MULTIPLE SOLUTIONS FOR LOGARITHMIC SCHRODINGER EQUATIONS WITH CRITICAL GROWTH

摘要

In this paper, we establish the existence of positive ground state solution and least energy sign-changing solution for the following logarithmic Schrodinger equation -Delta u + V(x)u = u log u(2) + vertical bar u vertical bar(2)*(-2) u, x is an element of R-N. It is known that the corresponding variational functional is not well defined in H-1(R-N). Via direction derivative and constrained minimization method, we first prove the existence of positive ground state solution and least energy sign-changing solution for the following subcritical problem -Delta u + V(x)u = u log u(2) + vertical bar u vertical bar(p-2) u, x is an element of R-N. Then, we analyze the behavior of solutions for subcritical problem and pass the limit as the exponent p approaches to 2*.