Universidade de Brasília
Departamento de Matemática
70910-900 Brasília-DF
email mfurtado@unb.br
Universidade de Brasília
Departamento de Matemática
70910-900 Brasília-DF, Brazil
email: guimaraesmelothiago@gmail.com
Marcelo F. Furtado, Thiago G. Melo
Abstract:
We consider the Henon-type equation
$$
\Delta^2 u = [W(z)]^\ell f(u) \text{ in } B, \quad
u= \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial B,
$$
where \(B\) is the unit ball in
\(\mathbb{R}^{N} = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}\),
the weight function \(W(z)\) behaves like \(|x||y|\) for \(x\in \mathbb{R}^{N_1}\),
\(y \in \mathbb{R}^{N_2}\), and the nonlinearity \(f\) is allowed to exhibit
supercritical growth.We establish a new radial-type lemma adapted to the weight
\(|x|^{(N_1 - 2)/2}\,|y|^{(N_2 - 2)/2}\), which yields a weighted Sobolev embeddings
for our functional framework into \(L^p\) spaces, with exponents \(p > 1\),
possibly within the supercritical range. Finally, we prove the existence of a weak
solution to the problem.
Submitted July 23, 2025. Published December 28, 2025.
Math Subject Classifications: 35J30, 35J35.
Key Words: Henon equation; variational methods; elliptic equations; supercritical problems; biharmonic operator.
DOI: 10.58997/ejde.2025.120
Show me the PDF file (403 KB), TEX file for this article.
|
Marcelo F. Furtado Universidade de Brasília Departamento de Matemática 70910-900 Brasília-DF email mfurtado@unb.br |
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|
Thiago G. Melo Universidade de Brasília Departamento de Matemática 70910-900 Brasília-DF, Brazil email: guimaraesmelothiago@gmail.com |
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