Infinitely Many Solutions for Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations

2017 ◽  
Vol 60 (4) ◽  
pp. 1003-1020 ◽  
Author(s):  
Hongxue Song ◽  
Caisheng Chen
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Biharmonic Operator ◽  
Kirchhoff Type ◽  
Symmetric Mountain Pass Theorem ◽  
Image Position ◽  
Fourth Order Elliptic Equations ◽  
Biharmonic Problems

AbstractThis paper deals with the class of Schrödinger–Kirchhoff-type biharmonic problemswhere Δ2 denotes the biharmonic operator, and f ∈ C(ℝN × ℝ, ℝ) satisfies the Ambrosetti–Rabinowitz-type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.

Multiple solutions for a fourth-order elliptic equation of Kirchhoff type with variable exponent

2019 ◽  
Vol 13 (05) ◽  
pp. 2050096 ◽  
Author(s):  
Nguyen Thanh Chung
Keyword(s):  
Continuous Function ◽  
Fourth Order ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Multiplicity Result ◽  
Hölder Continuous ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Elliptic Equations

In this paper, we consider a class of fourth-order elliptic equations of Kirchhoff type with variable exponent [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is the operator of fourth-order called the [Formula: see text]-biharmonic operator, [Formula: see text] is the [Formula: see text]-Laplacian, [Formula: see text] is a log-Hölder continuous function and [Formula: see text] is a continuous function satisfying some certain conditions. A multiplicity result for the problem is obtained by using the mountain pass theorem and Ekeland’s variational principle provided [Formula: see text] is small enough.

Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings

2015 ◽  
Vol 4 (2) ◽  
pp. 135-151 ◽  
Author(s):  
Reginaldo Demarque ◽  
Olimpio H. Miyagaki
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Continuous Functions ◽  
Existence Results ◽  
Compact Embedding ◽  
Biharmonic Operator ◽  
Radial Solutions ◽  
Sobolev Embeddings ◽  
Fourth Order Elliptic Equations

AbstractWe deal with a class of inhomogeneous elliptic problems involving the biharmonic operator Δ2u + V(|x|)|u|q-2u = Q(|x|)f(u), u ∈ D02,2(ℝN), where Δ2 is the biharmonic operator and V, Q are singular continuous functions. Compact embedding results are established and by using these facts some existence results are obtained.

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