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 Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1792
Author(s):  
Yun-Ho Kim
Keyword(s):  
Potential Function ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Laplacian Operator ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Continuous Potential ◽  
The Given ◽  
Laplacian Equations

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

Multiple waves with energy sign-changing for Schrödinger–Poisson system of critical potential

2016 ◽  
Vol 09 (02) ◽  
pp. 1650028
Author(s):  
Khalid Iskafi
Keyword(s):  
Mountain Pass Theorem ◽  
Physical Space ◽  
Critical Growth ◽  
Mountain Pass ◽  
Poisson System ◽  
Critical Potential ◽  
Nonlinear Potential

We consider a nonhomogeneous Schrödinger–Poisson system in the whole physical space involving nonlinear potential of critical growth, and we prove existence of two waves with energy sign-changing by using the Mountain-Pass theorem.

Small perturbations of elliptic problems with variable growth in RN

2021 ◽  
Vol 477 (2249) ◽  
Author(s):  
Bin Ge ◽  
Hai-Cheng Liu ◽  
Bei-Lei Zhang
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Ekeland’S Variational Principle ◽  
Mountain Pass ◽  
Ekeland's Variational Principle ◽  
Small Perturbations ◽  
Whole Space ◽  
Laplacian Equations

In this paper, we study the existence of at least two non-trivial solutions for a class of p ( x )-Laplacian equations with perturbation in the whole space. Using Ekeland’s variational principle and the mountain pass theorem, under appropriate assumptions, we prove the existence of two solutions for the equations.

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 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.

Schrödinger–Poisson system with potential of critical growth

2016 ◽  
Vol 09 (04) ◽  
pp. 1650086
Author(s):  
Abdessamad Hassani ◽  
Khalid Iskafi
Keyword(s):  
Variational Principle ◽  
Positive Solution ◽  
Mountain Pass Theorem ◽  
Critical Growth ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Positive Energy ◽  
Poisson System ◽  
Nonlinear Potential

In this paper, we consider a Schrödinger–Poisson system with nonlinear potential of critical growth, and we prove existence of positive solution with positive energy by using the Ekeland variational principle and the Mountain-Pass theorem.

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.

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