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At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for $$ \begin{cases} -\Delta u = u^p & \Omega \subset \mathbb{R}^n \\ u=0 & \partial \Omega \end{cases} $$ for $ 1 < p < \frac{n+2}{n-2}$.

But I don't have any idea about the sublinear case, namely, $ p<1$. Also, I don't have any information about the case $p<0$, too.

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  • $\begingroup$ for $ 0<p<1$ consider the energy that one would normally use to apply the Mountain Pass Theorem to obtain a solution when $p>1$ (so i mean not the contrained minimization approach). You can just minimize this energy to obtain a nonzero solution (when $p>1$ this minimization would give you $- \infty$) $\endgroup$
    – Math604
    Commented May 13, 2019 at 1:06
  • $\begingroup$ for $p<0$ you might want to look at some papers or books by Marius Ghergu (i believe he has done a bunch of this stuff) $\endgroup$
    – Math604
    Commented May 13, 2019 at 1:08

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