Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth

2022

Self Cite

In the present paper, we deal with the existence of nontrivial solutions and Nehari-type ground state solutions to the Kirchhoff type elliptic equations of the form:whereand f (x, s) has critical exponential growth on s which behaviors as e 𝛼 0 s 2 with 𝛼 0 > 0. Based on a deep analysis and some detailed estimate, we can determine a fine upper bound for the minimax level under weaker assumption on lim inf t→∞ t𝑓 (x,t) exp(𝛼 0 t N∕(N−1) ) . Our results generalize and improve the ones in Chen et al. (2021) (Lemma 3.1) and Figueiredo and Severo (2016) (Theorems 1.3 and 1.4) for N = 2 and Goyal et al. (2016) (Theorem 1.3) for N ≥ 2.

Section: Problem (11) Is Called Nonlocal Because Of the Presence Of T...mentioning

confidence: 99%

Section: Problem (11) Is Called Nonlocal Because Of the Presence Of T...mentioning

confidence: 99%

“…The above condition was introduced in Chen et al, 32 in which the existence of positive ground state solution for Equation () with

N = 2

is obtained under (F4 ′ ) and some other assumptions. (F4 ′ ) weakens the following assumption:

\lim_{t \to \infty} \frac{t f false(x, t false)}{e^{α_{0} t^{2}}} \geq κ_{0} > \frac{2}{α_{0} d^{2}} M ((), \frac{4 π}{α_{0}})

uniformly on x ∈ Ω, where d is radius of largest open ball contained in Ω; …”

Section: Introductionmentioning

confidence: 99%

({}, u_{n}) \subset W_{0}^{1, N} false(normalΩ false)

at the mountain pass level c ∗ > 0.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On critical N‐Kirchhoff type equations involving Trudinger–Moser nonlinearity

2022

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“…Remark Since

\lim_{t \to + \infty} \frac{t^{2} F_{i} (x, t)}{e^{α_{0} t^{2}}} = \lim_{t \to + \infty} \frac{2 F_{i} (x, t) + t f_{i} (x, t)}{2 α_{0} e^{α_{0} t^{2}}} .

Hence, (F5) can be deduced by the following hypothesis:

{lim inf}_{t \to + \infty} \frac{t f_{i} false(x, t false)}{e^{α_{0} t^{2}}} \geq β_{0} > \frac{1}{α_{0}}

uniformly on

x \in ℝ^{2}

. Clearly, (F5 ′ ′ ) is much weaker than (F5 ′ ) in do Ó and de Albuquerque 17 . In our arguments, we adopt the approach used in Chen, Tang and Wei 40 to estimate precisely the mountain pass level c ∗ ; see Lemma 3.1.…”

Section: Introductionmentioning

confidence: 99%

Ground state solutions for planar coupled system involving nonlinear Schrödinger equations with critical exponential growth

Wei

Lin

2021

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We consider the following two coupled nonlinear Schrödinger system: −normalΔu+u=f1false(x,ufalse)+λfalse(xfalse)v,x∈ℝ2,−normalΔv+v=f2false(x,vfalse)+λfalse(xfalse)u,x∈ℝ2, where the coupling parameter satisfies 0 < λ(x) ≤ λ0 < 1 and the reactions f1, f2 have critical exponential growth in the sense of Trudinger–Moser inequality. Using non‐Nehari manifold method together with the Lions's concentration compactness and the Trudinger‐Moser inequality, we show that the above system has a Nehari‐type ground state solution and a nontrivial solution. Our results improve and extend the previous results.

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

Wei

2022

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In this paper, we prove the following nonlocal Kirchhoff problem of the type {left leftarray−a+b∫ℝ2|∇u|2dxΔu+V(x)u=f(x,u),arrayx∈ℝ2;arrayu∈H1(ℝ2)array$$ \left\{\begin{array}{ll}-\left(a+b\underset{{\mathbb{R}}^2}{\int }{\left|\nabla u\right|}^2\mathrm{d}x\right)\Delta u+V(x)u=f\left(x,u\right),& x\in {\mathbb{R}}^2;\\ {}u\in {H}^1\left({\mathbb{R}}^2\right)& \end{array}\right. $$ has a Nehari‐type ground state solution when Vfalse(xfalse)$$ V(x) $$ and ffalse(x,ufalse)$$ f\left(x,u\right) $$ are periodic on x$$ x $$ and ffalse(x,ufalse)$$ f\left(x,u\right) $$ has critical exponential growth in the sense of Trudinger–Moser inequality on u$$ u $$. We develop some new approaches to estimate precisely the minimax level of the energy functional and to recover the compactness of Cerami sequences of the associated Euler–Lagrange functional. With this in hand, we can overcome difficulties arising from the appearance of the nonlocal term ∫ℝ2false|∇ufalse|2normaldx$$ {\int}_{{\mathbb{R}}^2}{\left|\nabla u\right|}^2\mathrm{d}x $$ and the nonlinearity which is of critical growth of Trudinger–Moser type in whole Euclidean space ℝ2$$ {\mathbb{R}}^2 $$.