scholarly journals Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

2021 ◽  
Vol 26 (2) ◽  
pp. 227-240
Author(s):  
Lin Li ◽  
Donal O’Regan
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Fourth Order ◽  
Local Minimum ◽  
Elliptic Problems ◽  
Order Problem ◽  
Critical Nonlinearities ◽  
Fourth Order Problem ◽  
Minimum Theorem ◽  
Sublinear Perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

SOME REMARKS ON THE CURVATURE TYPE PROBLEM ON

2014 ◽  
Vol 97 (1) ◽  
pp. 127-143
Author(s):  
SANJIBAN SANTRA
Keyword(s):  
Positive Solutions ◽  
Fourth Order ◽  
Type Problem ◽  
Order Problem ◽  
Fourth Order Problem ◽  
Multiplicity Of Positive Solutions ◽  
Curvature Type

AbstractIn this paper, we prove the existence, uniqueness and multiplicity of positive solutions of a nonlinear perturbed fourth-order problem related to the$Q$curvature.

scholarly journals A local minimum theorem and critical nonlinearities

2016 ◽  
Vol 24 (2) ◽  
pp. 67-86
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D’Aguì ◽  
Donal O’Regan
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Positive Solutions ◽  
Local Minimum ◽  
Numerical Estimate ◽  
Main Tool ◽  
Real Parameter ◽  
Fundamental Result ◽  
Critical Nonlinearities ◽  
Minimum Theorem

Abstract In this paper the existence of two positive solutions for a Dirichlet problem having a critical growth, and depending on a real parameter, is established. The approach is based on methods which are totally variational, unlike the fundamental result of Ambrosetti, Brezis and Cerami where a clever combination of topological and variational methods is used in order to obtain the same conclusion. In addition, a numerical estimate of real parameters, for which the two solutions are obtained, is provided. Our main tool is a local minimum theorem.

scholarly journals S-Shaped Connected Component for Nonlinear Fourth-Order Problem of Elastic Beam Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Jinxiang Wang ◽  
Ruyun Ma ◽  
Jin Wen
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Elastic Beam ◽  
Connected Components ◽  
Beam Equation ◽  
Connected Component ◽  
Order Problem ◽  
Fourth Order Problem

We investigate the existence of S-shaped connected component in the set of positive solutions of the fourth-order boundary value problem: u′′′′x=λhxfux, x∈(0,1),u(0)=u(1)=u′′0=u′′1=0, where λ>0 is a parameter, h∈C[0,1], and f∈C[0,∞) with f0≔lims→0⁡(f(s)/s)=∞. We develop a bifurcation approach to deal with this extreme situation by constructing a sequence of functions f[n] satisfying f[n]→f and (f[n])0→∞. By studying the auxiliary problems, we get a sequence of unbounded connected components C[n], and, then, we find an unbounded connected component C in the set of positive solutions of the fourth-order boundary value problem which satisfies 0,0∈C⊂lim⁡sup⁡C[n] and is S-shaped.

Fractional problems with critical nonlinearities by a sublinear perturbation

2020 ◽  
Vol 23 (2) ◽  
pp. 484-503 ◽  
Author(s):  
Lin Li ◽  
Stepan Tersian
Keyword(s):  
Critical Exponent ◽  
Variational Approach ◽  
Local Minimum ◽  
Numerical Estimate ◽  
Nontrivial Solutions ◽  
The Real ◽  
Critical Nonlinearities ◽  
Minimum Theorem ◽  
Sublinear Perturbation

AbstractIn this paper, the existence of two nontrivial solutions for a fractional problem with critical exponent, depending on real parameters, is established. The variational approach is used based on a local minimum theorem due to G. Bonanno. In addition, a numerical estimate on the real parameters is provided, for which the two solutions are obtained.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Some nonexistence theorems for semilinear fourth-order equations

2018 ◽  
Vol 149 (03) ◽  
pp. 761-779 ◽  
Author(s):  
M. Á. Burgos-Pérez ◽  
J. García-Melián ◽  
A. Quaas
Keyword(s):  
Fourth Order ◽  
Exterior Domains ◽  
Order Problem ◽  
Previous Condition ◽  
Fourth Order Problem ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Fourth Order Equations

AbstractIn this paper, we analyse the semilinear fourth-order problem ( − Δ)2 u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu &gt; 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ &gt; 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu &gt; 0 in ℝN, and they do not exist when the previous condition fails.

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