[Paper Review] Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method

Claudianor O. Alves, Chao Ji|arXiv (Cornell University)|Jun 18, 2019

Nonlinear Partial Differential Equations25 references4 citations

TL;DR

This paper establishes the existence and concentration of positive solutions for a logarithmic Schrödinger equation with a potential $ V(x) $ satisfying a local condition using a penalization method. By adapting techniques from del Pino and Felmer, the authors prove that solutions concentrate at minimum points of $ V $ as $ \epsilon \to 0 $, overcoming the lack of compactness and non-smoothness of the associated energy functional.

ABSTRACT

In this article we are concerned with the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} ight. $$ where $ε>0, N \geq 1$ and $V:\mathbb{R}^{N} ightarrow \mathbb{R}$ is a continuous potential. Under a local assumption on the potential $V$, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.

Motivation & Objective

• To establish the existence of positive solutions for a logarithmic Schrödinger equation with a potential $ V(x) $ satisfying a local condition.
• To prove concentration of solutions at minimum points of $ V $ as the parameter $ \epsilon \to 0 $, extending results beyond global assumptions.
• To overcome the technical challenges arising from the non-smooth and ill-defined nature of the logarithmic energy functional in $ H^1(\mathbb{R}^N) $.

Proposed method

• Adapt the penalization method of del Pino and Felmer to a logarithmic Schrödinger equation with a nonlinearity $ u \log u^2 $.
• Use a change of variables to transform the original problem into a form amenable to variational analysis.
• Apply variational methods to the penalized problem, proving the existence of a critical point in $ H^1(\mathbb{R}^N) $.
• Establish uniform $ L^\infty $ bounds on the sequence of solutions using Moser iteration and covering arguments.
• Prove concentration by showing the maximum points of the rescaled solutions converge to a minimum point of $ V $.
• Use a subsequence argument and the continuity of $ V $ to conclude that the concentration points satisfy $ V(y_0) = V_0 $.

Experimental results

Research questions

• RQ1 Under what conditions on the potential $ V(x) $ can positive solutions exist for the logarithmic Schrödinger equation?
• RQ2 Can the penalization method be adapted to handle the logarithmic nonlinearity, which leads to a non-smooth and ill-defined energy functional?
• RQ3 Do solutions concentrate at local minimum points of $ V $ as $ \epsilon \to 0 $, even under a local assumption rather than a global one?
• RQ4 How can uniform $ L^\infty $ bounds be established for solutions of the penalized problem in the presence of logarithmic growth?
• RQ5 What is the asymptotic behavior of the maximum points of the solutions as $ \epsilon \to 0 $?

Key findings

• The authors prove the existence of a positive solution $ u_\epsilon $ to the penalized problem for all $ \epsilon > 0 $, under the local assumption (V2) and $ V_0 > -1 $.
• Solutions concentrate at a point $ y_0 \in \Lambda $ such that $ V(y_0) = V_0 $, the global infimum of $ V $, as $ \epsilon \to 0 $.
• The maximum points of the rescaled solutions $ v_\epsilon(x) = u_\epsilon(x/\epsilon) $ converge to a point where $ V(y_0) = V_0 $, confirming concentration at a minimum of $ V $.
• Uniform $ L^\infty $ bounds on the sequence $ u_\epsilon $ are established via Moser iteration and covering arguments, ensuring the solutions remain bounded.
• The proof relies on a subsequence argument where the maximum points of $ u_\epsilon $ lie within a fixed ball around $ y_n $, and $ \epsilon y_n \to y_0 $, with $ V(y_0) = V_0 $.
• The result extends previous existence and concentration results from global assumptions (e.g., Rabinowitz) to local assumptions, broadening the applicability of the penalization method to logarithmic Schrödinger equations.