Bull. Austral. Math. Soc. 72(3) pp.349--370, 2005.

Bifurcation of positive entire solutions for a semilinear elliptic equation

Tsing-San Hsu

Huei-Li Lin

Received: 25th April, 2005



In this paper, we consider the nonhomogeneous semilinear elliptic equation
(*)$\scriptstyle \lambda $ - $\displaystyle \Delta $u + u = $\displaystyle \lambda $K(x)up + h(x) in $\displaystyle \mathbb {R}$N, u > 0 in $\displaystyle \mathbb {R}$N  , u $\displaystyle \in $ H 1($\displaystyle \mathbb {R}$N ),

where $ \lambda $ $ \geq $ 0, 1 < p < (N+2)/(N-2), if N$ \ge $3, 1 < p < $ \infty $, if N = 2, h(x) $ \in $ H-1($ \mathbb {R}$N ), 0 $ \not \equiv $h(x) $ \geq $ 0 in $ \mathbb {R}$N, K(x) is a positive, bounded and continuous function on $ \mathbb {R}$N. We prove that if K(x) $ \geq $ K$\scriptstyle \infty $ > 0 in $ \mathbb {R}$N, and $ \lim \limits _{{\vert x\vert \rightarrow \infty }}^{}$K( x) = K$\scriptstyle \infty $, then there exists a positive constant $ \lambda ^{*}_{}$ such that (*)$\scriptstyle \lambda $ has at least two solutions if $ \lambda $ $ \in $ (0,$ \lambda ^{*}_{}$) and no solution if $ \lambda $ > $ \lambda ^{*}_{}$. Furthermore, (*)$\scriptstyle \lambda $ has a unique solution for $ \lambda $ = $ \lambda ^{*}_{}$ provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (*)$\scriptstyle \lambda $ at $ \lambda $ = $ \lambda ^{*}_{}$.

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(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199637 Z'blatt-MATH: 1097.35051


  1. H. Amann;
    Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,
    SIAM. Rev. 18 (1976), pp. 620--709. MR415432
  2. A. Ambrosetti and P.H. Rabinowitz;
    Dual variational method in critical point theory and applications,
    J. Funct. Anal. 14 (1973), pp. 349--381. MR370183
  3. A. Bahri and P.L. Lions;
    On the existence of a positive solution of semilinear elliptic equations in unbounded domains,
    Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), pp. 365--413. MR1450954
  4. V. Benci and G. Cerami;
    Positive solutions of some nonlinear elliptic problems in unbounded domains,
    Arch. Rational Mech. Anal. 99 (1987), pp. 283--300. MR898712
  5. H. Berestycki and P.L. Lions;
    Nonlinear scalar field equations, I. Existence of a ground state,
    Arch. Rational. Mech. Anal. 82 (1983), pp. 313--345. MR695535
  6. D.M. Cao and Z.H. Zhou;
    Multiple positive solutions of nonhomogeneous elliptic equations in RN,
    Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), pp. 443--463. MR1386873
  7. M.G. Crandall and P.H. Rabinowitz;
    Bifurcation, perturbation of simple eigenvalues and lineaized stability,
    Arch. Rational Mech. Anal. 52 (1973), pp. 161--180. MR341212
  8. W.Y. Ding and W.M. Ni;
    On the existence of entire solution of a semilinear elliptic equation,
    Arch. Rational Mech. Anal. 91 (1986), pp. 283--308. MR807816
  9. L. Ekeland;
    Nonconvex minimization problems,
    Bull. Amer. Math. Soc. 1 (1979), pp. 443--474. MR526967
  10. D. Gilbarg and N.S. Trudinger;
    Elliptic partial differential equations of second order (Springer-Verlag, New York, 1983). MR737190
  11. T.S. Hsu;
    Existence and bifurcation of the positive solutions for a semilinear elliptic problem in exterior domains,
  12. P.L. Korman, Y. Li, and T.-C. Ouyang;
    Exact multiplicity results for boundary value problems with nonlinearities generalizing cubic,
    Proc. Royal Soc. Edinbburgh Ser. A 126 (1996), pp. 599--616. MR1396280
  13. P.L. Lions;
    The concentration-compactness principle in the calculus of variations. The locally compact case,
    Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), pp. 109--145 and 223--283. MR778970, MR778974
  14. P.L. Lions;
    On positive solutions of semilinear elliptic equations in unbounded domains,
    in Nonlinear Diffusion Equations and Their Equilibrium States,
    (W.M. Ni, L.A. Peletier and J. Serrin, Editors) (Springer-Verlag, New York, Berlin, 1988). MR956083
  15. W. Strauss;
    Existence of solitary waves in higher dimensions,
    Comm. Math. Phys. 55 (1977), pp. 149--162. MR454365
  16. X.P. Zhu;
    A perturation result on positive entire solutions of a semilinear elliptic equation,
    J. Differential Equations 92 (1991), pp. 163--178. MR1120901
  17. X.P. Zhu and D.M. Cao;
    The concentration-compactness principle in nonlinear elliptic equation,
    Acta. Math. Sci. 9 (1989), pp. 307--328. MR1043058
  18. X.P. Zhu and H.S. Zhou;
    Existence of multiple positive solutions of inhomogenous semilinear elliptic problems in unbounded domains,
    Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), pp. 301--318. MR1069524

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