Electron. J. Differential Equations, Vol. 2020 (2020), No. 118, pp. 1-25.

Polyharmonic systems involving critical nonlinearities with sign-changing weight functions

Anu Rani, Sarika Goyal

Abstract:
This article concerns the existence of multiple solutions of the polyharmonic system involving critical nonlinearities with sign-changing weight functions

where $(-\Delta)^m$ denotes the polyharmonic operators, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial \Omega$, $m\in \mathbb N$, $N\geq {2m+1}$, $1<r<2$ and $\beta>1$, $\gamma>1$ satisfying $2<\beta+\gamma\leq 2_{m}^{*}$ with $2_{m}^{*}=\frac{2N}{N-2m}$ as a critical Sobolev exponent and $\lambda$, $\mu>0$. The functions f, g and $h:\overline{\Omega}\to \mathbb R$ are sign-changing weight functions satisfying f, $g\in L^{\alpha}(\Omega)$ and $h\in L^{\infty}(\Omega)$ respectively. Using the variational methods and Nehari manifold, we prove that the system admits at least two nontrivial solutions with respect to parameter $(\lambda, \mu)\in \mathbb R^2_{+} \setminus \{(0, 0)\}$.

Submitted August 31, 2020. Published December 10, 2020.
Math Subject Classifications: 35A15, 35B33, 35J91.
Key Words: Polyharmonic operator system; sign-changing weight functions; critical exponent; Nehari manifold; concave-convex nonlinearities.
DOI: 10.58997/ejde.2020.119

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  Anu Rani
Department of Mathematics
Bennett University
Greater Noida, Uttar Pradesh, India
email: ar4091@bennett.edu.in
Sarika Goyal
Department of Mathematics
Bennett University
Greater Noida, Uttar Pradesh, India
email: sarika1.iitd@gmail.com

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