A Class Of Quasilinear Elliptic Equations With Weights

In this paper, we consider the quasilinear elliptic equationwhere ft is a bounded domain in RN(N 3) with smooth boundary 3ft, pu = div (| u|p-2 u) is the p-Laplacian, 1 < p < N, u = max{ u,0}, u = u+ - u-, the weight function a(x) > 0 a.e. in 0 and a(x) Lr( )(r N/p). If f(x,u) = 0, then Equation (A1.1) turns to the equationThe Fu ik spectrum of the p-Laplacian on W01,p( ) is defined as the set p of those ( , ) R2 such that Equation (A1.2) has a nontrivial solution. If = in (A1.2), then the Pucik spectrum Ep is the subset of R1 with the form of (A, A) € M2, where A is the eigenvalue of problemBy [9], there exists a sequence of eigenvalues of (A1.3) {An} : 0 < 1 < 2 < < k , and k - (k - ).In our paper, first, we study the existence of the first nontrivial curve C in p and it's some properties by using methods of [10] and a variant of the Mountain-pass Lemma given by [16]. Then under suitable conditions for f(x,u), we consider the solvability of (Al.l) in the cases of ( , ) p and ( , ) p by using the saddle point theorem and other variational methods.Our results can be included in the following:Theorem A1.1 The first nontrivial Fu ik spectrum curve C in p exists, moreover C is continuous and strictly decreasing.Theorem A1.2 If f(x,u) satisfies the following hypotheses: Then Equation (Al.l) has a nontrivial solution.Theorem A1.3 Let 6 > 0 be such that If f(x, u) satisfiesAnd suppose thatThen Equation (A1.1) has at least one weak solution.Theorem A1.4 If( , ) Cl. If a(x) satisfies(a) There exists a constant k > 0, such that a(x) > k, and a(x) Lr( ), f(x,u) satisfies (/) of the theorem A1.3, and assume thatwhere C[(l 2) are nontrivial Fucik spectrum curves, E( ) is the set of nontrivial solutions of (A1.2) with (, ) Cl p; F( ) and F( ) are defined as in Theorem Al.3. Then Equation (Al.l) has at least one nontrivial solution.