We study the logarithmic Schrödinger equation with the sign-changing potential function. It is known that the corresponding functional is not well defined in $H^1({\mathbb{R}}^N)$, but by imposing some condition on V(x), we can show that the functional is well defined in a subspace of $H^1({\mathbb{R}}^N)$. Then, the existence and multiplicity of solutions is obtained by using variational methods. We remark that the existence of solutions is deeply influenced by the sign of Q(x).

中文翻译：

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带势对数薛定谔方程解的存在性和多重性

我们研究具有符号变化势函数的对数薛定谔方程。众所周知，对应的泛函在$H^1({\mathbb{电阻}}^N)$，但是通过对V ( x )施加一些条件，我们可以证明泛函在$H^1({\mathbb{电阻}}^N)$. 然后，使用变分方法获得解的存在性和多重性。我们注意到解的存在性深受Q ( x )符号的影响。