Browsing by Author "Mohammed, Ahmed (Professor)"

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    On Entire Solutions of Quasilinear Elliptic Equations
    (Addis Ababa University, 2017-06-28) Araya, Ataklti; Mohammed, Ahmed (Professor)
    In this thesis, we investigate entire solutions of the quasilinear equation (y) __u = h(x; u) where __u := div(_(jruj)ru): Under suitable assumptions on the right-hand side we will show the existence of in_nitely many positive solutions that are bounded and bounded away from zero in RN: All these solutions converge to a positive constant at in_nity. The analysis that leads to these results is based on a _xed-point theorem attributed to Shcauder-Tychono_. Under appropriate assumptions on h(x; t), we will also study ground state solutions of (y) whose asymptotic behavior at in_nity is the same as a fundamental solution of the _-Laplacian operator __: Ground state solutions are positive solutions that decay to zero at in_nity. An investigation of positive solutions of (y) that converge to prescribed positive constants at in_nity will be considered when the right-hand side in (y) assumes the form h(x; t) = a(x)f(t): After establishing a general result on the construction of positive solutions that converge to positive constants, we will present simple su_cient conditions that apply to a wide class of continuous functions f : R ! R so that the equation __u = a(x)f(u) admits positive solutions that converge to prescribed positive constants at in_nity. We will also study Cauchy-Liuoville type problems associated with the equation __u = f(u) in RN: More speci_cally, we will study su_cient conditions on f : R ! R in order that the equation __u = f(u) admits only constant positive solution provided that f has at least one real root. Our result in this direction can best be illustrated by taking _(t) = ptp2 + qtq2 for some 1 < p < q which leads to the so called (p; q)-Laplacian, _(p;q)u := _pu + _qu: