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

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.

scholarly journals Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Junjun Zhou ◽  
Xiangyun Hu ◽  
Tiaojie Xiao
Keyword(s):  
Critical Exponent ◽  
Positive Solutions ◽  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Mountain Pass ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
Kirchhoff Type Problems

In this paper, we consider the following Kirchhoff-type problems involving critical exponent −a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u, x∈Ωu>0, x∈Ωu=0, x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

A class of asymptotically periodic fractional Schrödinger equations with critical growth

2018 ◽  
Vol 20 (03) ◽  
pp. 1750011
Author(s):  
Manassés de Souza ◽  
Yane Lísley Araújo
Keyword(s):  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Positive Potential ◽  
Fractional Schrödinger Equations ◽  
Asymptotically Periodic ◽  
Sobolev Exponent

In this paper, we study a class of fractional Schrödinger equations in [Formula: see text] of the form [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is the critical Sobolev exponent, [Formula: see text] is a positive potential bounded away from zero, and the nonlinearity [Formula: see text] behaves like [Formula: see text] at infinity for some [Formula: see text], and does not satisfy the usual Ambrosetti–Rabinowitz condition. We also assume that the potential [Formula: see text] and the nonlinearity [Formula: see text] are asymptotically periodic at infinity. We prove the existence of at least one solution [Formula: see text] by combining a version of the mountain-pass theorem and a result due to Lions for critical growth.

scholarly journals Multiple solutions for a problem with resonance involving thep-Laplacian

1998 ◽  
Vol 3 (1-2) ◽  
pp. 191-201 ◽  
Author(s):  
C. O. Alves ◽  
P. C. Carrião ◽  
O. H. Miyagaki
Keyword(s):  
Variational Principle ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Mountain Pass Theorem ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Three Solutions ◽  
Bounded Functions

In this paper we will investigate the existence of multiple solutions for the problem(P)                                                         −Δpu+g(x,u)=λ1h(x)|u|p−2u,     in     Ω,    u∈H01,p(Ω)whereΔpu=div(|∇u|p−2∇u)is thep-Laplacian operator,Ω⫅ℝNis a bounded domain with smooth boundary,handgare bounded functions,N≥1and1<p<∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).

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