Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems

2000 ◽  
Vol 81 (1) ◽  
pp. 373-396 ◽  
Author(s):  
Shujie Li ◽  
Zhi-Qiang Wang
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Dirichlet Problems ◽  
Semilinear Elliptic ◽  
Order Intervals

Decaying Solutions Of 2mth Order Elliptic Problems

1991 ◽  
Vol 43 (3) ◽  
pp. 449-460 ◽  
Author(s):  
W. Allegretto ◽  
L. S. Yu
Keyword(s):  
Positive Solution ◽  
Elliptic Problem ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Weighted Spaces ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Regularity Estimates ◽  
Open Question

AbstractWe consider a semilinear elliptic problem , (n > 2m). Under suitable conditions on f, we show the existence of a decaying positive solution. We do not employ radial arguments. Our main tools are weighted spaces, various applications of the Mountain Pass Theorem and LP regularity estimates of Agmon. We answer an open question of Kusano, Naito and Swanson [Canad. J. Math. 40(1988), 1281-1300] in the superlinear case: , and improve the results of Dalmasso [C. R. Acad. Sci. Paris 308(1989), 411-414] for the case .

scholarly journals Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Lizhen Chen ◽  
Anran Li ◽  
Chongqing Wei
Keyword(s):  
Variational Methods ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Negative Energy ◽  
Mountain Pass ◽  
Poisson System ◽  
Fountain Theorem ◽  
Symmetric Mountain Pass Theorem ◽  
Dual Fountain Theorem

We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to 0. These results extend some known results in previous papers.

scholarly journals Multiple Solutions for a Class ofN-Laplacian Equations with Critical Growth and Indefinite Weight

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Guoqing Zhang ◽  
Ziyan Yao
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Critical Growth ◽  
Mountain Pass ◽  
Indefinite Weight ◽  
Laplacian Equations

Using the suitable Trudinger-Moser inequality and the Mountain Pass Theorem, we prove the existence of multiple solutions for a class ofN-Laplacian equations with critical growth and indefinite weight-div∇uN-2∇u+VxuN-2u=λuN-2u/xβ+fx,u/xβ+ɛhx,x∈ℝN,u≠0,x∈ℝN, where0<β<N,V(x)is an indefinite weight,f:ℝN×ℝ→ℝbehaves likeexp⁡αuN/N-1and does not satisfy the Ambrosetti-Rabinowitz condition, andh∈(W1,N(ℝN))*.

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).

scholarly journals Existence of Multiple Solutions for a Class of Biharmonic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Chunhan Liu ◽  
Jianguo Wang
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Biharmonic Equations ◽  
Symmetric Mountain Pass Theorem ◽  
Type Boundary

By a symmetric Mountain Pass Theorem, a class of biharmonic equations with Navier type boundary value at the resonant and nonresonant case are discussed, and infinitely many solutions of the equations are obtained.

scholarly journals Critical Concave Convex Ambrosetti–Prodi Type Problems for Fractional 𝑝-Laplacian

2020 ◽  
Vol 20 (4) ◽  
pp. 847-865
Author(s):  
H. P. Bueno ◽  
E. Huerto Caqui ◽  
O. H. Miyagaki ◽  
F. R. Pereira
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Linking Theorem ◽  
First Eigenvalue ◽  
The First Eigenvalue

AbstractIn this paper, we consider a class of critical concave convex Ambrosetti–Prodi type problems involving the fractional 𝑝-Laplacian operator. By applying the linking theorem and the mountain pass theorem as well, the interaction of the nonlinearities with the first eigenvalue of the fractional 𝑝-Laplacian will be used to prove existence of multiple solutions.

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

scholarly journals A Variant of the Mountain Pass Theorem and Variational Gluing

2020 ◽  
Vol 88 (2) ◽  
pp. 347-372
Author(s):  
Piero Montecchiari ◽  
Paul H. Rabinowitz
Keyword(s):  
Differential Equations ◽  
Recent Work ◽  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Elliptic Pdes ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Pdes

AbstractThis paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on $$R^n$$ R n and to a family of Hamiltonian systems involving double well potentials will also be discussed.

Semilinear elliptic equations of the Hénon-type in hyperbolic space

2016 ◽  
Vol 18 (02) ◽  
pp. 1550026 ◽  
Author(s):  
P. C. Carrião ◽  
L. F. O. Faria ◽  
O. H. Miyagaki
Keyword(s):  
Unit Ball ◽  
Positive Solution ◽  
Hyperbolic Space ◽  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Equations ◽  
Mountain Pass ◽  
Sobolev Embedding ◽  
Radial Functions ◽  
Semilinear Elliptic

This paper deals with a class of the semilinear elliptic equations of the Hénon-type in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in [Formula: see text] Combining this compactness embedding with the Mountain Pass Theorem, a result of the existence of positive solution is established.

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