Kirchhoff–Schrödinger equations in ℝ2 with critical exponential growth and indefinite potential

2020 ◽  
pp. 2050030
Author(s):  
Marcelo F. Furtado ◽  
Henrique R. Zanata
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Exponential Growth ◽  
Nonlocal Problem ◽  
Type Inequality ◽  
Ground State Solution ◽  
Type Function ◽  
Local Case ◽  
Critical Exponential Growth ◽  
Indefinite Potential

We prove the existence of ground state solution for the nonlocal problem [Formula: see text] where [Formula: see text] is a Kirchhoff type function, [Formula: see text] may be negative and noncoercive, [Formula: see text] is locally bounded and the function [Formula: see text] has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case [Formula: see text]. In the proofs, we apply Variational Methods besides a new Trudinger–Moser type inequality.

scholarly journals Ground state solutions for a Kirchhoff type elliptic systems involving critical exponential growth nonlinearities

2021 ◽  
Author(s):  
Shengbing Deng ◽  
Tingting Huang
Keyword(s):  
Ground State ◽  
Exponential Growth ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
State Solution ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Critical Exponential Growth ◽  
Growth Nonlinearities

The aim of this paper is to study the ground state solution for a Kirchhoff type elliptic systems without the Ambrosetti-Rabinowitz condition.

scholarly journals Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

2020 ◽  
Vol 10 (1) ◽  
pp. 152-171
Author(s):  
Sitong Chen ◽  
Xianhua Tang ◽  
Jiuyang Wei
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Ground State ◽  
Exponential Growth ◽  
Sobolev Inequality ◽  
Critical Case ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Choquard Equation ◽  
Critical Exponential Growth

Abstract This paper deals with the following Choquard equation with a local nonlinear perturbation: $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$ where α ∈ (0, 2), Iα : ℝ2 → ℝ is the Riesz potential and f ∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponent $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ and the critical exponential growth of f(u).

Adams Type Inequality and Application for a Class of Polyharmonic Equations with Critical Growth

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
João Marcos do Ó ◽  
Abiel Costa Macedo
Keyword(s):  
Sobolev Space ◽  
Exponential Growth ◽  
Type Inequality ◽  
Critical Growth ◽  
Polyharmonic Equations ◽  
Critical Exponential Growth

AbstractIn this paper we give a new Adams type inequality for the Sobolev space W(−Δ)where the nonlinearity is “superlinear” and has critical exponential growth at infinite.

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