Class Of Fourth-order Elliptic Equations Research Articles

Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type

Abstract This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent Δ p ( x ) 2 u − M ( ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x ) Δ p ( x ) u + | u | p ( x ) − 2 u = λ f ( x , u ) + μ g ( x , u ) in Ω , u = Δ u = 0 on ∂ Ω , $$\begin{array}{} \left\{\begin{array}{} \Delta^2_{p(x)}u-M\bigg(\displaystyle\int\limits_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,\text{d} x \bigg)\Delta_{p(x)} u + |u|^{p(x)-2}u = \lambda f(x,u)+\mu g(x,u) \quad \text{ in }\Omega,\\ u=\Delta u = 0 \quad \text{ on } \partial\Omega, \end{array}\right. \end{array}$$ where p − := inf x ∈ Ω ¯ p ( x ) > max 1 , N 2 , λ > 0 $\begin{array}{} \displaystyle p^{-}:=\inf_{x \in \overline{\Omega}} p(x) \gt \max\left\{1, \frac{N}{2}\right\}, \lambda \gt 0 \end{array}$ and μ ≥ 0 are real numbers, Ω ⊂ ℝ N (N ≥ 1) is a smooth bounded domain, Δ p ( x ) 2 u = Δ ( | Δ u | p ( x ) − 2 Δ u ) $\begin{array}{} \displaystyle \Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u) \end{array}$ is the operator of fourth order called the p(x)-biharmonic operator, Δ p(x) u = div(|∇u| p(x)–2∇u) is the p(x)-Laplacian, p : Ω → ℝ is a log-Hölder continuous function, M : [0, +∞) → ℝ is a continuous function and f, g : Ω × ℝ → ℝ are two L 1-Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.

Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N

We consider a class of fourth-order elliptic equations of Kirchhoff type with critical growth in \(R^N\). By using constrained minimization in the Nehari manifold, weestablish sufficient conditions for the existence of nodal (that is, sign-changing) solutions.
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A class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$

In this article, we study the multiplicity of solutions for a class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$. Under the appropriate assumption, we prove that there are at least two solutions for the equation by Nehari manifold and Ekeland variational principle, one of which is the ground state solution.

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 of three nontrivial solutions for a class of fourth-order elliptic equations

The existence of three nontrivial solutions is established for a class of fourth-order elliptic equations. Our technical approach is based on Linking Theorem and ($\nabla$)-Theorem.

Non-uniformly asymptotically linear fourth-order elliptic problems

The existence of multiple solutions for a class of fourth-order elliptic equations with respect to the non-uniformly asymptotically linear conditions is established by using the minimax method and Morse theory.

Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations

This paper is concerned with the following fourth-order elliptic equation $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \Delta ^{2}u-\Delta u+V(x)u=|u|^{p-1}u,\,\mathrm{in}\,\mathbb {R}^{N},\\ u\in H^{2}\left( \mathbb {R}^{N}\right) , \end{array} \right. \end{aligned}$$ where \(p\in (2,\,2_{*}-1),\,u{\text {:}}\,\mathbb {R}^{N}\rightarrow \mathbb R.\) Under some appropriate assumptions on potential V(x), the existence of nontrivial solutions and the least energy nodal solution are obtained by using variational methods.

Multiple solutions for biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory.

Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

The main purpose of this paper is to establish the existence of two nontrivial solutions and the existence of infinitely many solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using a suitable version of the mountain pass theorem and the symmetric mountain pass theorem.

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].

Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

Existence of solutions for a class of biharmonic equations with the Navier boundary value condition

In this paper, the existence of at least one nontrivial solution for a class of fourth-order elliptic equations with the Navier boundary value conditions is established by using the linking methods.

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

In this paper, we study the existence and multiplicity of nontrivial solutions for the following fourth-order elliptic equations {△2u−△u+λV(x)u=f(x,u),in RNu∈H2(RN). via variational methods. Two main theorems are obtained.

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].