Existence and concentration of positive solutions for a Schrödinger logarithmic equation
The paper is concerned with the Schrödinger logarithmic equation \[ -\varepsilon ^2 \Delta u +V(x)u= u \log u^2 \quad \text{in }\mathbb{R}^N, \quad u\in H^1(\mathbb{R}^N),\tag{1} \] where \(\varepsilon >0\), \(N\geq 3\) and the continuous function \(V:\mathbb{R}^N \to \mathbb{R}\) satisfies \[ \lim_{|x| \to \infty} V(x)> \inf_{x\in \mathbb{R}^N} V(x)=:V_0. \] The authors prove that (1) has a positive solution \(v_{\varepsilon}\) for all sufficiently small \(\varepsilon>0\), and for each sequence \(\varepsilon _n \to 0\), up to a subsequence, the family \(\{v_{\varepsilon_{n}}\}\) concentrates around a point \(y_0\in \mathbb{R}^N\) such that \(V(y_0)=V_0\). The approach relies on a mountain pass theorem without Palais-Smale condition in the frame of critical point theory for \(C^1\) perturbations of convex lower semicontinuous functionals. The case when \(V\) is periodic or asymptotically periodic is also addressed.
scientific article; zbMATH DE number 6968554
scientific article