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The paper is concerned with a class of linear elliptic equations in a Gauss-Sobolev space setting. They arise from the stationary solutions of the corresponding parabolic equations. For nonhomogeneous elliptic equations, under appropriate conditions, the existence and uniqueness theorem for strong solutions is given. Then it is shown that the associated resolvent operator is compact. Based on this result, we shall prove a Fredholm Alternative theorem for the elliptic equation and adoi:10.31390/cosa.6.3.08 fatcat:ki3j6qkasfbtrijbnzvzjwhsse