a positive potential function. Under the classic Ambrosetti–Rabinowitz condition, nontrivial solutions are obtained via the symmetric mountain pass theorem and the mountain pass theorem. In our paper, the nonlinearity F can also change sign and does not need to satisfy any 4-superlinear condition. We extend and improve some existing results to some extent.

Existence and multiplicity of non-trivial solutions for the fractional Schr\xf6dinger\u2013Poisson system with superlinear terms

In this paper, we study the following fractional Schrödinger–Poisson system with superlinear terms \t\t\t{(−Δ)su+V(x)u+K(x)ϕu=f(x,u),x∈R3,(−Δ)tϕ=K(x)u2,x∈R3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\Delta )^{s}u+V(x)u+K(x)\\phi u=f(x,u), & x \\in \\mathbb{R}^{3}, \\\\ (-\\Delta )^{t}\\phi =K(x)u^{2}, & x \\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where s,tin (0,1), 4s+2t>3. Under certain assumptions of external potential V(x), nonnegative density charge K(x) and superlinear term f(x,u), using the symmetric mountain pass theorem, we obtain the existence and multiplicity of non-trivial solutions.

Infinitely many solutions for fractional Kirchhoff-Schrödinger-Poisson systems

In this paper, we study the existence of infinitely many solutions for a fractional Kirchhoff–Schrödinger–Poisson system. Based on variational methods, especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case, we obtain infinitely many solutions for the system under certain assumptions. The novelties of this article lie in the appearance of the possibly degenerate Kirchhoff function and weak assumptions on the nonlinear term which are quite mild.

Existence and multiplicity of solutions for the fractional Schrödinger equation with steep potential well and perturbation

In the present paper, we investigate the solutions of a fractional Schrodinger equation with sublinear perturbation and steep potential well. First, we obtain the existence of at least one solution by the minimisation result due to Mawhin and Willem. Second, by virtue of the symmetric mountain pass theorem, the existence criteria of infinitely many solutions are established. Lastly, an example is given to demonstrate the main results.

Multiple solutions for a critical Kirchhoff system

We consider the nonlocal system −m∫Ω|∇u|2dxΔu=Fu(x,u,v)+μ1|u|4u,inΩ,−l∫Ω|∇v|2dxΔv=Fv(x,u,v)+μ2|v|4v,inΩ,with positive potentials m and l. The nonlinearity F is subcritical and locally superlinear at infinity. By using the Symmetric Mountain Pass Theorem we obtain multiple solutions for small values of μ1 and μ2.

Infinitely many solutions for magnetic fractional problems with critical Sobolev‐Hardy nonlinearities

In this paper, we study the existence of infinitely many solutions of the following degenerate magnetic fractional problem involving critical Sobolev‐Hardy nonlinearities: urn:x-wiley:mma:media:mma5317:mma5317-math-0001 where s ∈ (0,1), N > 2s, is the fractional Hardy‐Sobolev critical exponent with α ∈ [0,2s), λ is a positive real parameter, is a Kirchhoff function, is a magnetic potential function, and is the fractional magnetic operator. By using the new version of symmetric mountain pass theorem of Kajikiya, we prove that the problem admits infinitely many solutions for the suitable value of λ.

Multiple solutions for a class of superquadratic fractional Hamiltonian systems

In this paper, we are concerned with the existence of solutions for a class of fractional Hamiltonian systems \[\left\{ \begin{array}{l} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in\mathbb{R}\\ u\in H^{\alpha}(\mathbb{R},\ \mathbb{R}^{N}), \end{array}\right. \] where $_{t}D_{\infty}^{\alpha}$ and $_{-\infty}D^{\alpha}_{t}$ are the Liouville-Weyl fractional derivatives of order $\frac{1}{2}<\alpha<1$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix-valued function and $W(t,x)\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many solutions for (1) when $L$ is not required to be either uniformly positive definite or coercive and $W(t,x)$ satisfies some weaker superquadratic conditions at infinity in the second variable but does not satisfy the well-known Ambrosetti-Rabinowitz superquadratic growth condition.