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

scholarly journals Concentration Phenomena for Fractional Elliptic Equations Involving Exponential Critical Growth

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Claudianor O. Alves ◽  
João Marcos do Ó ◽  
Olímpio H. Miyagaki
Keyword(s):  
Elliptic Problem ◽  
Elliptic Equations ◽  
Critical Growth ◽  
Concentration Phenomena ◽  
Exponential Critical Growth

AbstractIn this paper, we deal with the singular perturbed fractional elliptic problem

Multiplicity of Multi-Bump Type Nodal Solutions for A Class of Elliptic Problems with Exponential Critical Growth in ℝ2

2016 ◽  
Vol 60 (2) ◽  
pp. 273-297 ◽  
Author(s):  
Claudianor O. Alves ◽  
Denilson S. Pereira
Keyword(s):  
Continuous Function ◽  
Elliptic Problems ◽  
Critical Growth ◽  
Nodal Solutions ◽  
Existence And Multiplicity ◽  
Exponential Critical Growth ◽  
Image Position

AbstractIn this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problemswhere λ ∈ (0, ∞), f is a continuous function with exponential critical growth and V : ℝ2→ ℝ is a continuous function verifying some hypotheses.

scholarly journals Morse index of radial nodal solutions of Hénon type equations in dimension two

2017 ◽  
Vol 19 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Ederson Moreira dos Santos ◽  
Filomena Pacella
Keyword(s):  
Elliptic Equations ◽  
Morse Index ◽  
Radial Solution ◽  
Index Formula ◽  
Connected Components ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Semilinear Elliptic ◽  
Bounded Sets ◽  
Least Energy

We consider non-autonomous semilinear elliptic equations of the type [Formula: see text] where [Formula: see text] is either a ball or an annulus centered at the origin, [Formula: see text] and [Formula: see text] is [Formula: see text] on bounded sets of [Formula: see text]. We address the question of estimating the Morse index [Formula: see text] of a sign changing radial solution [Formula: see text]. We prove that [Formula: see text] for every [Formula: see text] and that [Formula: see text] if [Formula: see text] is even. If [Formula: see text] is superlinear the previous estimates become [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] denotes the number of nodal sets of [Formula: see text], i.e. of connected components of [Formula: see text]. Consequently, every least energy nodal solution [Formula: see text] is not radially symmetric and [Formula: see text] as [Formula: see text] along the sequence of even exponents [Formula: see text].

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 ◽  
Image Position

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.

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