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!






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    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







    share|cite|improve this question


























      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







      share|cite|improve this question















      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 7 hours ago

























      asked 2 days ago









      mnmn1993

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