NORMALIZED SOLUTION FOR A QUASILINEAR SCHRODINGER EQUATION WITH POTENTIALS AND GENERAL NONLINEARITIES

This paper is concerned with the existence and nonexistence of the minimizer of a L-2-constraint minimization problem: e(alpha) = inf {E(u) : u is an element of H, integral(RN) u(2)vertical bar del u vertical bar(2) < infinity and vertical bar vertical bar u vertical bar vertical bar(2)(2) = alpha}. Here E(u) := 1/2 integral(RN) vertical bar del u vertical bar(2) + V (x)u(2) + integral(RN) u(2)vertical bar del u vertical bar(2) - integral(RN) F(u), V(x) is an element of C(R-N), 0 not equal V (x) <= 0, V (x) -> 0 as vertical bar x vertical bar -> infinity and F(u) = integral(u)(0) f(t)dt. We show that there exists alpha(0) >= 0 such that e(alpha) is attained if alpha > alpha(0) and e(alpha) is not attained if 0 < alpha < alpha(0). Some sufficient conditions for alpha(0) = 0 and alpha(0) > 0 are also discussed.