A sign-changing solution for elliptic equation with sublinear term at origin

Abstract

It is proved that the semilinear elliptic problem with zero boundary value --Au=au--lu [q ]u has a changing sign solution, as qE (0,1) and 2~).z, where J, z is the second eigenvalue of the operator --A in the space H~(I2).This note is a sequel of paper [1]. We study the problem --Au = 2u -- [ulq-lu, in /2,where~QCR n is a bounded domain with smooth boundary~and the constant qE (0,1).In [-1], in a general case, we prove that the functional corresponding to the problem (1) verifies the P.S. condition and the geometry conditions of Mountain Pass Lemma.Then the problem (1) has at least one nontrivial solution by the Mountain Pass Lemma.Moreover, we can get one non-negative solution and one non-positive solution by the usual truncation technique (see [2]). We now show that there is also a changing sign solution for (1). There are many papers on this kind problem, such as [3,4-] and the references there in. We would like to mention that in [-5-], Struwe proved the existence of at least three nontrivial solutions for problem (1) as q~l.Let E=H~o ([2) equipped with the usual norm II u II 2 = [ 1~7ul z and denote its inner dn product by (., .). Let g(u)= ]u ]q-lu. The corresponding functional of (1) is • (u) = ywhere G is the primitive of the function g. The critical point of the functional~in E will be a weak solution of (1). By the regularity arguments, it in fact is a classical one. Suppose a>a2, define two sets as follows:andwhere ¢] is the normalized eigenfunction corresponding to the first eigenvalue~and ¢2 is the normalized eigenfunction corresponding to the second eigenvalue~2 of the operator --A in E. It is easy to see that~=(~)IU(~)2,whereAs the constant O%p%R, the two sets are linking (see, for example, [6]). Moreover, as p is sufficiently small and R is large enough, there is a constant a~0 such thatand a constant M~0 such thatProof of (5). For any fixed p~l, there is a constant cp~0 such that2 It gives (5) for p sufficiently small.Proof of (6). As 2~22, the second term on the right hand side of (2) is subquadratie and the first term is quadratic and negative definite on the set (~Q)z. In fact, for any given M 0, 1~ElY (t,¢, + t~)I~I~1 (t,~,-F t~¢2)IZ]-F IaG(t,¢,-Ftz~2)< 1 y[(1~)t~-F (1--~)t~]-F c(Itllq+l ÷ lt21~+l)~--M t2"t2--R 2 large enough with t2~0. It gives (6). as lI 2--Following the ideas and arguments in [3,4], we show that there exists an element from which the negative gradient flow tends to a critical point of the functional corresponding to problem (1) and the flownever runs into the set PU (-P),where P is the nonnegative function cone in the space C01(g]). However the nonlinear function lu Iq-lu(0~q~l) is only H61der continuous now, we have to do some smooth approximations first. Then by a result in [4], we get a sequence of changing sign solution. Finally we show that the sequence (precisely, up to a subsequence) has a limit which is also a changing sign function and a solution of (1). Let X-=C0~(~),P~:{uEE;u~O} ,P:PENX: {u~O,uEC~} ,P=P\{0}. Then for u,vE X, we say u~vVe~u--vE P~u~vC:~u-vEP....