It is known that the problem $ -\Delta u = u^p$ in $\mathbb{R}^n $ (which is known as Lane-Emden equation) has a positive solution of the form $$ u(x)=\dfrac{c_n}{(1+|x|^2)^{(n-2)/2}} $$ where $\Delta u=div(\nabla u)$ is the Laplacian in $\mathbb{R}^n$, $n > 2$ and $p= 2^* − 1 = \frac{n-2}{n+2}$, being $2^*$ the critical Sobolev exponent. The function $$ is called standard bubble in the elliptic theory of PDEs.
It is also known that there are others "bubble" solutions for different equations, for example $$ -\Delta u + u = u^p \quad \text{ in } \mathbb{R}^n, $$ or $$ (-\Delta)^2u = u^p \quad \text{ in } \mathbb{R}^n, $$ where $(-\Delta u)^2$ in the bi-Laplacian operator and $p$ is some special exponent.
Now I am interested in finding a bubble solution to the equations $$ \Delta u + u= u^\alpha \quad \text{ in } \mathbb{R}^n $$ or $$ \Delta u = u^\alpha \quad \text{ in } \mathbb{R}^n $$ with $\alpha\in (0,1)$. Note that I am taking $\Delta$ instead of $-\Delta$.
Do you know any paper where some of these equations (or similar) are studied? Thanks in advance.