Some new results on semilinear elliptic equations

This thesis is divided into three parts.; In the first part, some general results on the stability of droplets with zero contact angle for thin film type equations including those satisfying the power law are established. Instability of droplets with non-zero contact angle and configuration of droplets is also obtained. As applications, finite or infinite time touchdown of solution of the thin film type equations and an asymptotic stability result for droplets with zero contact angle are established.; In the second part, steady states including droplet shaped solutions of the thin film type equation are studied on the plane. Results including the existence of radial droplets with zero contact angles and the radial symmetry of droplets with zero contact angles are established.; In the third part, we study the scalar curvature equation on $S2$. We give two explicit constructions of the scalar curvature functions by perturbing constant with suitable spherical harmonic polynomials. Multiplicity results for the corresponding scalar curvature equations are proved through the calculation of the topological degree of the associated map G for each scalar curvature functions.