scholarly journals Multiplicity results for a class of quasilinear equations with exponential critical growth

2017 ◽  
Vol 291 (2-3) ◽  
pp. 222-244 ◽  
Author(s):  
Claudianor O. Alves ◽  
Luciana R. de Freitas
Keyword(s):  
Critical Growth ◽  
Quasilinear Equations ◽  
Multiplicity Results ◽  
Exponential Critical Growth

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

2021 ◽  
pp. 1-28
Author(s):  
B. B. V. Maia ◽  
O. H. Miyagaki
Keyword(s):  
Riesz Potential ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Exponential Critical Growth ◽  
Beta 2 ◽  
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In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

Solutions for a class of singular quasilinear equations involving critical growth in R2

2022 ◽  
Author(s):  
Manassés X. de Souza ◽  
Uberlandio B. Severo ◽  
Gilberto F. Vieira
Keyword(s):  
Critical Growth ◽  
Quasilinear Equations

Nodal solutions for fractional elliptic equations involving exponential critical growth

2020 ◽  
Vol 43 (6) ◽  
pp. 3650-3672
Author(s):  
Manassés Souza ◽  
Uberlandio Batista Severo ◽  
Thiago Luiz do Rêgo
Keyword(s):  
Elliptic Equations ◽  
Critical Growth ◽  
Nodal Solutions ◽  
Exponential Critical Growth
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