\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions

Abstract Related Papers Cited by
  • We study positive radial solutions to the boundary value problem \begin{eqnarray} -\Delta u = \lambda K(|x|)f(u), \quad x \in \Omega, \\ \frac{\partial u}{\partial \eta}+\tilde{c}(u)u = 0, \quad |x|=r_0, \\ u(x) \rightarrow 0, \quad |x|\rightarrow \infty, \end{eqnarray} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x \in \mathbb{R}^N| N>2, |x|> r_0 \mbox{ with }r_0>0\}$, $K:[r_0, \infty)\rightarrow(0,\infty)$ is a continuous function such that $\lim_{r \rightarrow \infty}K(r)=0$, $\frac{\partial}{\partial \eta}$ is the outward normal derivative, and $\tilde{c}:[0,\infty) \rightarrow (0,\infty)$ is a continuous function. We consider various $C^1$ classes of the reaction term $f:[0,\infty) \rightarrow \mathbb{R}$ that are sublinear at $\infty$ $(i.e. \lim_{s \rightarrow \infty}\frac{f(s)}{s}=0)$. In particular, we discuss existence and multiplicity results for classes of $f$ with $(a)$ $f(0)>0$, $(b)$ $f(0)<0$, and $(c)$ $f(0)=0$. We establish our existence and multiplicity results via the method of sub-super solutions. We also discuss some uniqueness results.
    Mathematics Subject Classification: Primary: 35J66, 34B18; Secondary: 35J60.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117 (1993), 775-782.doi: 10.2307/2159143.

    [2]

    H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Rev., 18 (1976), 620-709.

    [3]

    A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory, Differential Integral Equations, 7 (1994), 655-663.

    [4]

    K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal. TMA, 5 (1981), 475-486.doi: 10.1016/0362-546X(81)90096-1.

    [5]

    A. Castro, M. Hassanpour and R. Shivaji, Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity, Comm. Partial Differential Equations, 20 (1995), 1927-1936.doi: 10.1080/03605309508821157.

    [6]

    A. Castro, L. Sankar and R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437.doi: 10.1016/j.jmaa.2012.04.005.

    [7]

    A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems, Proc. Amer. Math. Soc., 106 (1989), 735-740.doi: 10.2307/2047429.

    [8]

    A. Castro and R. Shivaji, Uniqueness of positive solutions for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 267-269.doi: 10.1017/S0308210500013445.

    [9]

    P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1987), 97-121.

    [10]

    D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations, 7 (1970), 217-226.

    [11]

    E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 439-452.doi: 10.1112/plms/s3-53.3.429.

    [12]

    E. N. Dancer and J. Shi, Uniqueness and nonexistence of positive solutions to semipositone problems, Bull. London Math. Soc., 38 (2006), 1033-1044.doi: 10.1112/S0024609306018984.

    [13]

    P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion, Nonlinear Anal. Real World Appl., 15 (2014), 51-57.doi: 10.1016/j.nonrwa.2013.05.005.

    [14]

    F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221.doi: 10.1512/iumj.1982.31.31019.

    [15]

    H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16 (1967), 1361-1376.

    [16]

    E. Ko, E. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain, Discrete Contin. Dyn. Syst, Ser. A, 33 (2013), 5153-5166.doi: 10.3934/dcds.2013.33.5153.

    [17]

    C. Maya and R. Shivaji, Multiple positive solutions for a class of semilinear elliptic boundary value problems, Nonlinear Anal. TMA, 38 (1999), 497-504.doi: 10.1016/S0362-546X(98)00211-9.

    [18]

    S. Oruganti, J. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting. I. Steady states, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.doi: 10.1090/S0002-9947-02-03005-2.

    [19]

    P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J., 23 (1973/74), 174-186.

    [20]

    L. Sankar, S. Sasi and R. Shivaji, Semipositone problems with falling zeros on exterior domains, J. Math. Anal. Appl., 401 (2013), 146-153.doi: 10.1016/j.jmaa.2012.11.031.

    [21]

    R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear analysis and applications, Lecture Notes in Pure and Appl. Math., 109 (1987), 561-566.

    [22]

    R. Shivaji, Uniqueness results for a class of positone problems, Nonlinear Anal. TMA, 7 (1983), 223-230.doi: 10.1016/0362-546X(83)90084-6.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(129) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return