scholarly journals Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Said Taarabti
2014 ◽  
Vol 16 (06) ◽  
pp. 1450034 ◽  
Author(s):  
Xiaoyu Zeng ◽  
Yimin Zhang ◽  
Huan-Song Zhou

We are concerned with positive solutions of a quasilinear Schrödinger equation with Hardy potential and critical exponent. Different from the semilinear equation, the Hardy term in our equation is not only singular, but also nonlinear. It seems unlikely to get solutions for our equation in H1(ℝN) ∩ L∞(ℝN) by using Nehari method as Liu–Liu–Wang [Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 46 (2013) 641–669]. In this paper, by transforming the quasilinear equation to a semilinear equation, we established the existence of positive solutions for the quasilinear Schrödinger equation in H1(ℝN) under suitable conditions.


1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).


Author(s):  
Shaya Shakerian

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].


2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjie Liu ◽  
Xiao Fu ◽  
Liangping Qi

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.


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