In this paper, we are concerned with the following class of elliptic problems:where 2 = 2N/（N-2） is the critical Sobolev exponent, 2 < q < 2 , 0≤μ < μˉ=（N-2）24 , a（x）, k（x） ∈ C（RN ）. Through a compactness analysis of the functional corresponding to the problems , we obtain the existence of positive solutions for this problem under certain assumptions on a（x） and k（x）.
In this article, the authors study the exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction by hyperbolic tangent function expansion method, hyperbolic secant expansion method, and Jacobi elliptic function ex- pansion method. They obtain more exact traveling wave solutions including trigonometric function solutions, rational function solutions, and more generally solitary waves, which are called classical bright soliton, W-shaped soliton, and M-shaped soliton.
