Here, $ H ^ 1 ( mathbb {R} ^ 2) $ is the standard Sobolev space for $ L ^ 2 ( mathbb {R} ^ 2) $ Functions whose weak derivation is one of them $ L ^ 2 ( mathbb {R} ^ 2). $

My question in the title comes from the variational calculation. It is usually the case that a minimizer of a particular energy function is defined $ H ^ 1 ( mathbb {R} ^ 2) $ is known as continuous (or even continuous) $ C ^ 2 ( mathbb {R} ^ 2)) $. I want to know the behavior of this minimizer in infinity.

If $ u in L ^ 2 ( mathbb {R} ^ 2), $ then is known $ liminf_ {| x | to infty} u (x) = 0. $ But it cannot say $ limsup_ {| x | to infty} u (x) = 0 $ because counterexamples exist.

If we accept $ u in H ^ {1+ epsilon} ( mathbb {R} ^ 2) $ for some $ epsilon> 0, $ then the inequality of the classic Morrey can imply a uniform older continuity of $ u. $ So we can close $ limsup_ {| x | to infty} u (x) = 0 $ by proof by contradiction.

So my problem is the case $ epsilon = 0. $ That is when

$$ u in H ^ 1 ( mathbb {R} ^ 2) cap C ( mathbb {R} ^ 2) $$

(or $ u in H ^ 1 ( mathbb {R} ^ 2) cap C ^ 2 ( mathbb {R} ^ 2)), $ is it true that

$$ limsup_ {| x | to infty} u (x) = 0? $$

With evidence of contradiction, I think this should be true. Here is my non-rigorous evidence.

Let's not assume there is $ epsilon> 0 $ and $ x_n in mathbb {R} ^ 2 $ so that $ | x_n | to infty $ and $ | u (x_n) | geq 2 epsilon. $ Because of the continuity there is $ r_n> 0 $ so that $ | u (x) | geq epsilon $ for all $ x in B (x_n, r_n). $

Since $ u in L ^ 2, r_n to 0 $ how $ n to infty. $

**I don't think so rigorously**

$$ int_ {B (x_n, r_n)} | nabla u | ^ 2 gtrsim int_ {B (x_n, r_n)} ( frac { epsilon} {r_n}) ^ 2 = epsilon ^ 2 $ $

for big $ n $ and

$$

int _ { mathbb {R} ^ 2} | nabla u | ^ 2 geq sum_ {n , text {is large}} int_ {B (x_n, r_n)} | nabla u | ^ 2.

$$

So they imply a contradiction $ int _ { mathbb {R} ^ 2} | nabla u | ^ 2 = infty $

I am happy about every discussion.