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- Title
Mixed local and nonlocal supercritical Dirichlet problems.
- Authors
Amundsen, David; Moameni, Abbas; Temgoua, Remi Yvant
- Abstract
In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem$ \begin{equation} Lu = |u|^{p-2}u+\mu |u|^{q-2}u\quad\text{in}\; \; \Omega, \quad\quad u = 0\quad\text{in}\; \; \mathbb{R}^N\setminus\Omega, ~~~(1) \end{equation} $where $ L: = -\Delta +(-\Delta)^s $ for $ s\in(0, 1) $ and $ \Omega\subset\mathbb{R}^N $ is a bounded domain. Precisely, we show that problem (1) for $ 1<q<2<p $ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $ \mu $. Here $ u $ can be either a scalar function, or a vector valued function so that (1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system$ Lu = |v|^{p-2}v, \qquad Lv = |u|^{d-2}u+\mu |u|^{q-2}u $with the Dirichlet boundary condition on $ \Omega $ where $ 1<q<2<p, d $. Our method is variational, and in both problems the lack of compactness for the supercritical problem is recovered by working on a closed convex subset of an appropriate function space.
- Subjects
DIRICHLET problem; SYMMETRIC domains; VECTOR valued functions; FUNCTION spaces; VARIATIONAL principles; HAMILTONIAN systems
- Publication
Communications on Pure & Applied Analysis, 2023, Vol 22, Issue 10, p1
- ISSN
1534-0392
- Publication type
Article
- DOI
10.3934/cpaa.2023104