Questions tagged [semi-linear-elliptic-pdes]

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Precise decay of solution fo fractional Schroedinger equations

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 $1<p<\frac{N+2s}{N-2s}$. ...
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2 votes
1 answer
72 views

$C^2$-solution of Lane-Emden equation with positive frequency

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 ...
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5 votes
0 answers
86 views

Semilinear elliptic equation

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 ...
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1 vote
1 answer
56 views

About the continuity of the integral on the boundary of a ball

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

Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

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

Does the theorem in Sobolev spaces hold on the subset?

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

Solution or existence for a second-order semilinear PDE

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 ...
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3 votes
0 answers
56 views

Existence of ground state solutions for the critical exponent

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 ...
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2 votes
0 answers
249 views

A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book "Partial Differential Equations III"

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

Regularity of weak solutions to semi-linear elliptic PDEs

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 ...
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