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I encountered a sentence which says it is well known that problem $$ \begin{cases} -\Delta u =|u|^{p-1} u & in \,\, \Omega \\ u=0 & on \,\, \partial \Omega \end{cases} $$ have a solution for $1<p<\frac{N+2}{N-2}$ and doesn't have any solution for $p>\frac{N+2}{N-2}$.

The existence is okay by mountain pass theorem. But how about the non-existence case? Can some one give a reference or hint?

Thanks alot indeed.

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    $\begingroup$ Did you try Pohozaev identity assuming Omega is star shaped. $\endgroup$
    – GabS
    Sep 6 at 12:13
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It is not true that this equation always has no solutions in the supercritical case $p > \frac{N + 2}{N - 2}$.

The simplest counterexample is on an annulus, say $\Omega = B_R \setminus B_1$: in this case one may search for radial solutions by separating variables, which reduces to solving the second-order ODE $$ -u'' - \frac{N - 1}{r} u' = u |u|^{p - 1} $$ with boundary condition $0$ at $R$ and $1$. Here the mountain pass or constrained optimization approaches (applied in 1D) will give the existence of a nonnegative solution, similarly to the ball, so long as $p > 2$. The key point is that $r > 1$, so the coefficients are nonsingular.

The most well-known nonexistence result for this equation is that there are no nontrivial solutions on star-shaped domains. This based on the (Rellich-)Pohozaev identity. Here is a sketch assuming also that the domain is smooth (and, WLOG, star-shaped around the origin):

Set $T = x |\nabla u|^2 - 2 (\nabla u \cdot x) \nabla u - 2 x \frac{|u|^{p+1}}{p+1} $. This is a vector field, and a computation shows that $$ \text{div} T = (N-2)|\nabla u|^2 - 2 N \frac{|u|^{p + 1}}{p+1}. $$ At this point one integrates over $\Omega$ and applies the divergence theorem: $$ \int_\Omega (N-2)|\nabla u|^2 - 2 N \frac{|u|^{p + 1}}{p+1} = \int_{\partial \Omega} T \cdot \nu, $$ where $\nu$ is the outward unit normal vector to $\partial \Omega$. As $u = 0$ on $\partial \Omega$, we have $\nabla u = \pm |\nabla u| \nu$, so after some simplification $$ \int_{\partial \Omega} T \cdot \nu = - \int_{\partial \Omega} |\nabla u|^2 x \cdot \nu. $$ This is a nonpositive quantity as the domain is star-shaped (indeed, with more effort one can check that it is strictly negative for nontrivial solutions).

There is a second, simpler identity available: multiply the PDE by $u$ and integrate by parts: $$ \int_{\Omega} |\nabla u|^2 = \int_{\Omega} |u|^{p+1}. $$ Substituting this in, $$ \int_{\Omega} (N - 2 - \frac{2N}{p + 1})|\nabla u|^2 \leq 0. $$ When $p > \frac{N + 2}{N - 2}$ the coefficient is positive and this implies that $u$ is constant $0$, a contradiction.

This general approach accounts for a large portion of nonexistence theorems about semilinear equations in the literature.

To answer the question more fully is difficult: generally one expects existence of positive solutions to be related to the topology of the domain. In the critical case ($p = \frac{N + 2}{N - 2}$) this is fairly well understood, see e.g. Bahri, Coron. In the supercritical case this is not understood nearly as well.

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