Sign-changing tower of bubbles to an elliptic subcritical equation

Dammak, Yessine; Ghoudi, Rabeh

doi:10.1142/s0219199718500529

Cited by 10 publications

(3 citation statements)

References 18 publications

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“…, xn axes, problem (1.9) has a sign-changing solution with the shape of a tower of bubbles with alternate signs, centered at the center of symmetry of the domain. Finally, we mention the work in [8], where the authors studied the existence of bubble towers changing sign solutions for the counterpart of (1.9) when the Laplacian operator is replaced by the biharmonic one.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem

Fourti,

Ghoudi

2024

Nonlinear Differ. Equ. Appl.

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In this paper, we deal with the boundary value problem −∆u = |u| 4/(n−2) u/[ln(e + |u|)] ε in a bounded smooth domain Ω in R n , n ≥ 3 with homogenous Dirichlet boundary condition. Here ε > 0. Clapp et al. in Journal of Diff. Eq. (Vol 275) built a family of solution blowing up if n ≥ 4 and ε small enough.They conjectured in their paper the existence of sign changing solutions which blow up and blow down at the same point. Here we give a confirmative answer by proving that our slightly subcritical problem has a solution with the shape of sign changing bubbles concentrating on a stable critical point of the Robin function for ε sufficiently small.

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Section: Introduction and Resultsmentioning

confidence: 99%

Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem

Fourti,

Ghoudi

2024

Nonlinear Differ. Equ. Appl.

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“…When the Laplacian operator in (1.3) is replaced by the biharmonic one, the existence of multispike and bubble towers changing sign solutions for the counterpart of (1.3) have been studied in [6] and [8] respectively. Going back to (Pε), the existence of multispike solutions has not been studied yet and the main purpose of this paper is to focus on this issue.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Multispike Solutions for a slightly subcritical elliptic problem with non-power nonlinearity

Ayed¹,

Fourti²,

Ghoudi³

2022

Preprint

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In this paper, we are concerned with the following elliptic equation −∆u = |u| 4/(n−2) u/[ln(e + |u|)] ε in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded open domain in R n , n ≥ 3 and ε > 0. Clapp et al. in Journal of Diff. Eq. (Vol 275) proved that there exists a single-peak positive solution for small ε if n ≥ 4.Here we construct positive as well as changing sign solutions concentrated at several points at the same time.

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“…This program has been adjusted to various situations (see for example [22]) and other equations (see for example [11] (for K = 1) and [14] (for ∆ 2 instead of ∆)).…”

Section: Introductionmentioning

confidence: 99%

Blowing-up solutions for a supercritical elliptic equation

Dammak¹

2022

CPAA

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<p style='text-indent:20px;'>This paper concerns the existence of solutions of the following supercritical PDE: <inline-formula><tex-math id="M1">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M2">\begin{document}$ -\Delta u = K|u|^{\frac{4}{n-2}+\varepsilon}u\; \mbox{ in }\Omega, \; u = 0 \mbox{ on }\partial\Omega, $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M7">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> positive function and <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is a small positive real. Our method is inspired from the work of Bahri-Li-Rey. It consists to reduce the existence of a critical point to a finite dimensional system. Using a fixed-point theorem, we are able to construct positive solutions of <inline-formula><tex-math id="M9">\begin{document}$ (P_\varepsilon) $\end{document}</tex-math></inline-formula> having the form of two bubbles with non comparable speeds and which have only one blow-up point in <inline-formula><tex-math id="M10">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. That means that this blow-up point is non simple. This represents a new phenomenon compared with the subcritical case.</p>

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Sign-changing tower of bubbles to an elliptic subcritical equation

Cited by 10 publications

References 18 publications

Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem

Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem

Multispike Solutions for a slightly subcritical elliptic problem with non-power nonlinearity

Blowing-up solutions for a supercritical elliptic equation

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