EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION OF A SEMILINEAR ELLIPTIC EQUATION IN Rn

Abstract

Abstract. We study here semilinear elliptic indefinite-weight problems defined in R n , n > 3. We show that under some conditions, these problems have exactly one positive solution. IntroductionThe aim of the present paper is to prove the existence of one positive solution for a semilinear elliptic equation defined in R n , n > 3.In this research we examine the equation considered in the variational formwhere the potential q is positive and the weight function g changes sign in R n . The nonlinearity / : R n x R -> R is a Caratheodory function, verifying some hypotheses.The research of positive solutions of Equation (I) represents a great physical interest because they are the stationary solutions of the Problem We begin our work by formulating a Maximum principle for a perturbation of Schrodinger's operator and we give a necessary condition for the existence of the solution. In the end, we give a construction of this solution and we prove its uniqueness. NotationFor given measurable function h, we denote by h± = max(±/i,0) the positive and negative paxt of h, i.e. h = h+ -h-.We introduce the following functions in R n <T(A) : the spectrum of the operator A.(1.5) Let UQ, U° be fixed elements from the space V.for a.e. x 6 R n .(1.6) B R = {x e R" : \x\ < R}, B' R = W 1 -B R HypothesesWe suppose that (1.7) 3ci > 0, 3/3 > 1 : 0 < q(x) < c x p 0 {x) for a.e. x G R n .(1.8) 3C 2 >0, 3a > 1 : |j(x)| < c 2 p a (x) for a.e. x G R n .(1.9) / is a Caratheodory function, i.e /(.,u) is measurable for all u G R, /(x,.) is continuous everywhere in R n .(1.10) f(x, 0) > 0 for a.e. x € R". (1.13) The limit lim -= Ooo(x) u-»+oo u exists uniformly for x € R n .(1.14) f(x,u) < aoo(x)u + ij)(x) for all u > 0 and a.e. x in R n .(1.14-a) Ooo(x) < q{x) for a.e. in R n .(1.14-b) Ooo 6 L" "(r). Pi T (1.14-c) 0 < i\> e L 2 _i(IR n ). Pi Recall some fundamental results which we can find in [8]. Under the hypotheses (1.7) and (1.8) the eigenvalue problem|x|-»+oo has a double sequence of eigenvalues. These eigenvalues denoted by \f(q) are given by the Min-Max principles, i.e. It is shown that Af(q), the first positive eigenvalue of Problem (III), is principal ( i.e. has a positive eigenfunction).