Mixing
Semilinear elliptic equation $Delta u = P(u)$ with $P$ being polynomial of degree 3
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 2 days ago
mnmn1993
447413
add a comment |
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 2 days ago
mnmn1993
447413
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 2 days ago
mnmn1993
447413
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 2 days ago
mnmn1993
447413
asked 2 days ago
mnmn1993
447413
edited 7 hours ago
asked 2 days ago
mnmn1993
447413
asked 2 days ago
mnmn1993
447413
asked 2 days ago
mnmn1993
447413
447413
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3001480%2fsemilinear-elliptic-equation-delta-u-pu-with-p-being-polynomial-of-degr%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
