scholarly journals Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term

2015 ◽  
Vol 06 (14) ◽  
pp. 2248-2256
Author(s):  
Mohammed El Mokhtar Ould El Mokhtar
Keyword(s):  
Critical Exponent ◽  
Nontrivial Solutions ◽  
Critical Potential ◽  
Concave Term

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

scholarly journals Existence of Nontrivial Solutions for Perturbed Elliptic System inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Juan Jiang
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Elliptic System ◽  
Existence Result ◽  
Nonlinear Elliptic System ◽  
Nontrivial Solutions ◽  
Nonlinear Elliptic ◽  
Sobolev Critical Exponent

We consider the perturbed nonlinear elliptic system-ε2Δu+V(x)u=K(x)|u|2*-2u+Hu(u,v),  x∈ℝN,-ε2Δv+V(x)v=K(x)|v|2*-2v+Hv(u,v),  x∈ℝN, whereN≥3,2*=2N/(N-2)is the Sobolev critical exponent. Under proper conditions onV,H, andK, the existence result and multiplicity of the system are obtained by using variational method providedεis small enough.

scholarly journals Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
Mohammed El Mokhtar Ould El Mokhtar
Keyword(s):  
Variational Principle ◽  
Elliptic Equation ◽  
Critical Exponent ◽  
Elliptic Equations ◽  
Local Minimizer ◽  
Ekeland’S Variational Principle ◽  
Nontrivial Solutions ◽  
Ekeland's Variational Principle

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.

scholarly journals Nontrivial solutions for critical potential elliptic systems

2011 ◽  
Vol 250 (8) ◽  
pp. 3398-3417 ◽  
Author(s):  
Ezequiel R. Barbosa ◽  
Marcos Montenegro
Keyword(s):  
Elliptic Systems ◽  
Nontrivial Solutions ◽  
Critical Potential

scholarly journals A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality

2018 ◽  
Vol 20 (04) ◽  
pp. 1750037 ◽  
Author(s):  
Fashun Gao ◽  
Minbo Yang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Nontrivial Solution ◽  
Sobolev Inequality ◽  
Schrodinger Equation ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Choquard Equation ◽  
Semilinear Equation ◽  
Nonlinear Choquard Equation

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

scholarly journals On the Existence of Solutions for the Critical Fractional Laplacian Equation inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zifei Shen ◽  
Fashun Gao
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Lower Order ◽  
Fractional Laplacian ◽  
Critical Power ◽  
Existence Of Solutions ◽  
Order Perturbation ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Laplacian Equation

We study existence of solutions for the fractional Laplacian equation-Δsu+Vxu=u2*s-2u+fx, uinℝN,u∈Hs(RN), with critical exponent2*s=2N/(N-2s),N>2s,s∈0, 1, whereVx≥0has a potential well andf:ℝN×ℝ→ℝis a lower order perturbation of the critical poweru2*s-2u. By employing the variational method, we prove the existence of nontrivial solutions for the equation.

scholarly journals Infinitely many solutions for a nonlocal elliptic system of $(p_1,\ldots,p_n)$-Kirchhoff type with critical exponent

2021 ◽  
Vol 39 (5) ◽  
pp. 199-221
Author(s):  
Ghasem A. Afrouzi ◽  
Giuseppe Caristi ◽  
Amjad Salari
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Critical Point ◽  
Elliptic System ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Nontrivial Solutions ◽  
Kirchhoff Type

The existence of infinitely many nontrivial solutions for a nonlocal elliptic system of $(p_1,\ldots,p_n)$-Kirchhoff type with critical exponent is investigated. The approach is based on variational methods and critical point theory.

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