Positive solutions for the p-Laplacian: application of the fibrering method

1997 ◽  
Vol 127 (4) ◽  
pp. 703-726 ◽  
Author(s):  
Pavel Drábek ◽  
Stanislav I. Pohozaev
Keyword(s):  
Differential Operator ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Main Part ◽  
Second Order ◽  
Multiple Positive Solutions ◽  
Quasilinear Problems

Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.

scholarly journals The growth of the positive solutions of Lu = 0 near the boundary of an inner NTA domain

1988 ◽  
Vol 110 ◽  
pp. 129-135
Author(s):  
Katsunori Shimomura
Keyword(s):  
Differential Operator ◽  
Euclidean Space ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Partial Differential Operator ◽  
Second Order ◽  
Partial Differential ◽  
Hölder Continuous

Let D be a bounded domain in the Euclidean space Rn (n ≧ 2) and L a uniformly elliptic partial differential operator of second order with α-Hölder continuous coefficients (0 < α ≦ 1) on D.

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

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].

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