[Paper Review] Infinitely many positive solutions for nonlinear equations with non-symmetric potential

Weiwei Ao, Juncheng Wei|arXiv (Cornell University)|Oct 31, 2012

Advanced Mathematical Physics Problems28 references4 citations

TL;DR

This paper establishes the existence of infinitely many positive solutions for a nonlinear Schrödinger equation with a non-symmetric potential using a localized energy method and Liapunov-Schmidt reduction. By generalizing Cerami-Passaseo-Solimini's results to broader nonlinearities and providing a new variational proof, it proves that for sufficiently small $δ > 0$, the equation admits infinitely many bound states under mild decay and nondegeneracy conditions on the potential and nonlinearity.

ABSTRACT

We consider the following nonlinear Schrodinger equation [{l} Δu-(1+δV)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying some nondegeneracy condition. Using localized energy method, we prove that there exists some $δ_0$ such that for $0

Motivation & Objective

• To generalize the existence of infinitely many positive solutions for nonlinear Schrödinger equations beyond periodic or symmetric potentials.
• To extend the results of Cerami-Passaseo-Solimini [11] to a broader class of superlinear nonlinearities satisfying nondegeneracy conditions.
• To provide a new variational proof using localized energy methods and Liapunov-Schmidt reduction, avoiding reliance on concentration-compactness or periodicity.
• To establish the existence of infinitely many bound states for elliptic systems through the same framework.

Proposed method

• Employing a localized energy method to construct solutions with multiple peaks in the non-symmetric potential regime.
• Applying Liapunov-Schmidt reduction to decouple the problem into a finite-dimensional variational problem and a remainder equation.
• Using a $C^{1+σ}$ nonlinearity $f(u)$ that vanishes for $u \leq 0$ and satisfies a nondegeneracy condition on the ground state solution of the limiting problem.
• Defining a configuration set $\Lambda_k$ of $k$-point configurations with separation $\rho > \rho_0$ to control interactions between peaks.
• Constructing a formal solution as a sum of translated ground states $U_{Q_i}, V_{Q_i}$, then correcting with a remainder term via the implicit function theorem.
• Proving that the energy functional attains a minimum over $\Lambda_k$, and showing that the maximum of the energy functional occurs in the interior of the configuration space, excluding boundary maximization.

Experimental results

Research questions

• RQ1 Can the existence of infinitely many positive solutions be established for nonlinear Schrödinger equations with non-symmetric potentials without assuming periodicity or small parameters?
• RQ2 Does the localized energy method combined with Liapunov-Schmidt reduction yield a new proof for the multiplicity result of Cerami-Passaseo-Solimini [11] under more general nonlinearities?
• RQ3 What conditions on the nonlinearity $f(u)$ and potential $V(x)$ ensure the existence of infinitely many bound states in the non-symmetric case?
• RQ4 How can the interaction between multiple peaks in the solution be controlled to ensure the existence of distinct solutions for arbitrarily large $k$?

Key findings

• For any $k \in \mathbb{N}$, there exists $\delta_0 > 0$ such that for all $0 < \delta < \delta_0$, the equation has at least $k$ distinct positive solutions.
• The energy level $\mathcal{C}_{k+1}$ of the $(k+1)$-peak solution satisfies $\mathcal{C}_{k+1} \geq \mathcal{C}_k + I(U,V)$, where $I(U,V)$ is the energy of a single peak solution.
• The maximum of the energy functional over the configuration space $\Lambda_k$ is attained in the interior $\Lambda_k^\circ$, ruling out boundary concentration.
• The solution structure is stable under perturbation: the correction terms $\phi_{\mathbf{Q}}$, $\psi_{\mathbf{Q}}$ exist as $C^1$ maps in the configuration parameters $\mathbf{Q}$.
• The method applies to elliptic systems, establishing the existence of infinitely many positive bound states for such systems.
• The nondegeneracy of the ground state solution $w$ of the limiting problem $\Delta w - w + f(w) = 0$ is essential for the implicit function theorem application.