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

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

MathSciNet review:
1290720

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The semilinear elliptic equation $\Delta u + f(u) = 0$ in ${R^n}$ with the condition ${\lim _{|x| \to \infty }}u(x) = 0$ is studied, where $n \geqslant 2$ and $f(u)$ has a superlinear and subcritical growth at $u = \pm \infty$. For example, the functions $f(u) = |u{|^{p - 1}}u - u\;(1 < p < \infty \;{\text {if}}\;n = 2,\;1 < p < (n + 2)/(n - 2)\;{\text {if}}\;n \geqslant 3)$ and $f(u) = u\log |u|$ are treated. The ${L^2}$ and ${H^1}$ norm estimates ${C_1}{(k + 1)^{n/2}} \leqslant ||u|{|_{{L^2}}} \leqslant ||u|{|_{{H^1}}} \leqslant {C_2}{(k + 1)^{n/2}}$ are established for any radially symmetric solution $u$ which has exactly $k \geqslant 0$ zeros in the interval $0 \leqslant |x| < \infty$. Here ${C_1},\;{C_2} > 0$ are independent of $u$ and $k$.

- H. Berestycki and P.-L. Lions,
*Nonlinear scalar field equations. I. Existence of a ground state*, Arch. Rational Mech. Anal.**82**(1983), no. 4, 313–345. MR**695535**, DOI https://doi.org/10.1007/BF00250555 - M. Grillakis,
*Existence of nodal solutions of semilinear equations in ${\bf R}^N$*, J. Differential Equations**85**(1990), no. 2, 367–400. MR**1054554**, DOI https://doi.org/10.1016/0022-0396%2890%2990121-5 - Philip Hartman,
*Ordinary differential equations*, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR**658490** - 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**, DOI https://doi.org/10.1137/0517059 - Ryuji Kajikiya,
*Sobolev norms of radially symmetric oscillatory solutions for superlinear elliptic equations*, Hiroshima Math. J.**20**(1990), no. 2, 259–276. MR**1063363** - Ryuji Kajikiya,
*Radially symmetric solutions of semilinear elliptic equations, existence and Sobolev estimates*, Hiroshima Math. J.**21**(1991), no. 1, 111–161. MR**1091434** - 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** - Kevin McLeod, W. C. Troy, and F. B. Weissler,
*Radial solutions of $\Delta u+f(u)=0$ with prescribed numbers of zeros*, J. Differential Equations**83**(1990), no. 2, 368–378. MR**1033193**, DOI https://doi.org/10.1016/0022-0396%2890%2990063-U
S. I. Pohozaev, - Walter A. Strauss,
*Existence of solitary waves in higher dimensions*, Comm. Math. Phys.**55**(1977), no. 2, 149–162. MR**454365**

*Eigenfunctions of the equation*$\Delta u + \lambda f(u) = 0$, Soviet Math. Dokl.

**5**(1965), 1408-1411.

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

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

Article copyright:
© Copyright 1995
American Mathematical Society