On a logarithmic Hartree equation
The authors establish a multiplicity result of weak solutions for the logarithmic Hartree equation \[ -\Delta u +au-\frac{1}{2\pi}\left[\ln \frac{1}{|x|}\star |u|^2\right]u=f(x,u), \text{ in } \mathbb{R}^2, \tag{1} \] where \(a>0\) and \(f:\mathbb{R}^2\times \mathbb{R}\rightarrow \mathbb{R}\) is a Carathéodory function such that \(f(x,\cdot)=0\) and \(f(x,\cdot)=f(|x|,\cdot)\), for a.e. \(x\in \mathbb{R}^2\). The authors impose the following additional conditions on \(f\) (here \(F(x,t):=\int_0^t f(x,\sigma)d\sigma\)): \begin{itemize} \item[1)] there exist \(c\in L^{p}(\mathbb{R}^2)\), with \(1\)<\(p\)<\(2\), \(d>0\), and \(q>2\) such that \[ |f(x,t)|\leq c(x)+d|t|^{q-1}, \ \ \text{for a.e.} \ \ x\in\mathbb{R}^2 \text{ and for all } t\in \mathbb{R}; \] \item[2)] \(\lim_{|t|\rightarrow 0}\frac{f(x,t)}{t}=0\) and \(\lim_{|t|\rightarrow +\infty}\frac{F(x,t)}{t^2}=+\infty\), uniformly for a.e. \(x\in \mathbb{R}^2\); \item[3)] there exists \(\mathcal{M}^*\in L_+^1(\mathbb{R}^2)\) such that \[f(x,s)s - 2F(x,s)\leq f(x,t)t - 2F(x,t)+\mathcal{M}^*(x),\] \ \ \ \ for a.e. \(x\in\mathbb{R}^2\), and for all \(s,t\in \mathbb{R}\), with \(0 \leq s \leq t\) or \(t \leq s \leq 0\); \item[4)] there exists \(\tilde{u}\in H^1(\mathbb{R}^2)\), with \(\displaystyle{\int_{\mathbb{R}^2}|\tilde{u}(x)|^2\ln(1+|x|)dx<\infty}\), such that \[ \lim_{y\rightarrow +\infty}\frac{1}{y^4\ln y}\int_{\mathbb{R}^2}F(x,y^2\tilde{u}^{\pm}(xy))dx=+\infty. \] \end{itemize} Then they prove, via a mountain pass argument, the existence of nontrivial nonnegative and nontrivial nonpositive solutions to equation \((1)\).
scientific article; zbMATH DE number 7136853
scientific article