Department of Mathematics and Computer Science
Physical Science and Earth Science
University of Messina
Viale F. Stagno d'Alcontres 31, Italy
email: ganello@unime.it
Giovanni Anello
Abstract:
We study the nonlocal critical Kirchhoff problem
$$\displaylines{
-\Big(a+b\int_\Omega |\nabla u|^2dx\Big)\Delta u
=|u|^{2^*-2}u +\lambda f(x,u), \quad \text{in } \Omega,\cr
u=0, \quad \text{on } \partial\Omega,
}$$
where \(\Omega\) is a bounded smooth domain in \(R^N\), \(N>4\), \(a,b>0\),
\(\lambda\in R\), \(2^*:=\frac{2N}{N-2}\) is the critical exponent for the
Sobolev embedding, and \(f:\Omega\times R\to R\) is a
Caratheodory function with subcritical growth. We establish the
existence of global minimizers for the energy functional associated to
this problem. In particular, we improve a recent result proved by Faraci
and Silva [3] under more strict conditions on the nonlinearity f
and under additional conditions on a and b.
Submitted January 27, 2025. Published May 6, 2025.
Math Subject Classifications: 35J20, 35J25.
Key Words: Nonlocal problem; Kirchhoff equation; weak solution; critical growth;
approximation; variational methods.
DOI: 10.58997/ejde.2025.46
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Giovanni Anello Department of Mathematics and Computer Science Physical Science and Earth Science University of Messina Viale F. Stagno d'Alcontres 31, Italy email: ganello@unime.it |
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