# Non-existence result for $p>\frac{N+2}{N-2}$

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

• Did you try Pohozaev identity assuming Omega is star shaped.
– GabS
Sep 6, 2021 at 12:13

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.