scholarly journals On ground state and nodal solutions of Schrödinger–Poisson equations with critical growth

2015 ◽  
Vol 428 (1) ◽  
pp. 387-404 ◽  
Author(s):  
Jian Zhang
Keyword(s):  
Ground State ◽  
Critical Growth ◽  
Nodal Solutions ◽  
Poisson Equations

On solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity

2021 ◽  
pp. 2150002
Author(s):  
Manassés de Souza ◽  
Uberlandio B. Severo ◽  
Thiago Luiz do Rêgo
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
State Level ◽  
Open Interval ◽  
Ground State Solution ◽  
Nodal Solutions ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Fractional Kirchhoff Type Problems ◽  
Kirchhoff Type Problems

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] is a bounded open interval, [Formula: see text] is a continuous potential, the nonlinear term [Formula: see text] has exponential growth of Trudinger–Moser type, [Formula: see text] and [Formula: see text] denotes the standard Gagliardo seminorm of the fractional Sobolev space [Formula: see text]. More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or sign-changing) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard [Formula: see text] as a positive parameter, we study the behavior of the nodal solutions as [Formula: see text].

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