On Entire Solutions of Quasilinear Elliptic Equations
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Date
2017-06-28
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Addis Ababa University
Abstract
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:
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Keywords
On Entire Solutions, of Quasilinear Elliptic