scholarly journals Nontrivial solutions for discrete Kirchhoff-type problems with resonance via critical groups

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Jinping Yang ◽  
Jinsheng Liu
Keyword(s):  
Critical Groups ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problems

scholarly journals On the Existence of Nontrivial Solutions for Nonlocal Elliptic Kirchhoff Type Problems with Nonlinear Boundary Conditions

2015 ◽  
Vol 55 (1) ◽  
pp. 183-188
Author(s):  
S. H. Rasouli ◽  
B. Salehi
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Type Equation ◽  
Nonlinear Boundary Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Kirchhoff Type Problems

Abstract In this paper, by using the Mountain Pass Lemma, we study the existence of nontrivial solutions for a nonlocal elliptic Kirchhoff type equation together with nonlinear boundary conditions.

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.

scholarly journals Nontrivial Solutions of a Class of Kirchhoff Type Problems via Local Linking Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Anmin Mao ◽  
Lin Zhang
Keyword(s):  
Nontrivial Solutions ◽  
Local Linking ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problems

The existence of nontrivial solutions of Kirchhoff type problems is proved by using the local linking method, and the new results do not require classical compactness conditions.

Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ruichang Pei
Keyword(s):  
Exponential Growth ◽  
Polynomial Growth ◽  
Mountain Pass Theorem ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Critical Exponential Growth ◽  
Trudinger Inequality ◽  
Fractional Kirchhoff Type Problems ◽  
Kirchhoff Type Problems ◽  
Subcritical Polynomial Growth

Abstract The main aim of this paper is to investigate the existence of nontrivial solutions for a class of fractional Kirchhoff-type problems with right-hand side nonlinearity which is subcritical or critical exponential growth (subcritical polynomial growth) at infinity. However, it need not satisfy the Ambrosetti–Rabinowitz (AR) condition. Some existence results of nontrivial solutions are established via Mountain Pass Theorem combined with the fractional Moser–Trudinger inequality.

scholarly journals Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ruichang Pei ◽  
Jihui Zhang
Keyword(s):  
Nontrivial Solution ◽  
Mountain Pass Theorem ◽  
Existence Results ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Positive Semiaxis ◽  
Asymmetric Growth ◽  
Trudinger Inequality ◽  
Kirchhoff Type Problems

We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at+∞and−∞inℝN(N=2,3). Namely, it is 4-linear at−∞and 4-superlinear at+∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality.

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