In this thesis, we investigate entire solutions of the quasilinear equationrn(y) __u = h(x; u)rnwhere __u := div(_(jruj)ru): Under suitable assumptions on the right-hand sidernwe will show the existence of in_nitely many positive solutions that are bounded andrnbounded away from zero in RN: All these solutions converge to a positive constant atrnin_nity. The analysis that leads to these results is based on a _xed-point theorem attributedrnto Shcauder-Tychono_.rnUnder appropriate assumptions on h(x; t), we will also study ground state solutions ofrn(y) whose asymptotic behavior at in_nity is the same as a fundamental solution of thern_-Laplacian operator __: Ground state solutions are positive solutions that decay tornzero at in_nity.rnAn investigation of positive solutions of (y) that converge to prescribed positive constantsrnat in_nity will be considered when the right-hand side in (y) assumes the formrnh(x; t) = a(x)f(t): After establishing a general result on the construction of positivernsolutions that converge to positive constants, we will present simple su_cient conditionsrnthat apply to a wide class of continuous functions f : R ! R so that the equationrn__u = a(x)f(u) admits positive solutions that converge to prescribed positive constantsrnat in_nity.rnWe will also study Cauchy-Liuoville type problems associated with the equation __u =rnf(u) in RN: More speci_cally, we will study su_cient conditions on f : R ! R in orderrnthat the equationrn__u = f(u)rnadmits only constant positive solution provided that f has at least one real root. Ourrnresult in this direction can best be illustrated by taking _(t) = ptp2 + qtq2 for somern1 < p < q which leads to the so called (p; q)-Laplacian, _(p;q)u := _pu + _qu: