A new Trudinger–Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space $$ {\mathbb {R}}^2 $$

Aouaoui, Sami

doi:10.1007/s00013-019-01386-7

Cited by 6 publications

(3 citation statements)

References 20 publications

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“…Clearly, when σ = 0, we obtain an extension of Theorem 1.5 to the whole space R N . This result with those established in [9,10,11] are the first attempts to extend the inequalities established by M. Calanchi and B. Ruf in Theorem 1.3, Theorem 1.4, Theorem 1.5 and Theorem 1.6 to the whole space R N . For the value α α N,β + σ N = 1, we could not prove or disprove that the supremum in (11) is finite.…”

supporting

confidence: 71%

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

Aouaoui¹,

Jlel²

2022

DCDS

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<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>

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supporting

confidence: 71%

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

Aouaoui¹,

Jlel²

2022

DCDS

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“…By using truncation and comparison techniques, two positive solutions are obtained. In [3], by obtaining a Trudinger-Moser type inequality, the author obtains a nonnegative solution of the problem…”

mentioning

confidence: 99%

A log–exp elliptic equation in the plane

Figueiredo¹,

Montenegro²,

Stapenhorst³

2022

DCDS

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<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ u = 0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^{2} $\end{document}</tex-math></inline-formula>. We replace the singular function <inline-formula><tex-math id="M7">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> by a function <inline-formula><tex-math id="M8">\begin{document}$ g_\epsilon(u) $\end{document}</tex-math></inline-formula> which pointwisely converges to -<inline-formula><tex-math id="M9">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. When the parameter <inline-formula><tex-math id="M11">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is small enough, the corresponding energy functional to the perturbed equation <inline-formula><tex-math id="M12">\begin{document}$ -\Delta u + g_\epsilon(u) = \lambda f(u) $\end{document}</tex-math></inline-formula> has a critical point <inline-formula><tex-math id="M13">\begin{document}$ u_\epsilon $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M14">\begin{document}$ H_0^1(\Omega) $\end{document}</tex-math></inline-formula>, which converges to a nontrivial nonnegative solution of the original problem as <inline-formula><tex-math id="M15">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>

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“…The variational method was used in such papers as well as in [1–3, 10, 41, 45]. Consult also [5, 25] where the Trudinger–Moser inequality should be built ad hoc for the problem. Nonlinearities with exponential growth are also important in conformal metrics [12], Gaussian curvature [13], and Trudinger–Moser inequality on manifolds with boundary [31].…”

Section: Introductionmentioning

confidence: 99%

A singular Liouville equation on planar domains

Figueiredo

Montenegro²,

Stapenhorst³

2023

Mathematische Nachrichten

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We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, −Δ𝑢 = (log 𝑢 + 𝑓(𝑢))𝜒 {𝑢>0} in Ω ⊂ ℝ 2 with Dirichlet boundary condition. The function 𝑓 has exponential growth, which can be subcritical or critical with respect to the Trudinger-Moser inequality. We study the energy functional 𝐼 𝜖 corresponding to the perturbed equation −Δ𝑢 + 𝑔 𝜖 (𝑢) = 𝑓(𝑢), where 𝑔 𝜖 is well defined at 0 and approximates − log 𝑢. We show that 𝐼 𝜖 has a critical point 𝑢 𝜖 in 𝐻 1 0 (Ω), which converges to a legitimate nontrivial nonnegative solution of the original problem as 𝜖 → 0. We also investigate the problem with 𝑓(𝑢) replaced by 𝜆𝑓(𝑢), when the parameter 𝜆 > 0 is sufficiently large.

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A new Trudinger–Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space $$ {\mathbb {R}}^2 $$

Cited by 6 publications

References 20 publications

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

A log–exp elliptic equation in the plane

A singular Liouville equation on planar domains

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