The Sign-Changing Solution for Fractional (p,q)-Laplacian Problems Involving Supercritical Exponent - Jianwen Zhou; Chengwen Gong; Wenbo Wang (Fractal and Fractional, 2024)
In this article, we consider the following fractional <semantics>(p,q)</semantics>-Laplacian problem <semantics><msubsup>(−Δ)ps1</msubsup>u+<msubsup>(−Δ)qs2</msubsup>u+V(x)(|u|p−2u+|u|q−2u)=f(u)+λ|u|r−2u,</semantics> where <semantics>x∈RN</semantics>, <semantics><msubsup>(−Δ)ps1</msubsup></semantics> is the fractional p-Laplacian operator (<semantics><msubsup>(−Δ)qs2</msubsup></semantics> is similar), <semantics>0<s1<s2<1<p<q<Ns2</semantics>, <semantics><msubsup>qs2*</msubsup>=NqN−s2q</semantics>, <semantics>r≥<msubsup>qs2*</msubsup></semantics>, f is a <semantics>C1</semantics> real function and V is a coercive function. By using variational methods, we prove that the above problem admits a sign-changing solution if <semantics>λ>0</semantics> is small.