Abstract:
The major theme of this thesis is the study of multiplicity results for fractional
elliptic equations and the system of equations. The thesis is mainly divided
into three parts. In the first part, the existence and multiplicity of positive solutions
for perturbed nonlocal scalar field equation with subcritical nonlinearity
and nonhomogeneous terms have been studied, and the global compactness
result has been proved. The second part deals with Fractional Hardy-Sobolev
equation involving critical nonlinearity and nonhomogeneous term. The existence
of at least two positive solutions is obtained provided the corresponding
nonhomogeneous terms are small enough in the dual space norm. Besides
the profile decomposition for the Palais-Smale sequences of the associated
energy functional has been accomplished. The third part comprises of the study
of nonhomogeneous weakly coupled nonlocal system of equations with critical
and subcritical nonlinearities. Firstly, the existence of a positive solution
exploiting the local geometry of the associated functional near the origin
is achieved. Then proving the global compactness result (which gives the
complete description of the associated Palais Smale sequences for the system),
the existence of two positive solutions is obtained under some suitable
conditions on the nonhomogeneous terms. In addition, considering the corresponding
homogeneous system, uniqueness for the ground state solution has
been proved.
