Normalized solutions for the p-Laplacian equation with a trapping potential

In this article, we are concerned with normalized solutions for the p -Laplacian equation with a trapping potential and L-r-supercritical growth, where r = p or 2. The solutions correspond to critical points of the underlying energy functional subject to the L-r-norm constraint, namely, integral(RN)|u|(r)dx = c N for given c > 0. When r = p, we show that such problem has a ground state with positive energy for c small enough. When r = 2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.