Speaker
Description
*Abstract
In this paper, we study a class of fourth-order nonlinear elliptic equations on compact Riemannian manifolds. Using variational methods and critical point theory, we establish the existence of at least three distinct solutions in the constant coefficient case.
Keywords
Elliptic equations, Critical Sobolev exponent , Compact Rienmannin manifolds, Plais-Smale condition, Nehari manifold.
Introduction*
Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with $n \geq 5$, where $g$ denotes the Riemannian metric and $\mathrm{Scal}_g$ represents its scalar curvature. We consider the Sobolev space $H^{2}_{2}(M)$, defined as the completion of $C^{\infty}(M)$ with respect to the norm
$$
\|u\|_{H^{2}_{2}(M)} = \|\nabla^{2}_{g} u\|_{2} + \|\nabla_{g} u\|_{2} + \|u\|_{2}.
$$
Here, $\Delta_g = \mathrm{div}_g(\nabla_g)$ denotes the Laplace--Beltrami operator associated with the metric $g$.
In this paper, we study the existence and multiplicity of solutions for the following class of fourth-order non-linear elliptic equations:
\begin{equation}
\Delta_{g}^{2} u + \mathrm{div}_{g}(a(x)\nabla_{g}u) + b(x)u = f(x)|u|^{N-2}u + \lambda |u|^{q-2}u,
\end{equation}
where $a, b, f \in C^{\infty}(M)$ with $f > 0$, $N = \frac{2n}{n-4}$ is the critical Sobolev exponent, $1 < q < 2$, and $\lambda > 0$ is a real parameter.
*Mais results*
The main results of this paper are stated as follows.
**Theorem 01**
Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 6$, and let $f \in C^{\infty}(M)$ be a positive smooth function. Assume that the operator
$$
P(u) = \Delta_{g}^{2} u + \mathrm{div}_{g}(a(x)\nabla_{g}u) + b(x)u
$$
is coercive. Let $x_0 \in M$ be a point where $f$ achieves its maximum.
If $n > 6$ and the following inequality holds at $x_0$:
$$
\frac{n(n^2+4n-20)}{2(n-6)(n^2-4)}\mathrm{Scal}_{g}(x_0) + \frac{n(n-1)}{(n-6)(n^2-4)}a(x_0) - \frac{n}{8(n-2)}\frac{\Delta_g f(x_0)}{f(x_0)} > 0,
$$
then there exists $\Lambda > 0$ such that for any $\lambda \in (0,\Lambda)$, the problem has a non-trivial solution $u \in C^{4,\theta}(M)$ with $\theta \in (0,1)$.