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Positive solution for a semilinear elliptic equation which is almost linear at infinity. (English) Zbl 0916.35036

Summary: We consider the following elliptic equation:

-Δu+λu=f(x,u)u,x N ,uH 1 ( N ),N2,

where λ>0, f(x,u)=f(|x|,u)Q(x)>0 as u+, Q(x)Const. or Q(x)L ( N ). The nonlinear term f(x,u)u here no longer satisfies the usual condition:

F(x,u) 0 u f(x,s)sds1 2+θf(x,u)u 2 ,forθ>0,and|u|islarge,

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.

35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods