ON THE EXISTENCE, UNIQUENESS AND POSITIVITY PRESERVING PROPERTY FOR ENTIRE WEAK SOLUTION TO SEMILINEAR LAPLACIAN AND BIHARMONIC PROBLEMS WITH SINGULAR TERMS

In this paper we study on the existence, uniqueness and positivity preserving property for weaksolution of the following problem(−1)∆u=u|x|2+b(x)f(u) +g(x);inRN:HereN≥2+1,= 1;2. For existence and uniqueness we assume thatb∈L∞(RN) andb(x)≥0a.e. inRN,g∈L2NN+2(RN),f:R→Ris continuous and|f(u)| ≤c|u|N+2N2for somec >0,uf(u)≤0 for allu∈R, (u−v)(f(u)−f(v))≤0 for allu; v∈Rand <ΛN; = 42(N+24)2(N24).We also address a positivity preserving property under some hypotheses, namely the property thatg(x)≥0 a.e. inRNandg̸≡0 a.e. inRNimplies that the weak solution is positive a.e. inRN.