Multiplicity and concentration of normalized solutions to p-Laplacian equations

In this paper, we study a type of p-Laplacian equation -Delta(p)u=lambda|u|(p-2)u+|u|(q-2)u, x is an element of R-N, with prescribed mass (integral(RN)|u|p)(1/p)=c>0,where 1<p<q<p(& lowast;):=(pN)/(N-p),p<N,lambda is an element of R is a Lagrange multiplier. Firstly, we prove the existence of normalized solutionsto p-Laplacian equations and provide accurate descriptions; secondly, we discuss the existence of ground states; finally, westudy the radial symmetry of normalized solutions in the mass supercritical case. Besides, we also study normalized solutionsto p-Laplacian equation with a potential function V(x) -Delta(p)u+V(x)|u|(p-2)u=lambda|u|(p-2)u+|u|(q-2)u, x is an element of R-N, under different assumptions onqand the constraint normc, we prove the existence, nonexistence, concentration phenomenonand exponential decay of normalized solutions.