Some bifurcation results and multiple solutions for the $p$-Laplacian equation

2021 ◽  
pp. 1-14
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Leiga Zhao
Keyword(s):  
Multiple Solutions ◽  
Laplacian Equation

scholarly journals Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

2015 ◽  
Vol 4 (1) ◽  
pp. 37-58 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Continuous Function ◽  
Weight Function ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Smooth Boundary ◽  
Existence Results ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Beta 2 ◽  
Fractional Laplace Operator

AbstractIn this article, we study the following p-fractional Laplacian equation: $ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.

scholarly journals MULTIPLE SOLUTIONS OF A $p(x)$-LAPLACIAN EQUATION INVOLVING CRITICAL NONLINEARITIES

2013 ◽  
Vol 17 (6) ◽  
pp. 2055-2082 ◽  
Author(s):  
Yuan Liang ◽  
Xianbin Wu ◽  
Qihu Zhang ◽  
Chunshan Zhao
Keyword(s):  
Multiple Solutions ◽  
Laplacian Equation ◽  
Critical Nonlinearities

Multiple solutions for the p-Laplacian equations with concave nonlinearities via Morse theory

2017 ◽  
Vol 19 (03) ◽  
pp. 1650014 ◽  
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Hongrui Cai
Keyword(s):  
Growth Condition ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Subcritical Growth ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Global Condition ◽  
Laplacian Equations

In this paper, by Morse theory, we study the existence and multiplicity of solutions for the [Formula: see text]-Laplacian equation with a “concave” nonlinearity and a parameter. In our results, we do not need any additional global condition on the nonlinearities, except for a subcritical growth condition.

Sign in / Sign up

Export Citation Format

Share Document