Existence and Concentration of Solutions for a Class of Elliptic Kirchhoff–Schrödinger Equations with Subcritical and Critical Growth

Abstract

This study focuses on the existence and concentration of ground state solutions for a class of fractional Kirchhoff–Schrödinger equations. We first study the problem

$$\left\{ \begin{array}{ll} M ([u]^{2}_{s} + \int_{\mathbb{R}^{N}} V(x)u^{2}) ((-{\Delta})^{s}u + V (x)u) = \bar{c}u + f(u)\, {\rm in}\,\, \mathbb{R}^N,\\ u > 0, u\, {\in} \, {H}^{s} (\mathbb{R}^N),\end{array} \right.$$

where $s \in (0,1), N > 2s, [\cdot]_s$ is the Gagliardo semi-norm, $\bar{c}$ is a suitable constant,M is a non-degenerate continuous Kirchhoff function that behaves like $t^{\alpha}, V(x) = {\lambda}a(x) + 1, {\rm with}\,\, a(x) \geq 0$ and a is identically zero on the bounded set ${\Omega}_{\Upsilon}$ , 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

$$\left\{\begin{array}{ll} M ([u]^2_s+\int_{\mathbb{R}^N}V(x)u^2) ((-\Delta)^s u + V(x)u)= h(x)u + u^{2^*_s -1} \ {\rm in} \ \mathbb{R}^N,\\ u>0, \quad u\in H^s (\mathbb{R}^N), \end{array}\right.$$

if the value of $\lambda$ is large enough and b₀ is small or m₀ is large.