[Paper Review] Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential

Claudianor O. Alves, Chao Ji|arXiv (Cornell University)|Apr 22, 2019

Nonlinear Partial Differential Equations17 references4 citations

TL;DR

This paper establishes the existence of a positive solution for a logarithmic Schrödinger equation with a saddle-like potential using variational methods. By applying Szulkin's approach to a transformed problem, the authors prove the existence of a critical point corresponding to a positive solution under geometric and boundedness conditions on the potential, extending prior results to the logarithmic nonlinearity case with $ c_0 > -1 $.

ABSTRACT

In this article we use the variational method developed by Szulkin \cite{szulkin} to prove the existence of a positive solution for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} ight. $$ where $ε>0, N \geq 1$ and $V$ is a saddle-like potential.

Motivation & Objective

• Address the existence of positive solutions for the logarithmic Schrödinger equation with a saddle-like potential, a case not fully covered in prior variational treatments.
• Overcome the challenge of the logarithmic nonlinearity making the energy functional ill-defined in $ H^1(\mathbb{R}^N) $ by employing a refined variational framework.
• Extend previous existence results for semilinear elliptic equations with power-type nonlinearities to the logarithmic case, particularly under saddle-like potential geometry.
• Establish the existence of a positive solution when the potential satisfies geometric conditions (V1)-(V3), including $ c_0 > -1 $, which generalizes earlier assumptions.
• Use a change of variables to transform the original problem into a form amenable to minimax methods, enabling the application of Szulkin's variational technique.

Proposed method

• Transform the original equation $ -\epsilon^2\Delta u + V(x)u = u\log u^2 $ into $ -\Delta u + V(\epsilon x)u = u\log u^2 $ via a change of variables, simplifying the analysis.
• Apply Szulkin's variational method to the transformed problem, which is suitable for functionals that are the sum of a convex and a $ C^1 $-functional.
• Define the energy functional $ J_{\epsilon}(u) = \frac{1}{2}\int_{\mathbb{R}^N} \left( |\nabla u|^2 + (V(\epsilon x) + 1)|u|^2 \right) dx - \frac{1}{2}\int_{\mathbb{R}^N} u^2 \log u^2 \, dx $, which is well-defined on $ H^1(\mathbb{R}^N) $ under the given conditions.
• Construct a minimax class $ \Gamma $ of paths in the Sobolev space $ H^1(\mathbb{R}^N) $, using a deformation retract $ \Phi $ from a bounded set $ Q $ to a subset $ K $, to capture critical points.
• Prove that the minimax value $ C_\epsilon = \inf_{h \in \Gamma} \sup_{x \in Q} J(h(x)) $ is a critical level of $ J $ by verifying the Palais-Smale condition and using homotopy invariance of the topological degree.
• Establish that the critical point at level $ C_\epsilon \in (m(c_0) + \sigma/2, 2m(c_0) - \sigma) $ is positive via the maximum principle, since the nonlinearity $ f(t) = t \log t^2 $ is odd and the solution does not change sign.

Experimental results

Research questions

• RQ1 Does a positive solution exist for the logarithmic Schrödinger equation with a saddle-like potential when the potential infimum $ c_0 $ satisfies $ c_0 > -1 $?
• RQ2 Can Szulkin's variational method be adapted to handle the logarithmic nonlinearity, which causes the energy functional to be formally ill-defined in $ H^1(\mathbb{R}^N) $?
• RQ3 Under what geometric and boundedness conditions on the potential $ V $ does the minimax method yield a critical point corresponding to a positive solution?
• RQ4 Is the critical level $ C_\epsilon $ strictly between $ m(c_0) + \sigma/2 $ and $ 2m(c_0) - \sigma $, ensuring the existence of a nontrivial solution?
• RQ5 Does the solution obtained via the minimax method remain positive, and can this be deduced from the structure of the nonlinearity and the maximum principle?

Key findings

• The paper proves the existence of a positive solution for the logarithmic Schrödinger equation with a saddle-like potential under conditions (V1)-(V3), even when $ c_0 > -1 $, extending prior results.
• The minimax value $ C_\epsilon $ lies in the interval $ (m(c_0) + \sigma/2, 2m(c_0) - \sigma) $, which ensures the Palais-Smale condition holds, allowing the application of critical point theory.
• By contradiction, the authors show that the minimax level $ C_\epsilon $ is a critical level of the energy functional $ J $, confirming the existence of a critical point.
• The critical point at level $ C_\epsilon $ is nonnegative and, by the maximum principle, strictly positive, as the nonlinearity $ f(t) = t \log t^2 $ is odd and the solution does not change sign.
• Using a deformation argument and homotopy invariance of the topological degree, the authors establish that the minimax value $ C_\epsilon $ is indeed a critical level, completing the existence proof.