Ground state solutions to a class of critical Schrödinger problem

Abstract We consider the following critical nonlocal Schrödinger problem with general nonlinearities − open="(" close=")"> ε 2 a + b ∫ R 3 | mathvariant="normal">∇ u class="MJX-tex-mathit" mathvariant="italic">Δ V stretchy="false">( x stretchy="false">) = f 5 , ∈ H 1 $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$ ( SK ε ) and study existence of semiclassical ground state solutions Nehari-Pohožaev type to ), where f u may behave like | q –2 for ∈ (2, 4] which is seldom studied. With some decay assumption on V , we establish an result improves exiting works only handle (4, 6). monotonicity condition also get a solution v̄ analysis its concentrating behaviour around global minimum x as → 0. Our results extend related works.