On the nonexistence of nontrivial solutions for some P-Laplacian equations

2017 ◽  
Vol 11 ◽  
pp. 1057-1066
Author(s):  
Jeng-Eng Lin
Keyword(s):  
Nontrivial Solutions ◽  
Laplacian Equations

N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu
Keyword(s):  
Bounded Domain ◽  
Exponential Growth ◽  
Nonnegative Solution ◽  
Nontrivial Solutions ◽  
Critical Exponential Growth ◽  
Trudinger Inequality ◽  
Laplacian Equations

AbstractLet Ω be a bounded domain in ℝwhen f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝ

NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH

2013 ◽  
Vol 11 (03) ◽  
pp. 1350005 ◽  
Author(s):  
ZHONG TAN ◽  
FEI FANG
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Orlicz Spaces ◽  
Nontrivial Solutions ◽  
Laplacian Equations

Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

Multiple Nontrivial Solutions for a Class of p-Laplacian Equations

2009 ◽  
Vol 110 (3) ◽  
pp. 1153-1167
Author(s):  
Lei Wei ◽  
Mingxin Wang ◽  
Jiang Zhu
Keyword(s):  
Nontrivial Solutions ◽  
Laplacian Equations

scholarly journals Three nontrivial solutions for nonlinear fractional Laplacian equations

2018 ◽  
Vol 7 (2) ◽  
pp. 211-226 ◽  
Author(s):  
Fatma Gamze Düzgün ◽  
Antonio Iannizzotto
Keyword(s):  
Boundary Value Problem ◽  
Morse Theory ◽  
Fractional Laplacian ◽  
Mountain Pass Theorem ◽  
Pseudodifferential Equation ◽  
Nontrivial Solutions ◽  
Reaction Term ◽  
Second Deformation Theorem ◽  
Type Boundary ◽  
Laplacian Equations

AbstractWe study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

scholarly journals Four Solutions for Fractional p-Laplacian Equations with Asymmetric Reactions

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Roberto Livrea
Keyword(s):  
Morse Theory ◽  
Critical Point Theory ◽  
Point Theory ◽  
Positive Parameter ◽  
Type Problem ◽  
Nontrivial Solutions ◽  
Comparison Result ◽  
Asymmetric Perturbation ◽  
Laplacian Equations ◽  
Dirichlet Type Problem

AbstractWe consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.

scholarly journals N-Laplacian equations inℝNwith critical growth

1997 ◽  
Vol 2 (3-4) ◽  
pp. 301-315 ◽  
Author(s):  
João Marcos B. do Ó
Keyword(s):  
Continuous Function ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Laplacian Equations

We study the existence of nontrivial solutions to the following problem:{u∈W1,N(ℝN),u≥0  and−div(|∇u|N−2∇u)+a(x)|u|N−2u=f(x,u)  in  ℝN(N≥2),whereais a continuous function which is coercive, i.e.,a(x)→∞ as |x|→∞and the nonlinearityfbehaves likeexp(α|u|N/(N−1))when|u|→∞.

scholarly journals Nontrivial solutions of superlinear nonlocal problems

2016 ◽  
Vol 28 (6) ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš ◽  
Raffaella Servadei
Keyword(s):  
Weak Solutions ◽  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Fountain Theorem ◽  
Nonlocal Equations ◽  
Laplacian Equations

AbstractWe study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

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