[Paper Review] A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities

Mousomi Bhakta|arXiv (Cornell University)|May 1, 2014

Nonlinear Partial Differential Equations14 references3 citations

TL;DR

This paper investigates the existence and non-existence of nontrivial weak solutions for a fourth-order elliptic equation with biharmonic operator and multiple critical nonlinearities involving singular potentials. Using Pohozaev-type identities, it proves non-existence for $1 < q < 2^{**}$, and for $q = 2^{**}$ and $-(N-2)^2 \leq \mu < \mu_1$, it establishes existence via the Mountain-Pass theorem by balancing competing critical nonlinearities through careful analysis of Palais-Smale sequences and conformal invariance under dilation.

ABSTRACT

We study the existence and non-existence of nontrivial weak solution of $$ {Δ^2u-μ\frac{u}{|x|^{4}} = \frac{|u|^{q_β-2}u}{|x|^β}+|u|^{q-2}u\quad extrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0

Motivation & Objective

• To analyze the existence and non-existence of nontrivial weak solutions for a semilinear biharmonic equation with multiple critical nonlinearities in $\mathbb{R}^N$, $N \geq 5$.
• To address the challenge posed by competing critical nonlinearities arising from $|u|^{q_\beta-2}u/|x|^\beta$ and $|u|^{2^{**}-2}u$ terms when $q = 2^{**}$.
• To overcome the lack of compactness due to conformal invariance under the dilation $u(x) \mapsto t^{(N-4)/2}u(tx)$ by constructing a Palais-Smale sequence at a suitable energy level.
• To extend variational methods to fourth-order equations with singular potentials, particularly when $\mu < \mu_1$ and $\mu \in [-(N-2)^2, 0)$.

Proposed method

• Employing a Pohozaev-type identity to prove non-existence of solutions when $1 < q < 2^{**}$.
• Using the Mountain-Pass theorem of Ambrosetti and Rabinowitz to establish existence of a nontrivial weak solution when $q = 2^{**}$ and $\mu \in [-(N-2)^2, \mu_1)$.
• Defining the energy functional $I(u) = \frac{1}{2}\|u\|^2 - \frac{1}{q_\beta}\int \frac{|u|^{q_\beta}}{|x|^\beta}dx - \frac{1}{2^{**}}\int |u|^{2^{**}}dx$ on $D^{2,2}(\mathbb{R}^N)$, which is $C^1$ due to Rellich and Sobolev inequalities.
• Constructing a Palais-Smale sequence at a critical level that avoids dominance of one nonlinearity over the other by balancing the energy contributions via dilation invariance.
• Using radial symmetry and the radial extremal constants $S_{\mu,0}^{rad}$ and $S_{\mu,\beta}^{rad}$ to recover existence in the case $\mu < 0$, particularly for $\mu \in [-(N-2)^2, 0)$.
• Applying Schauder estimates and regularity theory to show that the weak limit $v_0$ of a weakly convergent subsequence is a classical $C^4$ solution away from the origin.

Experimental results

Research questions

• RQ1 Under what conditions does the biharmonic equation $\Delta^2 u - \mu \frac{u}{|x|^4} = \frac{|u|^{q_\beta-2}u}{|x|^\beta} + |u|^{q-2}u$ in $\mathbb{R}^N$ admit nontrivial weak solutions?
• RQ2 Why does the standard Mountain-Pass theorem fail to produce a solution when $q = 2^{**}$, and how can this failure be overcome?
• RQ3 What role does the competition between the two critical nonlinearities play in the existence or non-existence of solutions?
• RQ4 How does the sign and magnitude of $\mu$ affect the existence of solutions, particularly when $\mu < 0$?
• RQ5 Can the conformal invariance under the dilation $u(x) \mapsto t^{(N-4)/2}u(tx)$ be exploited to construct a solution when both nonlinearities are critical?

Key findings

• Non-existence of nontrivial weak solutions holds for $1 < q < 2^{**}$, proven via a Pohozaev-type identity.
• For $q = 2^{**}$ and $\mu \in [-(N-2)^2, \mu_1)$, a nontrivial weak solution exists, established using the Mountain-Pass theorem.
• The critical energy level is chosen to balance the contributions of the two critical nonlinearities, preventing one from dominating the other.
• The Palais-Smale sequence converges weakly to a nontrivial limit $v_0$, which is shown to be a classical solution in $\mathbb{R}^N \setminus \{0\}$ via regularity theory.
• For $\mu < 0$, the existence result is extended by using radial extremals and the radial constants $S_{\mu,0}^{rad}$ and $S_{\mu,\beta}^{rad}$, which are always achieved.
• The solution $v_0$ is $C^4$-smooth away from the origin, and the nonlinearity's superlinear growth ensures local Lipschitz regularity of $\Delta^2 v_0$.