A Class Of Semilinear Elliptic Equations Dirichlet Problem Changing Solutions Its Concentration Phenomenon

In this paper, we study the existence and concentration behavior of a nodal solution to a kind of semilinear elliptic equation with Dirichlet boundary value by using minimax and truncation methods.Let Ω (?) R^N be a domain with smooth boundary. We consider the semilinear elliptic equationwhere V(x) is a Holder continuous function, satisfying(V₂) there exists a bounded domain A compactly contained in Ω such thatand f(s) ∈ C¹(R) satisfies (f₁) f(s) = o(|s|), as s→0. (f₂) there exists 1 < p < 2^* -1 such that(f₃) there exists 2 < θ ≤ p + 1 such thatIt is well known that if f(s)∈ C¹(R) satisfies (f₁) and (f₂), then the functionalis well-defined and I_ε ∈ C²(H, R). The derivatives of I_ε areTherefore, the weak solutions of (P_ε) are exactly the critical points of I_ε. The main result in this paper is in the following:Theorem: Let f(s) ∈ C¹(R) satisfy (f₁) ～ (f₄) and V(x) satisfy (V₁)～ (V₂). Then there exists ε₀ > 0 such that problem (P_ε) possesses a nodal solution u_ε∈ H₀¹(Ω) for every ε ∈ (0, ε₀). Moreover, u_ε has just one positive local maximum point P_ε¹ ∈ A (hence global) and one negative local minimum point P_ε²âˆˆ A (hence global). We also have andwhere M,β are positive constants.Remark: Under the hypotheses of the above theorem and assuming that f(s) is an odd function, CO. Alves and S.H.M. Soares obtained the above result in [3]. In this paper, we get the same conclusion without the oddness of f(s).