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

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 M([u]s2+∫RNV(x)u2)((-Δ)su+V(x)u)=c¯u+f(u)inRN,u>0,u∈Hs(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{array}{ll} M ([u]^{2}_{s} + \int_{\mathbb{R}^{N}} V(x)u^{2}) ((-{\Delta})^{s}u + V (x)u) = \bar{c}u + f(u)\, {\rm in}\,\, \mathbb{R}^N,\\ u > 0, u\, {\in} \, {H}^{s} (\mathbb{R}^N),\end{array} \right.$$\end{document} where s∈(0,1),N>2s,[·]s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1), N > 2s, [\cdot]_s$$\end{document} is the Gagliardo semi-norm, c¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{c}$$\end{document} is a suitable constant,M is a non-degenerate continuous Kirchhoff function that behaves like tα,V(x)=λa(x)+1,witha(x)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{\alpha}, V(x) = {\lambda}a(x) + 1, {\rm with}\,\, a(x) \geq 0$$\end{document} and a is identically zero on the bounded set ΩΥ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega}_{\Upsilon}$$\end{document} , 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 λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}. Moreover, assuming that M(t)=m0+b0tα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(t) = m_{0} + b_{0}t^{\alpha}$$\end{document} 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]s2+∫RNV(x)u2)((-Δ)su+V(x)u)=h(x)u+u2s∗-1inRN,u>0,u∈Hs(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll} M ([u]^2_s+\int_{\mathbb{R}^N}V(x)u^2) ((-\Delta)^s u + V(x)u)= h(x)u + u^{2^*_s -1} \ {\rm in} \ \mathbb{R}^N,\\ u>0, \quad u\in H^s (\mathbb{R}^N), \end{array}\right.$$\end{document} if the value of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} is large enough and b0 is small or m0 is large.