New solutions for Trudinger–Moser critical equations in R2

Abstract

Let Ω be a bounded, smooth domain in R 2 . We consider critical points of the Trudinger-Moser type functional J λ (u) = 1 2 Ω |∇u| 2 − λ 2 Ω e u 2 in H 1 0 (Ω), namely solutions of the boundary value problem u + λue u 2 = 0 with homogeneous Dirichlet boundary conditions, where λ > 0 is a small parameter. Given k 1 we find conditions under which there exists a solution u λ which blows up at exactly k points in Ω as λ → 0 and J λ (u λ ) → 2kπ. We find that at least one such solution always exists if k = 2 and Ω is not simply connected. If Ω has d 1 holes, in addition d + 1 bubbling solutions with k = 1 exist. These results are existence counterparts of one by Druet in [O. Druet, Multibump analysis in dimension 2: Quantification of blow-up levels, Duke Math. J. 132 (2) (2006) 217-269] which classifies asymptotic bounded energy levels of blow-up solutions for a class of nonlinearities of critical exponential growth, including this one as a prototype case.

Bubbling nodal solutions for a large perturbation of the Moser–Trudinger equation on planar domains

“…If one drops the radial requirement, Adimurthi and Yadava in [8] proved the existence of infinitely many sign-changing solutions in a ball whatever λ > 0 is. We point out that, in the case a = 0, the approach of Del Pino, Musso and Ruf [14] allows to find sign-changing solutions which blow-up positively and negatively at least at two different points in any domain Ω as λ → 0 (even if this is not explicitly said in their work).…”

“…a < 0, then there are no positive solutions to problem (1) when the domain Ω is a ball provided λ is small. The case a = 0 in a general domain Ω has been studied by Del Pino, Musso and Ruf [14] using a perturbative approach. Indeed they find multiplicity of positive solutions which blow-up in one or more points of Ω (depending on the geometry) as λ → 0.…”

Multiple blow-up solutions for an anisotropic Emden Fowler equation in ${\boldsymbol{\mathbb{R}}}^{2}$

“…In (1.8) o(1) → 0 as λ → 0 uniformly on compact sets of \ {ξ 1 , ξ 2 }. This has been treated in [5]. We refer the reader to [5] for further details.…”

Construction solutions for Neumann problem with Hénon term in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula>

“…If α = 0, the asymptotic behavior of blowing-up family of solutions for the problem (7) when p = 1 can be referred to the papers [1,3,16,17,18], and the existence of blow up solutions has been considered in [1,6,15] for p = 1, [11] for 0 < p < 2, and [7] for p = 2. Moreover, if α = 0, the blowing up analysis of solutions to (7) when p = 1 and λ → 0 is given by Bartolucci and Tarantello in [2]. Esposito [13] proved the existence of blow-up solutions, under a certain assumption of non degeneracy, in the spirit of [1].…”

“…The proof of Theorem 1.1 relies on a Lyapunov-schmidt reduction. This method has been used in many other papers, see for instance [6], [7], [9], [10], [11], [23] and references therein.…”