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Zeitschrift für angewandte Mathematik und Physik

, Volume 49, Issue 6, pp 896–906 | Cite as

Positive solution for a semilinear elliptic equation which is almost linear at infinity

  • H.-S. Zhou
Article

Abstract.

We consider the following elliptic equation: \(- \triangle u +\lambda u =f(x,u)u, x \in {R^N}, u \in {H^1} ({R^N}), N\geq 2,\) where \(\lambda >0, f(x,u)=f(|x|, u)\rightarrow Q(x) >0 \) as \(u\rightarrow +\infty, Q(x) \equiv Const.\;{\rm{or}}\;Q(x)\in {L^{\infty}} ({R^N})\) . The nonlinear term f(x,u)u here no longer satisfies the usual condition: \(F(x,u){\triangle\atop =} \int_0^u f(x,s)sds \leq {1\over{2+\theta}}f(x,u) {u^2}, \rm \,{for} \, \theta >0, \rm \,{and}\, |u| \rm \,\,{is\,large},\) which is important in using the Mountain Pass Theorem. The aim of this paper is to discuss how to use the Mountain Pass Theorem to show the existence of nontrivial solution to the present problem without the above condition.

Key words. Positive solution, semilinear elliptic equation, mountain pass. 

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Copyright information

© Birkhäuser Verlag, Basel, 1998

Authors and Affiliations

  • H.-S. Zhou
    • 1
  1. 1.The Young Scientist Laboratory of Mathematical Physics, Wuhan Institute of Physics and Mathematics, Academia Sinica, P.O. Box 71010, Wuhan 430071, P.R. China, e-mail: hszhou@wipm.whcnc.ac.cnChina

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