Osaka Journal of Mathematics

Non-local elliptic problem in higher dimension

Tosiya Miyasita

Full-text: Open access

Abstract

Non-local elliptic problem, $-\Delta v = \lambda \bigl(e^{v}\big/\bigl(\int_{\Omega}e^{v} dx \bigr)^{p}\bigr)$ with Dirichlet boundary condition is considered on $n$-dimensional bounded domain $\Omega$ with $n \geq 3$ for $p>0$. If $\Omega$ is the unit ball, $3 \leq n \leq 9$ and $2/n \leq p \leq 1$, we have infinitely many bendings in $\lambda$ of the solution set in $\lambda-v$ plane. Finally if $\Omega$ is an annulus domain and $p \geq 1$, we show that a solution exists for all $\lambda>0$.

Article information

Source
Osaka J. Math., Volume 44, Number 1 (2007), 159-172.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1174324329

Mathematical Reviews number (MathSciNet)
MR2313033

Zentralblatt MATH identifier
1213.35229

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory

Citation

Miyasita, Tosiya. Non-local elliptic problem in higher dimension. Osaka J. Math. 44 (2007), no. 1, 159--172. https://projecteuclid.org/euclid.ojm/1174324329


Export citation

References

  • J. Bebernes and D. Eberly: Mathematical Problems from Combustion Theory, Springer-Verlag, New York, 1989.
  • J.W. Bebernes and A.A. Lacey: Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Differential Equations 2 (1997), 927--953.
  • J.W. Bebernes and P. Talaga: Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal. 3 (1996), 836--844.
  • S. Chandrasekhar: An Introduction to the Study of Stellar Structure, Dover, New York, 1957.
  • I.M. Gel'fand: Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295--381.
  • B. Gidas, W.-M. Ni and L. Nirenberg: Symmetry and related properties via the maximal principle, Comm. Math. Phys. 68 (1979), 209--243.
  • D.D. Joseph and T.S. Lundgren: Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241--269.
  • T. Kato: Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
  • E.F. Keller and L.A. Segel: Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 36 (1970), 399--415.
  • A.A. Lacey: Thermal runaway in a nonlocal problem modelling Ohmic heating: Part I, European J. Appl. Math. 6 (1995), 127--144.
  • A.A. Lacey: Thermal runaway in a nonlocal problem modelling Ohmic heating: Part II, European J. Appl. Math. 6 (1995), 201--224.
  • C.-S. Lin and W.-M. Ni: A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc. 102 (1988), 271--277.
  • T. Miyasita and T. Suzuki: Non-local Gel'fand problem in higher dimension; in Nonlocal Elliptic and Parabolic Problems, Banach Center Publ. 66 Polish Acad. Sci., Warsaw, 2004, 221--235.
  • K. Nagasaki and T. Suzuki: Radial solutions for $\Delta u + \lambda e^u = 0$ on annuli in higher dimensions, J. Differential Equations 100 (1992), 137--161.
  • K. Nagasaki and T. Suzuki: Spectral and related properties about the Emden-Fowler equation $-\Delta u = \lambda e^u$ on circular domains, Math. Ann. 299 (1994), 1--15.
  • S.I. Pohozaev: Eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Dokl. 6 (1965), 1408--1411.
  • P.H. Rabinowitz: Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487--513.
  • P.H. Rabinowitz: Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161--202.
  • T. Suzuki: Semilinear Elliptic Equations, Gakkōtosho Co., Ltd., Tokyo, 1994.
  • G. Wolansky: A critical parabolic estimate and application to nonlocal equations arising in chemotaxis, Appl. Anal. 66 (1997), 291--321.