Norm estimates for radially symmetric solutions of semilinear elliptic equations

Author:
Ryuji Kajikiya

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1163-1199

MSC:
Primary 35J60; Secondary 34B15, 35B05, 35B45

MathSciNet review:
1290720

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The semilinear elliptic equation in with the condition is studied, where and has a superlinear and subcritical growth at . For example, the functions and are treated. The and norm estimates are established for any radially symmetric solution which has exactly zeros in the interval . Here are independent of and .

**[1]**H. Berestycki and P.-L. Lions,*Nonlinear scalar field equations. II. Existence of infinitely many solutions*, Arch. Rational Mech. Anal.**82**(1983), no. 4, 347–375. MR**695536**, 10.1007/BF00250556**[2]**M. Grillakis,*Existence of nodal solutions of semilinear equations in 𝑅^{𝑁}*, J. Differential Equations**85**(1990), no. 2, 367–400. MR**1054554**, 10.1016/0022-0396(90)90121-5**[3]**Philip Hartman,*Ordinary differential equations*, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR**658490****[4]**C. Jones and T. Küpper,*On the infinitely many solutions of a semilinear elliptic equation*, SIAM J. Math. Anal.**17**(1986), no. 4, 803–835. MR**846391**, 10.1137/0517059**[5]**Ryuji Kajikiya,*Sobolev norms of radially symmetric oscillatory solutions for superlinear elliptic equations*, Hiroshima Math. J.**20**(1990), no. 2, 259–276. MR**1063363****[6]**Ryuji Kajikiya,*Radially symmetric solutions of semilinear elliptic equations, existence and Sobolev estimates*, Hiroshima Math. J.**21**(1991), no. 1, 111–161. MR**1091434****[7]**Ryuji Kajikiya,*Nodal solutions of superlinear elliptic equations in symmetric domains*, Adv. Math. Sci. Appl.**3**(1993/94), no. Special Issue, 219–266. MR**1287930****[8]**Kevin McLeod, W. C. Troy, and F. B. Weissler,*Radial solutions of Δ𝑢+𝑓(𝑢)=0 with prescribed numbers of zeros*, J. Differential Equations**83**(1990), no. 2, 368–378. MR**1033193**, 10.1016/0022-0396(90)90063-U**[9]**S. I. Pohozaev,*Eigenfunctions of the equation*, Soviet Math. Dokl.**5**(1965), 1408-1411.**[10]**Walter A. Strauss,*Existence of solitary waves in higher dimensions*, Comm. Math. Phys.**55**(1977), no. 2, 149–162. MR**0454365**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35J60,
34B15,
35B05,
35B45

Retrieve articles in all journals with MSC: 35J60, 34B15, 35B05, 35B45

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1290720-4

Keywords:
Semilinear elliptic equation,
radially symmetric solution,
norm estimate

Article copyright:
© Copyright 1995
American Mathematical Society