Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Positive solutions of nonlinear elliptic equations--existence and nonexistence of solutions with radial symmetry in $ L\sb{p}({\bf R}\sp{N})$

Author: J. F. Toland
Journal: Trans. Amer. Math. Soc. 282 (1984), 335-354
MSC: Primary 35J60; Secondary 35B32, 35B45, 58E07
MathSciNet review: 728716
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that when $ r$ is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem

$\displaystyle - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\... ...athbf{R}^N},\qquad u \to 0\qquad {\text{as}}\,{\text{\vert x\vert}} \to \infty $

, contains a continuum $ \mathcal{C}$ of nontrivial solutions which is unbounded in $ \mathbf{R}\, \times \,{L_p}({\mathbf{R}^N})$ for all $ p \geq 1$. Various estimates of the $ {L_p}$ norm of $ u$ are obtained which depend on the relative values of $ \sigma$ and $ p$, and the Pohozaev and Sobolev embedding constants.

References [Enhancements On Off] (What's this?)

  • [1] C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9–95. MR 629699, 10.1007/BF00250799
  • [2] C. J. Amick and J. F. Toland, Nonlinear elliptic eigenvalue problems on an infinite strip—global theory of bifurcation and asymptotic bifurcation, Math. Ann. 262 (1983), no. 3, 313–342. MR 692860, 10.1007/BF01456013
  • [3] H. Berestycki and P.-L. Lions, Une méthode locale pour l’existence de solutions positives de problèmes semi-linéaires elliptiques dans 𝑅^{𝑁}, J. Analyse Math. 38 (1980), 144–187 (French). MR 600785
  • [4] H. Berestycki, P.-L. Lions, and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in 𝑅^{𝑁}, Indiana Univ. Math. J. 30 (1981), no. 1, 141–157. MR 600039, 10.1512/iumj.1981.30.30012
  • [5] Haïm Brézis, Positive solutions of nonlinear elliptic equations in the case of critical Sobolev exponent, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981) Res. Notes in Math., vol. 70, Pitman, Boston, Mass.-London, 1982, pp. 129–146. MR 670270
  • [6] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
  • [7] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in 𝑅ⁿ, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369–402. MR 634248
  • [8] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, 10.1002/cpa.3160340406
  • [9] -, A priori bounds for positive solutions of nonlinear equations, Comm. Partial Differential Equations (1981), 883-901.
  • [10] S. I. Pohozaev, Eigenfunctions of the equations $ \Delta u + \lambda \,f(u) = 0$, Soviet Math. Dokl. 5 (1965), 1408-1411.
  • [11] Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587
  • [12] Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 0454365
  • [13] C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. (3) 45 (1982), no. 1, 169–192. MR 662670, 10.1112/plms/s3-45.1.169
  • [14] -, Bifurcation from the continuous spectrum in the $ {L_2}$-theory of elliptic equations in $ {\mathbf{R}^N}$, (Lectures at S.A.F.A. IV, Naples, 1980), Recent Methods in Nonlinear Analysis and Applications, Liguori, Naples, 1981.
  • [15] J. F. Toland, Global bifurcation for Neumann problems without eigenvalues, J. Differential Equations 44 (1982), no. 1, 82–110. MR 651688, 10.1016/0022-0396(82)90026-2
  • [16] J. F. Toland, Solitary wave solutions for a model of the two-way propagation of water waves in a channel, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 2, 343–360. MR 620744, 10.1017/S0305004100058801

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J60, 35B32, 35B45, 58E07

Retrieve articles in all journals with MSC: 35J60, 35B32, 35B45, 58E07

Additional Information

Keywords: Global bifurcation, singular elliptic problem, a priori estimates
Article copyright: © Copyright 1984 American Mathematical Society