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I am not an expert in elliptic partial differential equations, but while studying the attractor structure of evolutionary PDEs, I frequently encounter problems related to elliptic equations. Specifically, I am interested in the number of non-trivial solutions for the semilinear elliptic equation defined on a smooth star-shaped bounded domain $\Omega\in\mathbb{R}^{3}$ with 0-Dirichlet boundary conditions. For instance, consider the equation

$−\Delta u=λu(u−1)(u+1)$,

where $\lambda$ is a parameter. I would like to understand under what conditions on $\lambda$ this equation has at most finite number of non-trivial solutions, and when it might have infinitely many. Most of the literature I have found uses variational methods, such as the mountain pass theorem, to show that there are at least a finite number of non-trivial solutions. However, I have not come across results providing upper bounds or exact counts for the number of solutions. Are there any articles that give estimates for the number of non-trivial solutions to elliptic equations, either upper or lower bounds? For general $\Delta u=f(u)$ with 0 Dirichlet boundary condition?

Additionally, what about the case of the biharmonic equation

$\Delta^{2}u=\lambda u-u^{p}$

with boundary conditions

$u=Δu=0$? Thanks in advance!

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  • $\begingroup$ This topic was fashionable in the 70s. An early paper (J. Leray ?) studied such an equation in the sphere. Depending upon the space dimension, there could exist infinitely many radial solutions. $\endgroup$
    – Denis Serre
    Commented Sep 16 at 11:43
  • $\begingroup$ Thank you Professor.@DenisSerre. There are many papers on this topic, but these are too advanced for me. But unfortunately, I'm not familiar with 70s' papers concerning this. I know that the situation is very complex and the complexity depends on the geometry of the domain(so I constrain it to star-shaped), the nonlinearity(so I consider the above special type concerning parameter $\lambda$) $\endgroup$
    – monotone operator
    Commented Sep 16 at 11:52

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See Theorem 1.5 in C. Brauner & B. Nicolaenko

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  • $\begingroup$ Thank you professor@DenisSerre! Except for this one, do you know other results related to this topic? Or suitable lecture notes? Thanks a lot! $\endgroup$
    – monotone operator
    Commented Sep 19 at 4:13

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