Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries

Bartsch, Thomas; DʼAprile, Teresa; Pistoia, Angela

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 1027-1047

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem

$$-\Delta u={\left|u\right|}^{2^{⁎}-2-ϵ}u\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$

where Ω is a smooth bounded domain in ${ℝ}^N$, $N⩾3$, $2^{⁎}=\frac{2N}{N-2}$ and $ϵ>0$ is a small parameter. In particular we prove that if Ω is convex and satisfies a certain symmetry, then a nodal four-bubble solution exists with two positive and two negative bubbles.

Classification : 35B40, 35J20, 35J65
Keywords: Slightly subcritical problem, Sign-changing solutions, Finite-dimensional reduction, Max–min argument

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