Electron. J. Differential Equations, Vol. 2018 (2018), No. 99, pp. 1-9.

Existence of solutions to biharmonic equations with sign-changing coefficients

Somayeh Saiedinezhad

Abstract:
In this article, we study the existence of solutions for the semi-linear elliptic equation
$$ \Delta^2 u-a(x)\Delta u=b(x)| u|^{p-2}u $$
with Navier boundary condition $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain with smooth boundary and $2<p<2^*$. We consider two different assumptions on the potentials $a$ and $b$, including the case of sign-changing weights. The approach is based on the Nehari manifold with variational arguments about the corresponding fibering map, which ensures the multiple results.

Submitted July 17, 2017. Published Aapril 28, 2018.
Math Subject Classifications: 35A01, 35J35, 35D30, 35J91.
Key Words: Bi-Laplacian operator; weak solution; Nehari manifold; fibering map.

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Somayeh Saiedinezhad
School of Mathematics
Iran University of Science and Technology
Narmak, Tehran, Iran
email: ssaiedinezhad@iust.ac.ir

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