Semilinear elliptic equation $Delta u = P(u)$ with $P$ being polynomial of degree 3
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Suppose that $B_1 subset mathbb{R}^3$ and $P(u)$ is a polynomial with degree 3. If $u in W^{1,2}(B_1)$ is a weak solution of $$Delta u = P(u) text{ in } B_1,$$
then can we obtain the smoothness of the solution?
I found that the theories in Trudinger's book can not be applied since the integrability of $P(u)$ is not good enough. And If the degree of $P$ is higher, I found there may not exist a smooth solution.
Is that true if I replace $P(u)$ by a smooth function $g(u)$ such that $lim_{x to infty}dfrac{g}{u^3}<infty$? May I have a reference of it? Thank you!
reference-request elliptic-equations
asked Nov 16 at 18:26
mnmn1993
397413
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reputation from mnmn1993 that ended 19 hours ago. Grace period ends in 4 hours
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add a comment |
up vote
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down vote
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Suppose that $B_1 subset mathbb{R}^3$ and $P(u)$ is a polynomial with degree 3. If $u in W^{1,2}(B_1)$ is a weak solution of $$Delta u = P(u) text{ in } B_1,$$
then can we obtain the smoothness of the solution?
I found that the theories in Trudinger's book can not be applied since the integrability of $P(u)$ is not good enough. And If the degree of $P$ is higher, I found there may not exist a smooth solution.
Is that true if I replace $P(u)$ by a smooth function $g(u)$ such that $lim_{x to infty}dfrac{g}{u^3}<infty$? May I have a reference of it? Thank you!
reference-request elliptic-equations
asked Nov 16 at 18:26
mnmn1993
397413
This question had a bounty worth +50
reputation from mnmn1993 that ended 19 hours ago. Grace period ends in 4 hours
Looking for an answer drawing from credible and/or official sources.
1
If $uin W^{1,2}$ then by embeddings $uin L^6$, thus $P(u)in L^2$. Then you get $uin H^2_{loc}$. What do you want to obtain here?
– daw
Nov 20 at 10:31
Then we can embed $H^2$ into $C^{0, alpha}$ and thus $Delta u in C^{0, alpha }$. Therefore, we have $u in C^{2, alpha}$. Is it correct?
– mnmn1993
Nov 20 at 17:02
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose that $B_1 subset mathbb{R}^3$ and $P(u)$ is a polynomial with degree 3. If $u in W^{1,2}(B_1)$ is a weak solution of $$Delta u = P(u) text{ in } B_1,$$
then can we obtain the smoothness of the solution?
I found that the theories in Trudinger's book can not be applied since the integrability of $P(u)$ is not good enough. And If the degree of $P$ is higher, I found there may not exist a smooth solution.
Is that true if I replace $P(u)$ by a smooth function $g(u)$ such that $lim_{x to infty}dfrac{g}{u^3}<infty$? May I have a reference of it? Thank you!
reference-request elliptic-equations
asked Nov 16 at 18:26
mnmn1993
397413
Suppose that $B_1 subset mathbb{R}^3$ and $P(u)$ is a polynomial with degree 3. If $u in W^{1,2}(B_1)$ is a weak solution of $$Delta u = P(u) text{ in } B_1,$$
then can we obtain the smoothness of the solution?
I found that the theories in Trudinger's book can not be applied since the integrability of $P(u)$ is not good enough. And If the degree of $P$ is higher, I found there may not exist a smooth solution.
Is that true if I replace $P(u)$ by a smooth function $g(u)$ such that $lim_{x to infty}dfrac{g}{u^3}<infty$? May I have a reference of it? Thank you!
reference-request elliptic-equations
reference-request elliptic-equations
asked Nov 16 at 18:26
mnmn1993
397413
asked Nov 16 at 18:26
mnmn1993
397413
edited Nov 19 at 7:15
asked Nov 16 at 18:26
mnmn1993
397413
asked Nov 16 at 18:26
mnmn1993
397413
asked Nov 16 at 18:26
mnmn1993
397413
397413
This question had a bounty worth +50
reputation from mnmn1993 that ended 19 hours ago. Grace period ends in 4 hours
Looking for an answer drawing from credible and/or official sources.
This question had a bounty worth +50
reputation from mnmn1993 that ended 19 hours ago. Grace period ends in 4 hours
Looking for an answer drawing from credible and/or official sources.
1
If $uin W^{1,2}$ then by embeddings $uin L^6$, thus $P(u)in L^2$. Then you get $uin H^2_{loc}$. What do you want to obtain here?
– daw
Nov 20 at 10:31
Then we can embed $H^2$ into $C^{0, alpha}$ and thus $Delta u in C^{0, alpha }$. Therefore, we have $u in C^{2, alpha}$. Is it correct?
– mnmn1993
Nov 20 at 17:02
add a comment |
1
If $uin W^{1,2}$ then by embeddings $uin L^6$, thus $P(u)in L^2$. Then you get $uin H^2_{loc}$. What do you want to obtain here?
– daw
Nov 20 at 10:31
Then we can embed $H^2$ into $C^{0, alpha}$ and thus $Delta u in C^{0, alpha }$. Therefore, we have $u in C^{2, alpha}$. Is it correct?
– mnmn1993
Nov 20 at 17:02
1
1
If $uin W^{1,2}$ then by embeddings $uin L^6$, thus $P(u)in L^2$. Then you get $uin H^2_{loc}$. What do you want to obtain here?
– daw
Nov 20 at 10:31
If $uin W^{1,2}$ then by embeddings $uin L^6$, thus $P(u)in L^2$. Then you get $uin H^2_{loc}$. What do you want to obtain here?
– daw
Nov 20 at 10:31
Then we can embed $H^2$ into $C^{0, alpha}$ and thus $Delta u in C^{0, alpha }$. Therefore, we have $u in C^{2, alpha}$. Is it correct?
– mnmn1993
Nov 20 at 17:02
Then we can embed $H^2$ into $C^{0, alpha}$ and thus $Delta u in C^{0, alpha }$. Therefore, we have $u in C^{2, alpha}$. Is it correct?
– mnmn1993
Nov 20 at 17:02
add a comment |
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1
If $uin W^{1,2}$ then by embeddings $uin L^6$, thus $P(u)in L^2$. Then you get $uin H^2_{loc}$. What do you want to obtain here?
– daw
Nov 20 at 10:31
Then we can embed $H^2$ into $C^{0, alpha}$ and thus $Delta u in C^{0, alpha }$. Therefore, we have $u in C^{2, alpha}$. Is it correct?
– mnmn1993
Nov 20 at 17:02