Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on the Unit Ball

Abstract

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u,\,|\nabla u|)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$

where \(\Omega =\{x\in \mathbb {R}^N:\;|x|<1\}\), \(N\ge 2\), \(f:[0,\,1]\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}\) are continuous, \(\mathbb {R}^+=[0,\,\infty )\). Under some inequality conditions, the existence results of positive radial solution are obtained. The proofs of the main results are based on the method of lower and upper solutions and truncating function technique.

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References

  1. 1.

    Esteban, M.J.: Multiple solutions of semilinear elliptic problems in a ball. J. Differ. Equ. 57, 112–137 (1985)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Castro, A., Kurepa, A.: Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball. Proc. Am. Math. Soc. 101, 57–64 (1987)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Grossi, M.: Radial solutions for the Brezis-Nirenberg problem involving large nonlinearities. J. Funct. Anal. 254, 2995–3036 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Korman, P.: Global solution curves for self-similar equations. J. Differ. Equ. 257, 2543–2564 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Korman, P.: On the multiplicity of solutions of semilinear equations. Math. Nachr. 229, 119–127 (2001)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Lin, S.S., Pai, F.M.: Existence and multiplicity of positive radial solutions for semilinear elliptic equations in annular domains. SIAM J. Math. Anal. 22, 1500–1515 (1991)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Arcoya, D.: Positive solutions for semilinear Dirichlet problems in an annulus. J. Differ. Equ. 94, 217–227 (1991)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Wang, H.: On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differ. Equ. 109, 1–7 (1994)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Stanczy, R.: Decaying solutions for sublinear elliptic equations in exterior domains. Topol. Methods Nonlinear Anal. 14, 363–370 (1999)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Butler, D., Ko, E., Shivaji, R.: Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete Contin. Dyn. Syst. Ser. S 7, 1181–1191 (2014)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Li, Y., Zhang, H.: Existence of positive radial solutions for the elliptic equations on an exterior domain. Ann. Polon. Math. 116, 67–78 (2016)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Li, Y.: Positive radial solutions for elliptic equations with nonlinear gradient terms in an annulus. Complex Var. Elliptic Equ. 63, 171–187 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Li, Y., Ding, Y., Ibrahim, E.: Positive radial solutions for elliptic equations with nonlinear gradient terms on an exterior domain, Mediterr. J. Math. 2018 15 (2018), Art. 83, 19 pp

  14. 14.

    Ruiz, D.: A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differential Equations 199, 96–114 (2004)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Faria, L.F.O., Miyagaki, O.H., Motreanu, D., Tanaka, M.: Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient. Nonlinear Anal. 96, 154–166 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)

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Correspondence to Yongxiang Li.

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Research was supported by NNSFs of China (11661071, 11761063).

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Li, Y. Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on the Unit Ball. Mediterr. J. Math. 17, 176 (2020). https://doi.org/10.1007/s00009-020-01615-2

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Mathematic Subject Classification

  • 35J25
  • 35J60
  • 47H11
  • 47N20

Keywords

  • Elliptic equation
  • nonlinear gradient term
  • positive radial solution
  • lower and upper solutions