$C^2$-solution of Lane-Emden equation with positive frequency

Question

Consider the Lane-Emden equation $$-\Delta u=u^{\frac{d+2}{d-2}} $$ in $\mathbb{R}^d$ with $d\geq 3$ and $u>0$ a positive $C^2$-solution. It is well-known, due to [Caffarelli et al., CPAM '89] that $u$ must take the form $$ u(x)=a\Big(\frac{b}{1+b^2|x-x_0|^2}\Big)^{\frac{d-2}{2}}$$ for some $a,b>0$ and $x_0\in\mathbb{R}^d$ (these are the so-called Aubin-Talenti bubble functions). Now let us consider $$-\Delta u+\beta u=u^{\frac{d+2}{d-2}}.$$ We will assume that $u$ is still a positive $C^2$-solution, and we also assume that $\beta>0$. Of course, when for instance assuming that $u$ is constant along one or more $x_i$-directions, the equation becomes the energy-subcritical semilinear elliptic equation and it is well-known that it has smooth solution. Therefore my questions would be:

1. Does the second equation have positive solution that is non-trivial along all directions?

2. When assuming $\beta$ is positive, is it necessary that $u$ must be constant along at least one $x_i$-direction?

Thank you for your ideas in advance!

Answer 1

The answer to my questions is indeed negative. In this paper it is shown that solutions to the second equation do exist, which are periodic in $x_1$ (but not constant) and decay to zero when $|x'|\to\infty$.