The Sign-Changing Solution for Fractional (<i>p</i>,<i>q</i>)-Laplacian Problems Involving Supercritical Exponent

Abstract

In this article, we consider the following fractional (p,q)-Laplacian problem (−Δ)ps1u+(−Δ)qs2u+V(x)(|u|p−2u+|u|q−2u)=f(u)+λ|u|r−2u, where x∈RN, (−Δ)ps1 is the fractional p-Laplacian operator ((−Δ)qs2 is similar), 0s1s21pqNs2, qs2*=NqN−s2q, r≥qs2*, f is a C1 real function and V is a coercive function. By using variational methods, we prove that the above problem admits a sign-changing solution if λ>0 is small.

Keywords

• fractional (<i>p</i>,<i>q</i>)-Laplacian problem
• supercritical exponent
• sign-changing solution