Radial solutions of a supercritical elliptic equation with Hardy potential. (English) Zbl 1247.35021
Summary: Various properties of radial solutions of the supercritical elliptic equation with Hardy potential
\[
-\Delta u+\mu\frac{u}{|x|^2}=|u|^{p-2}u\text{ in }\Omega\setminus\{0\},\qquad u=0\;\mathrm{on }\partial \Omega
\]
are studied, where \(\Omega=\int\{x\in{\mathcal R}^N|a\leq|x|< b\}\), which is a ball if \(0=a<b<+\infty\), an annulus if \(0<a<b<+\infty\), an exterior domain if \(0<a<b=+\infty\), and the whole space \({\mathcal R}^N\) if \(a=0\), \(b=+\infty\). We assume \(p\) is supercritical, that is, \(p>2^*\) with \(2^*=\frac{2N}{N-2}\) being the critical Sobolev exponent, and \(N\geq 3\).
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35J75 | Singular elliptic equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 70K05 | Phase plane analysis, limit cycles for nonlinear problems in mechanics |
References:
| [1] | Pucci, Existence non - existence and regularity of radial ground states for p - Laplacian equations with singular weights Poincare Non Line aire, Inst Anal 25 pp 505– (2008) · Zbl 1147.35045 |
| [2] | Ge, Bubble towers for supercritical semilinear elliptic equations, Funct Anal pp 221– (2005) · Zbl 1129.35379 |
| [3] | Bartsch, Infinitely many radial solutions of a semilinear elliptic problem on RN Rational, Arch Mech Anal pp 124– (1993) |
| [4] | Huidobro, Corta zar Garcı a - On the uniqueness of the second bound state solution of a semilinear equation Poincare Non Line aire to appear, Inst Anal |
| [5] | Johnson, Positive solution of supercritical elliptic equations and asymptotics, PDEs 18 pp 1153– (1993) |
| [6] | Farina, On the classification of solutions of the Lane - Emden equation on unbounded domains of RN Pures, Math Appl pp 87– (2007) · Zbl 1143.35041 |
| [7] | Dancer, Sign - changing solutions for supercritical elliptic problems in domains with small holes Manuscripta, Math pp 123– (2007) · Zbl 1387.35284 |
| [8] | Passaseo, Nontrivial solutions of elliptic equations with supercritical exponent in contractible do - mains Duke, Math J 92 pp 429– (1998) · Zbl 0943.35034 |
| [9] | Joseph, Quasilinear Dirichlet problems driven by positive sources Rational, Arch Mech Anal 49 pp 321– (1973) · Zbl 0266.34021 |
| [10] | Gidas, Global and local behavior of positive solutions of nonlinear elliptic equations Pure, Appl Math 34 pp 528– (1981) · Zbl 0465.35003 |
| [11] | Pino, Da vila del The supercritical Lane - Emden - Fowler equation in exterior do - mains, PDEs 32 pp 1225– (2007) · Zbl 1137.35023 · doi:10.1080/03605300600854209 |
| [12] | Gui, On the stsbility and instability of positive steady states of a semilinear heat equation in RN Pure, Appl Math 45 pp 1153– (1992) |
| [13] | Pino, del Nonlinear elliptic problems above criticality Milan, J Math 74 pp 313– (2006) |
| [14] | Rabinowitz, Some aspects of nonlinear eigenvalue problems Rocky Mount, Math 3 pp 161– (1973) |
| [15] | Hashimoto, Existence of nontrivial solutions for some elliptic equations with supercrit - ical nonlinearity in exterior domains, Disc Cont Dyn Sys 19 pp 323– (2007) · Zbl 1129.35417 · doi:10.3934/dcds.2007.19.323 |
| [16] | Rabinowitz, Some global results for nonlinear eigenvalue problems, Funct Anal 7 pp 487– (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9 |
| [17] | Brezis, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents Pure, Appl Math 36 pp 437– (1983) |
| [18] | Cao, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc Amer Math Soc pp 131– (2003) · Zbl 1075.35009 |
| [19] | Felmer, Uniqueness of radially sysmmetric positive solutions for u u up in an annulus Differential Equations Further studies on Emden s and similar differential equations, Quart Math 66 pp 245– (2008) |
| [20] | Smets, Nonlinear Schro dinger equations with Hardy potential and critical nonlinearities, Trans Amer Math Soc pp 357– (2004) |
| [21] | Bahri, On a nonlinear elliptic equation involving the critical Sobolev exponent : the effect of the topology of the domain Pure, Appl Math 41 pp 255– (1988) · Zbl 0649.35033 |
| [22] | Pino, del Supercritical elliptic problems in domains with small holes Poincare Non Line aire, Inst Anal 24 pp 507– (2007) |
| [23] | Ghoussoub, Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents, Trans Amer Math Soc pp 352– (2000) · Zbl 0956.35056 |
| [24] | Ni, Nonexistence theorems for singular solutions of quasilinear partial differential equations Pure, Appl Math 39 pp 379– (1986) · Zbl 0602.35031 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
