Uniqueness of positive solutions for singular problems involving the 𝑝-Laplacian

Poliakovsky, Arkady; Shafrir, Itai

doi:10.1090/S0002-9939-05-07290-4

Uniqueness of positive solutions for singular problems involving the $p$-Laplacian
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by Arkady Poliakovsky and Itai Shafrir PDF

Proc. Amer. Math. Soc. 133 (2005), 2549-2557 Request permission

Abstract:

We study existence and uniqueness of positive eigenfunctions for the singular eigenvalue problem: $-\Delta _p{u}-\lambda \eta (x)\frac {{u}^{p-1}}{|x|^p} = \mu \frac {{u}^{p-1}}{|x|^p}$ on a bounded smooth domain $\Omega \subset \mathbb {R}^N$ with zero boundary condition. We also characterize all positive solutions of $-\Delta _p{u}=|\frac {N-p}{p}|^p \frac {u^{p-1}}{|x|^p}$ in $\mathbb {R}^N\setminus \{0\}$.

References

Walter Allegretto and Yin Xi Huang, A Picone’s identity for the $p$-Laplacian and applications, Nonlinear Anal. 32 (1998), no. 7, 819–830. MR 1618334, DOI 10.1016/S0362-546X(97)00530-0
Aomar Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725–728 (French, with English summary). MR 920052
Haïm Brezis and Moshe Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 217–237 (1998). Dedicated to Ennio De Giorgi. MR 1655516
E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, DOI 10.1016/0362-546X(83)90061-5
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
Moshe Marcus, Victor J. Mizel, and Yehuda Pinchover, On the best constant for Hardy’s inequality in $\mathbf R^n$, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3237–3255. MR 1458330, DOI 10.1090/S0002-9947-98-02122-9
Moshe Marcus and Itai Shafrir, An eigenvalue problem related to Hardy’s $L^p$ inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 3, 581–604. MR 1817710
Arkady Poliakovsky, On minimization problems which approximate Hardy $L^p$ inequality, Nonlinear Anal. 54 (2003), no. 7, 1221–1240. MR 1995927, DOI 10.1016/S0362-546X(03)00130-5
James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, DOI 10.1007/BF02391014
Itai Shafrir, Asymptotic behaviour of minimizing sequences for Hardy’s inequality, Commun. Contemp. Math. 2 (2000), no. 2, 151–189. MR 1759788, DOI 10.1142/S0219199700000098
Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. MR 768629, DOI 10.1007/BF01449041