Multiple solutions for a class of fractional (p, q)–Laplacian system in RN

Abstract

In this work, the symmetric mountain pass lemma is employed to establish the existence of infinitely many solutions to the fractional (p, q)-Laplacian system: (−Δ)psu+V1(x)|u|p−2u=α−1Fu(x,u,v)+λb1(x)|u|m−2u and (−Δ)qsv+V2(x)|v|q−2v=α−1Fv(x,u,v)+μb2(x)|v|k−2v in RN, where (−Δ)ps and (−Δ)qs are the fractional p and q-Laplacian operators, respectively, and 0 < s < 1 < q ≤ p, sp < N, p<m≤α<qs*=qNN−qs,q<k≤α<qs*. The function F(x,u,v)∈C1(RN×R2) satisfies the condition (A4) below.

Existence of Continuous Eigenvalues for a Class of Parametric Problems Involving the (p,2)$(p,2)$-Laplacian Operator

“…where ∆ s denotes the s-Laplacian operator ∆ s u := ∇ • (|∇u| s−2 ∇u) for s ∈ (1, ∞). Differential equations of type (2) have attracted a great deal of attention in recent years, see, e. g., [4,8,31,32,30,5,6,13,18,3] for scalar equations, and [17,7] for systems. When p = q, equation (2) becomes −∆ p u = g(x, u), which has been studied extensively in the literature.…”