The Existence Of A Positive Solution For A Class Of Elliptic Equations With Discontinuous Nonlinearities

In the present paper, our main purpose is to study the existence of a positive solution for a class of elliptic equations with discontinuous nonlinearitieswhere r = |x|,N≥3,H_r¹(R^N)={u∈H¹(R_N) | u(x)=u(|x|)},f(x,u):R_N×R→R is a locally bounded measurable function, f(x,u)=f(|x|,u)→q(x)> 0, as u→+∞, q(x)=Constant or q(x)∈L^∞(R^N). Because the nonlinear term f(x,u)u is discontinuous, so the corresponding functional is not Frechet differentiable. Therefore, the problem (p) is belong to DNDE(differential equations with discontinuous nonlinearities). Moreover, the nonlinear term f(x,u)u here no longer satisfies the usual Condition (AR), that is, for someθ>0, M > 0,0≤F(x,u)=(?)₀^u f(x,s)sds≤(?)f(x,u)u²,(?)|u|>M. The condition is important in using the Mountain Pass Theorem. On the one hand, this condition is used to find a point such that the corresponding functional is nonpositive at this point; on the other hand, this condition ensure the (C)_c sequence obtained by Mountain Pass Theorem is bounded. In this paper, we use the DNDE theory which was published in J. Math. Anal. by Academician Chang in 1981. In order to solve the problem that f(x,u)u does not satisfies the usual Condition (AR), we let f(x,u) satisfy some suitable conditions. Using the Mountain Pass Lemma obtained by Professor Wu, we prove the existence of a positive solution for problem (p). This paper improve the results of semilinear elliptic equations in R^N which were obtained by Professor Zhou in 1998.