Sign-changing solution and ground state solution for a class of ( p , q ) $(p,q)$ -Laplacian equations with nonlocal terms on R N $\mathbb{R}^{N}$

Rui Li,Zhanping Liang

doi:10.1186/s13661-016-0624-5

Abstract

In the paper, we investigate the least energy sign-changing solution and the ground state solution of a class of $(p,q)$ -Laplacian equations with nonlocal terms on $\mathbb{R}^{N}$ . Applying the constraint variational method, the quantitative deformation lemma, and topological degree theory, we see that the equation has one least energy sign-changing solution u. Moreover, we regard c, d as parameters and give a convergence property of such a solution $u_{c,d}$ as $(c,d)\to 0$ . Finally, using the Lagrange multiplier method, we obtain a ground state solution of the equation and show that the energy of u is strictly larger than two times the ground state energy.

Highlights

In this paper, we discuss the existence of a least energy sign-changing solution and a ground state solution of the following equation:– a + c |∇u|p pu – b + d |∇u|q qu + h(x)|u|p– u + g(x)|u|q– u RN= f (u), x ∈ RN, ( . )where ≤ q < p < q∗, N < p, m = div(|∇u|m– ∇u) is the m-Laplacian operator, m∗ = ∞ for N ≤ m, and m∗ = Nm/(N – m) for N > m. a, b are positive constants, c, d ≥
Using the Lagrange multiplier method, we obtain a ground state solution of the equation and show that the energy of u is strictly larger than two times the ground state energy
1 Introduction In this paper, we discuss the existence of a least energy sign-changing solution and a ground state solution of the following equation:

Summary

Introduction

We discuss the existence of a least energy sign-changing solution and a ground state solution of the following equation:. For any sequence {(cn, dn)} with cn, dn ≥ , as (cn, dn) → , there exists a subsequence, still denoted by {(cn, dn)}, such that ucn,dn → u in W , and u is a least energy sign-changing solution of equation ). Because the embedding W ,m( ) → Ls( ) is continuous if s ∈ [ , m∗] and compact if s ∈ [ , m∗), we find solutions in the space W ,p( ) ∩ W ,q( ), and can obtain the same conclusions as Theorems .

Then we have lim

Then we get

Then un u