Existence and Concentration of Solutions for a Class of Elliptic Kirchhoff-Schrodinger Equations with Subcritical and Critical Growth

This study focuses on the existence and concentration of ground state solutions for a class of fractional Kirchhoff-Schrodinger equations. We first study the problem {M(vertical bar u vertical bar(2)(s)+integral V-RN(x)u(2))((-Delta)(s)u+V(x)u)=(c) over baru+f(u)in R-N, u>0, u is an element of H-s(R-N), where s is an element of (0, 1), N > 2s, [.] s is the Gagliardo semi-norm, (c) over bar is a suitable constant M is a non-degenerate continuous Kirchhoff function that behaves like t(alpha), V(x) = lambda a(x) + 1, with a(x) >= 0 and a is identically zero on the bounded set Omega(Gamma), and f denotes a continuous nonlinearity with subcritical growth at infinity. The proof relies on penalization arguments and variational methods to obtain the existence of a solution with minimal energy for a large value of lambda. Moreover, assuming that M(t)=m(0)+b(0)t(alpha) and utilizing the same techniques combined with a concentration-compactness lemma, we can establish the existence and concentration of solutions for the problem {M([u](s)(2)+ integral V-RN(x)u(2))((-Delta)(s)u+V(x)u)=h(x)u+us(2)*-1 in R-N, u>0,u is an element of H-s(R-N), if the value of lambda is large enough and b(0) is small or m(0) is large.