The existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R} \end{split} \end{align*} will be proved. Here $\mu>0$. we will show the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity is of exponential critical growth.
Infinitely many sign-changing solutions for a class of elliptic problem with exponential critical growth
Multiplicity of solutions to an elliptic problem with singularity and measure data
Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in $\mathbb{R}^{2}$
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