[Paper Review] Some continuation properties via minimax arguments

Louis Jeanjean|arXiv (Cornell University)|Apr 5, 2011

Nonlinear Partial Differential Equations13 references20 citations

TL;DR

This paper establishes continuity of solutions to nonlinear elliptic equations via minimax methods, proving that under suitable conditions, the solution branch $ u_\lambda $ depends continuously on a parameter $ \lambda $. It applies abstract minimax arguments to show global continuation of solutions and proves existence of positive solutions for nonhomogeneous problems even when the forcing term $ f $ changes sign, provided its $ L^q $-norm is sufficiently small.

ABSTRACT

This note is devotes to some remarks regarding the use of variational methods, of minimax type, to establish continuity type results

Motivation & Objective

• To establish continuity of solution branches for parameter-dependent elliptic equations using variational minimax methods.
• To prove existence of positive solutions for nonhomogeneous elliptic problems even when the nonlinearity $ f $ changes sign.
• To extend results on global continuation of solutions beyond the standard assumption of nonnegative $ f $.
• To provide abstract conditions under which minimax sequences converge to critical points of a limiting functional.
• To demonstrate that small $ L^q $-norm of $ f $ ensures existence of positive solutions without requiring $ f \geq 0 $.

Proposed method

• Formulates a one-parameter family of $ C^1 $-functionals $ I_\lambda(u) = A(u) - \lambda B(u) $ on a reflexive Banach space $ X $, with $ \lambda \to 1 $ as the parameter of interest.
• Imposes four abstract assumptions: boundedness of $ B $ and its derivative, existence of critical points $ u_\lambda $ with uniformly bounded energy levels $ c_\lambda $, boundedness of $ u_\lambda $ as $ \lambda \to 1 $, and Palais-Smale condition for the limiting functional $ I_1 $.
• Applies the mountain pass theorem to the limiting functional $ I_1 $ to ensure existence of a critical point at level $ c \in S $, where $ S $ is a bounded interval.
• Uses convergence of $ u_\lambda $ as $ \lambda \to 1 $ to show that the solution branch is continuous, under uniqueness of the limiting critical point.
• Applies the abstract framework to two classes of PDEs: autonomous equations on $ \mathbb{R}^N $ and nonhomogeneous problems on bounded domains.
• Employs Sobolev embeddings, uniform bounds, and elliptic regularity to verify the abstract assumptions in concrete settings.

Experimental results

Research questions

• RQ1 Under what conditions does the solution $ u_\lambda $ of a parameter-dependent elliptic equation depend continuously on $ \lambda $?
• RQ2 Can positive solutions be guaranteed for nonhomogeneous elliptic equations when the forcing term $ f $ is not nonnegative but has small $ L^q $-norm?
• RQ3 Does the abstract minimax framework ensure convergence of solution branches even when the parameter $ \lambda $ approaches a critical value?
• RQ4 Can the mountain pass characterization of the least energy level be used to prove global continuation of solutions?
• RQ5 Is it possible to obtain positive solutions for equations with sign-changing nonlinearities via variational methods without requiring $ f \geq 0 $?

Key findings

• The solution map $ \lambda \mapsto u_\lambda $ is continuous on $ \mathbb{R}^N $ for the autonomous equation $ -\Delta u + \lambda u = g(u) $, provided $ g $ satisfies superlinear and subcritical growth conditions and uniqueness of positive solutions holds for each $ \lambda $.
• For the nonhomogeneous problem $ -\Delta u = |u|^{p-1}u + f(x) $ on a bounded domain, there exists $ \alpha > 0 $ such that if $ \|f\|_q \leq \alpha $, a positive solution exists even when $ f $ changes sign.
• The abstract minimax framework ensures that any sequence $ u_{\lambda_n} $ with $ \lambda_n \to 1 $ converges (up to subsequence) to a critical point of the limiting functional $ I_1 $, provided the Palais-Smale condition holds.
• The existence of a positive solution for small $ \|f\|_q $ is established without assuming $ f \geq 0 $, relying on convergence of Palais-Smale sequences and the maximum principle in the limit.
• The proof relies on showing that a Palais-Smale sequence for $ I_f $ is bounded and converges strongly in $ H^1_0(\Omega) $, leading to a nontrivial nonnegative solution in the limit as $ f \to 0 $, which is then shown to be positive via the strong maximum principle.
• The method allows for the existence of a second solution as a local minimum, but this solution tends to zero as $ f \to 0 $, so it does not contribute to positivity.