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I want to prove the uniqueness of the solution of the following problem: $$\eqalign{ & - d\,\Delta u + u = {u^p} \text{ in } \Omega \cr & u > 0 \text{ in } \Omega \cr & \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial \Omega \cr} $$ with $\Omega$ is a bounded open in $R^n$, $d>0$ and $p>1$. I tried the classical methods but without any success. Thanx.

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I see no reason to expect uniqueness. On the contrary: $u=1$ is a trivial solution. Standard bifurcation theory shows that nontrivial solutions bifurcate when $(1-p)/d$ is an eigenvalue of the Neumann Laplacian.

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  • $\begingroup$ Thank you sir. What if ${1-p}/d$ is not an eigenvalue of the Neumann laplacian, can we prove the uniqueness?. Cordially. $\endgroup$ – Gustave Mar 31 '18 at 12:57
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To illustrate nonuniqueness for $n = 1$ via simple phase plane analysis, consider the system of ODEs $$ \begin{cases} u'(x) = v(x) \\ v'(x) = u(x) - (u(x))^p \end{cases} $$ with the initial conditions $$ \begin{cases} u(0) = a \\ v(0) = 0, \end{cases} $$ where $a$ is taken to belong to $(0, 1)$. The $v$-coordinate of the solution of the above IVP increases as long as its $u$-coordinate remains in $(0,1)$. After that, the $v$-coordinate decreases, and reaches $0$ at some finite $ A > 0$. For $x \in [0, A]$ the $u$-coordinate keeps increasing. We have thus obtained a nontrivial positive solution of the elliptic BVP $$ \begin{cases} -u''(x) + u(x) - (u(x))^p = 0 & \text{ on } (0, A) \\ u'(0) = u'(A) = 0. \end{cases} $$

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  • $\begingroup$ try googling Lin-Ni Conjecture... I think this might be related... $\endgroup$ – Math604 Mar 31 '18 at 16:11
  • $\begingroup$ But is there any conditions on p and q to expect uniqueness? $\endgroup$ – Gustave Apr 1 '18 at 0:01
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    $\begingroup$ It appears that, in the one-dimensional case (on $[0,A]$), the existence of a nontrivial solution is equivalent to the existence of a solution of $v'=u$, $v'=\frac{1}{d}(u-u^p)$ starting at the positive $u$-semiaxis and reaching the semiaxis for $x=A$. The system of ODEs is a Hamiltonian system, with $$H(u, v)=\frac{v^2}{2}-\frac{u^2}{2d} +\frac{u^{p+1}}{(p+1)d}.$$ And quite a lot is known on Hamiltonian systems (but I am no expert). To sum up, the problem boils down to that of (non)existence of periodic solutions of period $2A$. $\endgroup$ – user539887 Apr 1 '18 at 8:21

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