Solutions For Fourth-order Elliptic Equations Research Articles

Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain

Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain

Infinitely many small energy solutions for Fourth-Order Elliptic Equations with p-Laplacian in [formula omitted

In this paper, we study the following fourth-order elliptic equation with p-Laplacian, steep potential well and sublinear perturbation: Δ2u−Δpu+μV(x)u=f(x,u)+ξ(x)|u|q−2u,x∈RN,where N≥5, Δ2≔Δ(Δ) is the biharmonic operator, Δpu=div|∇u|p−2∇u with p>2, μ>0 is a parameter, f∈CRN×R,R, ξ∈L22−qRN with 1≤q<2, we have the potential V∈C(RN,R), and V−1(0) has nonempty interior. Under certain assumptions on V and f, we show the existence nontrivial solutions by virtue of variational methods, the existence criteria of infinitely many nontrivial small energy solutions are established.

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Multiplicity of Solutions for Fourth-Order Elliptic Equations with p-Laplacian and Mixed Nonlinearity

In the paper, we study the multiplicity of solutions for a class of fourth-order elliptic equations with p-Laplacian and mixed nonlinearity of the form: $$\begin{aligned} \left\{ \begin{array}{lll} \Delta ^2u-\Delta _{p}u+\lambda V(x)u=f(x,u)+\mu \xi (x)\vert u\vert ^{q-2}u, \ \ x\in {\mathbb {R}}^N, \\ u\in H^{2}({\mathbb {R}}^N). \\ \end{array} \right. \end{aligned}$$Unlike most other works, we replace the Laplacian with a p-Laplacian. Using the mountain pass theorem and Ekeland’s variational principle, we establish the existence of two nontrivial solutions. To overcome the difficulty of the convergence of the subsequences for the Palais–Smale sequences of the Euler–Lagrange functional, we consider Cerami sequences. Our results extend the recent results of Zhang et al. (Electron J Differ Equ 2017(250):1–15, 2017).

Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN

In this paper, we deal with the existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical nonlinearity: −ε4Δ2u+ε4a+b∫RN∇u2dxΔu+V(x)u=u2**−2u+h(x,u), (t, x) ∈ ℝ × ℝN. By using Lions’ second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS) condition holds locally and by variational method, we prove that it has at least one solution and for any m ∈ ℕ, it has at least m pairs of solutions.

Multiplicity of Solutions for Fourth-Order Elliptic Equations of Kirchhoff Type with Critical Exponent

In this paper, we deal with fourth-order elliptic equations of Kirchhoff type with critical exponent in bounded domains, the new results about existence, and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method.

Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory

In this paper, we study the following fourth-order elliptic equations of Kirchhoff type: , in , , where are constants, we have the potential and the nonlinearity . Under certain assumptions on and , we show the existence and multiplicity of negative energy solutions for the above system based on the genus properties in critical point theory. MSC: 35J20, 35J65, 35J60.

Infinitely many solutions for fourth-order elliptic equations with general potentials

This paper is concerned with the following fourth-order elliptic equations {Δ2u−Δu+V(x)u=f(x,u)in RN,u∈H2(RN), where u:RN→R. Under general assumption on potential V(x), and f(x,u) is indefinite sign and sublinear as |u|→∞, we establish the existence of infinitely many solutions by using the genus properties in critical point theory. Recent results from the literature are extended.

Existence and multiplicity of solutions for fourth-order elliptic equations in [formula omitted

In this paper, we study the fourth order elliptic equations Δ2u−Δu+λV(x)u=f(x,u) in RN, u∈H2(RN), where λ≥1 and the nonlinearity f is either superlinear or sublinear at infinity. We establish the existence and multiplicity results without the compactness of embedding of the working space, which unify and sharply improve the recent results of Liu, Chen and Wu [J. Liu, S. Chen, X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in RN, J. Math. Anal. Appl. 395 (2012) 608–615].

Infinitely many solutions for fourth-order elliptic equations

In this paper, we study a class of fourth-order elliptic equations setting on RN. Based on the minimax methods in critical point theory, we obtain the existence of infinitely many large-energy and small-energy solutions, which unifies and improves the recent results of Yin and Wu [Y. Yin, X. Wu, High energy solutions and nontrivial solutions for fourth-order elliptic equations, J. Math. Anal. Appl. 375 (2011) 699–705].

High energy solutions and nontrivial solutions for fourth-order elliptic equations

In this paper, we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth-order elliptic equations in R N via variational methods. Three main theorems are obtained.

Boundary decay estimates for solutions of fourth-order elliptic equations

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.