scholarly journals On Multiple Solutions of Quasilinear Equations Involving the Critical Sobolev Exponent

1999 ◽  
Vol 231 (1) ◽  
pp. 142-160 ◽  
Author(s):  
Yisheng Huang
Keyword(s):  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Quasilinear Equations ◽  
Sobolev Exponent

Multiplicity results for nonlinear elliptic equations involving critical Sobolev exponent

1986 ◽  
Vol 103 (3-4) ◽  
pp. 275-285 ◽  
Author(s):  
A. Capozzi ◽  
G. Palmieri
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Multiplicity Results ◽  
Variational Techniques ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisIn this paper we study the following boundary value problemwhere Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.

On inhomogeneous biharmonic equations involving critical exponents

1999 ◽  
Vol 129 (5) ◽  
pp. 925-946 ◽  
Author(s):  
Yinbin Deng ◽  
Gengsheng Wang
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Mountain Pass Lemma ◽  
Biharmonic Equations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Image Position

In this paper, we consider the existence of multiple solutions of biharmonic equations boundary value problemwhere Ω is a bounded smooth domain in ℝN, N ≥ 5; λ ∈ ℝ1 is a given constant; p = 2N/(N − 4) is the critical Sobolev exponent for the embedding ; Δ2 = ΔΔ denotes iterated N-dimensional Laplacian; f(x) is a given function. Some results on the existence and non-existence of multiple solutions for the above problem have been obtained by Ekeland's variational principle and the mountain-pass lemma under some assumptions on f(x) and N.

Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent

1994 ◽  
Vol 124 (6) ◽  
pp. 1177-1191 ◽  
Author(s):  
Dao-Min Cao ◽  
Gong-Bao Li ◽  
Huan-Song Zhou
Keyword(s):  
Variational Principle ◽  
Energy Balance ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Mountain Pass ◽  
Sobolev Exponent ◽  
Image Position

We consider the following problem:where is continuous on RN and h(x)≢0. By using Ekeland's variational principle and the Mountain Pass Theorem without (PS) conditions, through a careful inspection of the energy balance for the approximated solutions, we show that the probelm (*) has at least two solutions for some λ* > 0 and λ ∈ (0, λ*). In particular, if p = 2, in a different way we prove that problem (*) with λ ≡ 1 and h(x) ≧ 0 has at least two positive solutions as

Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

2016 ◽  
Vol 146 (6) ◽  
pp. 1167-1193 ◽  
Author(s):  
Wenjing Chen ◽  
Shengbing Deng
Keyword(s):  
Elliptic System ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Dirichlet Boundary Conditions ◽  
Bounded Set ◽  
Sobolev Exponent ◽  
Image Position ◽  
Two Parameters ◽  
The Nehari Manifold

We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:where Ω is a smooth bounded set in ℝn, n > 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ > 0 are two parameters; the exponent n/(n – 2s) ⩽ q < 2; α > 1, β > 1 satisfy is the fractional critical Sobolev exponent.

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