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Classification of positive radial solutions to a weighted biharmonic equation
School of Mathematical Sciences, East China Normal University, Shanghai 200241, China |
$ \Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u = |x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\}, $ |
$ n\geq 5 $ |
$ -n<\alpha<n-4 $ |
$ p>1 $ |
$ (p,\alpha,\beta,n) $ |
$ \frac{n+\alpha}{2}+\frac{n+\beta}{p+1} = n-2. $ |
$ \lambda $ |
$ \mu $ |
$ v(t): = |x|^{\frac{n-4-\alpha}{2}}u(|x|) $ |
$ t = -\ln |x| $ |
$ u $ |
$ v(t) $ |
$ \alpha \in (-2,n-4) $ |
$ \lambda $ |
$ \mu $ |
$ \alpha \in (-n,-2] $ |
$ v(t) $ |
References:
[1] |
M. Bhakta and R. Musina,
Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal., 75 (2012), 3836-3848.
doi: 10.1016/j.na.2012.02.005. |
[2] |
P. Caldiroli and G. Cora,
Entire solutions for a class of fourth-order semilinear elliptic equations with weights, Mediterr. J. Math., 13 (2016), 657-675.
doi: 10.1007/s00009-015-0519-1. |
[3] |
P. Caldiroli and R. Musina,
On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687.
doi: 10.1007/s00032-011-0167-2. |
[4] |
P. Caldiroli and R. Musina,
Rellich inequalities with weights, Calc. Var. Partial Differ. Equ., 45 (2012), 147-164.
doi: 10.1007/s00526-011-0454-3. |
[5] |
R. Frank and T. König,
Classification of positive solutions to a nonlinear biharmonic equation with critical exponent, Anal. Partial Differ. Equ., 12 (2019), 1101-1113.
doi: 10.2140/apde.2019.12.1101. |
[6] |
R. Frank and T. König,
Singular solution to a semilinear biharmonic equation with a general critical nonlinearity, Rend. Lincei Mat. Appl., 30 (2019), 817-846.
doi: 10.4171/RLM/871. |
[7] |
Z. M. Guo, X. Huang, L. P. Wang and J. C. Wei,
On Delaunay solutions of a biharmonic elliptic equation with critical exponent, J. Anal. Math., 140 (2020), 371-394.
doi: 10.1007/s11854-020-0096-5. |
[8] |
Z. M. Guo, F. S. Wan and L. P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation, Commun. Contemp. Math., 22 (2020), 1950057, 40 pp.
doi: 10.1142/S0219199719500573. |
[9] |
X. Huang and L. P. Wang,
Classification to the positive radial solutions with weighted biharmonic equation, Discrete Contin. Dyn. Syst., 40 (2020), 4821-4837.
doi: 10.3934/dcds.2020203. |
[10] |
X. Huang and D. Ye, Hardy-Rellich type equalities, preprint. |
[11] |
C. S. Lin,
A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^N$, Comment. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[12] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145. |
[13] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283. |
[14] |
R. Musina,
Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1626-1659.
doi: 10.1007/s10231-013-0348-4. |
show all references
References:
[1] |
M. Bhakta and R. Musina,
Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal., 75 (2012), 3836-3848.
doi: 10.1016/j.na.2012.02.005. |
[2] |
P. Caldiroli and G. Cora,
Entire solutions for a class of fourth-order semilinear elliptic equations with weights, Mediterr. J. Math., 13 (2016), 657-675.
doi: 10.1007/s00009-015-0519-1. |
[3] |
P. Caldiroli and R. Musina,
On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687.
doi: 10.1007/s00032-011-0167-2. |
[4] |
P. Caldiroli and R. Musina,
Rellich inequalities with weights, Calc. Var. Partial Differ. Equ., 45 (2012), 147-164.
doi: 10.1007/s00526-011-0454-3. |
[5] |
R. Frank and T. König,
Classification of positive solutions to a nonlinear biharmonic equation with critical exponent, Anal. Partial Differ. Equ., 12 (2019), 1101-1113.
doi: 10.2140/apde.2019.12.1101. |
[6] |
R. Frank and T. König,
Singular solution to a semilinear biharmonic equation with a general critical nonlinearity, Rend. Lincei Mat. Appl., 30 (2019), 817-846.
doi: 10.4171/RLM/871. |
[7] |
Z. M. Guo, X. Huang, L. P. Wang and J. C. Wei,
On Delaunay solutions of a biharmonic elliptic equation with critical exponent, J. Anal. Math., 140 (2020), 371-394.
doi: 10.1007/s11854-020-0096-5. |
[8] |
Z. M. Guo, F. S. Wan and L. P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation, Commun. Contemp. Math., 22 (2020), 1950057, 40 pp.
doi: 10.1142/S0219199719500573. |
[9] |
X. Huang and L. P. Wang,
Classification to the positive radial solutions with weighted biharmonic equation, Discrete Contin. Dyn. Syst., 40 (2020), 4821-4837.
doi: 10.3934/dcds.2020203. |
[10] |
X. Huang and D. Ye, Hardy-Rellich type equalities, preprint. |
[11] |
C. S. Lin,
A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^N$, Comment. Math. Helv., 73 (1998), 206-231.
doi: 10.1007/s000140050052. |
[12] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145. |
[13] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283. |
[14] |
R. Musina,
Weighted Sobolev spaces of radially symmetric functions, Ann. Mat. Pura Appl., 193 (2014), 1626-1659.
doi: 10.1007/s10231-013-0348-4. |
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