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.

scholarly journals Hemivariational inequalities with the potential crossing the first eigenvalue

2001 ◽  
Vol 64 (3) ◽  
pp. 381-393 ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Dirichlet Boundary ◽  
Mountain Pass Theorem ◽  
Principal Eigenvalue ◽  
Dirichlet Boundary Conditions ◽  
Hemivariational Inequalities ◽  
Linking Theorem ◽  
First Eigenvalue ◽  
The First Eigenvalue

In this paper we study a nonlinear hemivariational inequality involving the p-Laplacian. Our approach is variational and uses a recent nonsmooth Linking Theorem, due to Kourogenis and Papageorgiou (2000). The use of the Linking Theorem instead of the Mountain Pass Theorem allows us to assume an asymptotic behaviour of the generalised potential function which goes beyond the principal eigenvalue of the negative p-Laplacian with Dirichlet boundary conditions.

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).

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.

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

2016 ◽  
Vol 16 (1) ◽  
pp. 51-65 ◽  
Author(s):  
Salvatore A. Marano ◽  
Sunra J. N. Mosconi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Morse Theory ◽  
Multiple Solutions ◽  
First Eigenvalue ◽  
Reaction Term ◽  
The First Eigenvalue

AbstractThe existence of multiple solutions to a Dirichlet problem involving the ${(p,q)}$-Laplacian is investigated via variational methods, truncation-comparison techniques, and Morse theory. The involved reaction term is resonant at infinity with respect to the first eigenvalue of ${-\Delta_{p}}$ in ${W^{1,p}_{0}(\Omega)}$ and exhibits a concave behavior near zero.

Stable hypersurfaces via the first eigenvalue of the anisotropic Laplacian operator

2018 ◽  
Vol 64 (2) ◽  
pp. 427-436
Author(s):  
Jonatan Floriano da Silva ◽  
Henrique Fernandes de Lima ◽  
Marco Antonio Lázaro Velásquez
Keyword(s):  
Laplacian Operator ◽  
First Eigenvalue ◽  
The First Eigenvalue

scholarly journals Nonresonance conditions on the potential for a nonlinear nonautonomous Neumann problem

2019 ◽  
Vol 38 (3) ◽  
pp. 79-96 ◽  
Author(s):  
Ahmed Sanhaji ◽  
A. Dakkak
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Elliptic Problems ◽  
Neumann Boundary ◽  
Existence Results ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
The First Eigenvalue

The aim of this paper is to establish the existence of the principal eigencurve of the p-Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p-Laplacian operator.

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

2020 ◽  
pp. 2050031
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Binlin Zhang
Keyword(s):  
Critical Exponents ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Groups ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
The First Eigenvalue

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

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