A monotone property of the ground state energy to the scalar field equation and applications

We consider the ground state energy m ( p , λ ) for the classical nonlinear scalar field equation − Δ u + λ u = | u | p − 2 u , x ∈ R N , where N ⩾ 3 , λ > 0 , 2 < p < 2 ∗and2 ∗ = 2 N N − 2is the critical Sobolev exponent. It is well known that the equation has a unique ground state U p , λinH 1 ( R N )such thatU p , λ( 0 ) = max x ∈ R NU p , λ( x ) . In this paper, we uncover a monotone property of m ( p , λ ) in terms of p : m ( q , λ ) increases in q ∈ ( 2 , p ) when 0 < λ < ( U p , 1( 0 ) ) 2 − pis fixed and m ( q , λ ) decreases in q ∈ ( p , 2 ∗ ) when λ > e p − 2 p2 N − ( N − 2 ) p 2 pis fixed. For applications, we establish the existence and the concentration behavior of the least energy solutions (positive or sign changing) for the singularly perturbed elliptic equation with a variable exponent p ( x ) : − ε 2 Δ u + λ u = | u | p ( x ) − 2 u , u ∈ H 1 ( R N ) . Our results show that the concentration behavior of the least energy solutions of the above equation is quite different from that of the semi‐classical states of nonlinear Schrödinger equations with p being a constant: − ε 2 Δ u + V ( x ) u = | u | p − 2 u . Roughly speaking, for small λ , the least energy solution concentrates at the global minimum point of p ( x ) , and for large λ , the least energy solution concentrates at the global maximum point of p ( x ) . This is a striking new phenomenon which, to the best of our knowledge, has not been studied before.