scholarly journals Existence of solutions for a fractional Choquard-type equation in $$\mathbb {R}$$ with critical exponential growth

2021 ◽  
Vol 72 (1) ◽  
Author(s):  
Rodrigo Clemente ◽  
José Carlos de Albuquerque ◽  
Eudes Barboza
Keyword(s):  
Exponential Growth ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Critical Exponential Growth

Semiclassical ground state solutions for a Choquard type equation in ℝ2 with critical exponential growth

2018 ◽  
Vol 24 (1) ◽  
pp. 177-209 ◽  
Author(s):  
Minbo Yang
Keyword(s):  
Ground State ◽  
Exponential Growth ◽  
Riesz Potential ◽  
Type Equation ◽  
Singularly Perturbed ◽  
Maximum Point ◽  
Positive Parameter ◽  
Primitive Function ◽  
Critical Exponential Growth ◽  
Point Set

In this paper we study a nonlocal singularly perturbed Choquard type equation $$-\varepsilon^2\Delta u +V(x)u =\vr^{\mu-2}\left[\frac{1}{|x|^{\mu}}\ast \big(P(x)G(u)\big)\right]P(x)g(u)$$ in ℝ2, where ε is a positive parameter, \hbox{$\frac{1}{|x|^\mu}$} with 0 < μ < 2 is the Riesz potential, ∗ is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in ℝ2 in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.

scholarly journals (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

2020 ◽  
Vol 18 (1) ◽  
pp. 1423-1439
Author(s):  
Patrizia Pucci ◽  
Letizia Temperini
Keyword(s):  
Heisenberg Group ◽  
Exponential Growth ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Singular Behavior ◽  
Actual System ◽  
Exponential Term ◽  
Nonnegative Solutions ◽  
Critical Exponential Growth

Abstract The paper deals with the existence of solutions for (p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in {{\mathbb{H}}}^{n} .

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

scholarly journals Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

2020 ◽  
Vol 20 (2) ◽  
pp. 373-384
Author(s):  
Quoc-Hung Nguyen ◽  
Nguyen Cong Phuc
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Linear Range ◽  
Measure Data ◽  
Regularity Conditions ◽  
Compactly Supported ◽  
Singular Data ◽  
Quasilinear Elliptic

AbstractWe characterize the existence of solutions to the quasilinear Riccati-type equation\left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.with a distributional or measure datum σ. Here {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ({p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that {p>1} and {q>p}. For measure data, we assume that they are compactly supported in Ω, {p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range {p-1<q<1}. We also assume more regularity conditions on {\mathcal{A}} and on {\partial\Omega\Omega} in this case.

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.

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

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