Changing Solutions For A Class Of Quasilinear Elliptic Equations

In this paper, we study the number of nodal domains of nodal solutions for p-Laplacian equations using minimax methods.LetΩ(?) R^N be a bounded domain with smooth boundary. We consider the quasilinear elliptic equationwhere p > 1 andis the p-Laplacian of u.The aim of this paper is to estimate the number of nodal domains of nodal solutions of problem (1) with f being subcritical and f(x, t)/ |t|^p-2 being superlinear at t =∞. In order to formulate the hypotheses, we set h_t :=|t|^p-2 t and p^* = Np/(N - p) for p < N, p^* :=∞for p≥N. Letλ₁ be the first eigenvalue of -â–³_pu =λh_p(u), u∈X = W₀^1,p(Ω).(H₀) f∈C(Ω×R, R) and f(x, 0) = 0;(H₁) If 1< p < N, thenlim(?) f(x,t)/|t|^p*-1=0 uniformly in x∈Ω;if p = N thenlim(?) ln|f(x,t)|/|t|^N/(N-1)=0 uniformly in x∈Ω;(H₂) There existμ> p such that(H₃) lim(?)sup |f(x,t)|/|t|^p-1<λ₁, uniformly in x∈Ω;(H₄) f is odd in t, that is, f(x, -t) = -f(x,t); (H₅) For every x∈R^N, the function tâ†'f(x,t)/|t|^p-1 is nondecreasing.Under the assumptions above, T. Bartsch, Z. Liu, and T. Weth in [5] proved that the problem (1) has infinitely many nodal solutions. The purpose of this paper is to estimate the number of nodal domains of those nodal solutions. The main result of this paper on problem (1) is the following.Theorem: Assume (H₀)-(H₅) hold. Then problem (1) has a sequence {±u_n}_n≥1 of nodal solutions such that u_n has at most n + 1 nodal domains.