# Questions tagged [semi-linear-elliptic-pdes]

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10
questions

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0
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45
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Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$.
...

2
votes

1
answer

72
views

Consider the Lane-Emden equation
$$-\Delta u=u^{\frac{d+2}{d-2}} $$
in $\mathbb{R}^d$ with $d\geq 3$ and $u>0$ a positive $C^2$-solution. It is well-known, due to [Caffarelli et al., CPAM '89] that ...

5
votes

0
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86
views

Assume $u$ is a smooth solution for
$$
\Delta u + f(u)=0\qquad \hbox{in}\quad \Omega
$$
and $\Omega$ is a smooth convex domain in $\mathbb{R}^n$.
Is there a conjecture which are the weakest conditions ...

1
vote

1
answer

56
views

I’m considering a $H^1$ function u on a open domain D. Is the integral:
$$ \int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$
continuous with respect to x?
I tried to prove that it’s differential by ...

1
vote

1
answer

43
views

I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate.
I am reading a paper of Brezis and Oswald about existence ...

2
votes

0
answers

105
views

Here are some notations:
$W^{1,2}(\mathbb{R}^N)=H^1(\mathbb{R}^N)=\{ u\in L^2(\mathbb{R}^N) \,|\,\nabla u\in L^2(\mathbb{R}^N) \}$
$D^{1,2}(\mathbb{R}^N)=\{ u\in L^6(\mathbb{R}^N) \,|\, \nabla u\in L^...

3
votes

1
answer

218
views

Consider the following PDE:$$0=u_t+u_{yy}+u_{xx}+(x-y)u_y+y^{-\frac{3}{2}}u^2+1,$$ with $t \in [0,T], $ and a terminal condition $u_T=-1$ for all $x$ and $y.$ The domain for $x$ and $y$ can be bounded ...

3
votes

0
answers

56
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I have been recently reading Kwong's paper on the uniqueness of positive solutions for the equation $\Delta u-u+u^p=0$ in $\mathbb{R}^n$.
The authors show that the above equation has a unique positive ...

2
votes

0
answers

249
views

Let $M$ be a bounded domain in $\Bbb R^2$: under the assumption that
$$
\partial_{u} f(x, u)=0 \text { for }|u| \geq K\label{1}\tag{1.6}
$$
Michael E. Taylor said that (proposition (1.3))
For $k=1,2, \...

4
votes

1
answer

273
views

Suppose that $f:\Bbb R^2\to\Bbb R$ is a continuous non-linearity and consider the following semi-linear elliptic PDE given by:
$$-\Delta u=f(x,u),\;\;x\in\Omega\subset\Bbb R^n,\tag{1}\label{1}$$
To ...