Multiple Solutions For An Elliptic Boundary Value Problem Involving An Inhomogeneous Term

In this thesis, we will be concerned with the following semilinear elliptic boundary vaue problem with an inhomogeneous term and a critical term:where N≥3, 2^* =(?) is the Sobolev critical exponent, D^1,2(R^N) is the completion of C₀^∞(R^N) with respect to the norm || ? || = (∫_R^N |(?)u|² dx)^1/2,θ> 0 is a parameter, 0≤α(x)∈L^?(R^N)∩L^∞(R^N),0 < h(x)∈L^?(R^N).By using the variational methods, such as the critical point theory, we prove that (P_θ) has at least three nonzero solutions for sufficiently smallθ.Explicitly, we first obtain two nonzero solutions of the problem (P_θ) by the use of the Ljusternik-Schnirelman category mrthod;next, we prove the existence of an extremum point for the functional corresponding to the problem (P_θ) in an open ball in D^1,2(R^N), hence the existence of a nonzero solution of the problem (P_θ); finally we prove that the above three solutions are indeed different ifθis small enough,and consider the convergence behavior of the third solution.