Existence and multiplicity of solutions for fractional Schrödinger–Kirchhoff equations with Trudinger–Moser nonlinearity

Abstract

We study the existence and multiplicity of solutions for a class of fractional Schrödinger–Kirchhoff type equations with the Trudinger–Moser nonlinearity. More precisely, we consider M(‖u‖N∕s)(−Δ)N∕ssu+V(x)|u|Ns−1u=f(x,u)+λh(x)|u|p−2uinRN,‖u‖=∬R2N|u(x)−u(y)|N∕s|x−y|2Ndxdy+∫RNV(x)|u|N∕sdxs∕N,where M:[0,∞]→[0,∞) is a continuous function, s∈(0,1), N≥2, λ>0 is a parameter, 1<p<∞, (−Δ)N∕ss is the fractional N∕s-Laplacian, V:RN→(0,∞) is a continuous function, f:RN×R→R is a continuous function, and h:RN→[0,∞) is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when f satisfies exponential growth conditions and λ is large enough, and we prove that the solution converges to zero in WVs,N∕s(RN) as λ→∞. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when λ is small enough, and we show that the solution converges to zero in WVs,N∕s(RN) as λ→0. Furthermore, using the genus theory, infinitely many solutions are obtained when M is a special function and λ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function M(0)=0.