This paper is dedicated to studying the following Kirchhoff-type problem: $$ \textstyle\begin{cases} -m ( \Vert \nabla u \Vert ^{2}_{L^{2}(\mathbb{R} ^{N})} )\Delta u+V(x)u=f(u), & x\in \mathbb{R} ^{N}; \\ u\in H^{1}(\mathbb{R} ^{N}), \end{cases} $$ { − m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) Δ u + V ( x ) u = f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $N=1,2$ N = 1 , 2 , $m:[0,\infty )\rightarrow (0,\infty )$ m : [ 0 , ∞ ) → ( 0 , ∞ ) is a continuous function, $V:\mathbb{R} ^{N}\rightarrow \mathbb{R} $ V : R N → R is differentiable, and $f\in \mathcal{C}(\mathbb{R} ,\mathbb{R} )$ f ∈ C ( R , R ) . We obtain the existence of a ground state solution of Nehari–Pohozaev type and the least energy solution under some assumptions on V, m, and f. Especially, the existence of nonlocal term $m(\|\nabla u\|^{2}_{L^{2}(\mathbb{R} ^{N})})$ m ( ∥ ∇ u ∥ L 2 ( R N ) 2 ) and the lack of Hardy’s inequality and Sobolev’s inequality in low dimension make the problem more complicated. To overcome the above-mentioned difficulties, some new energy inequalities and subtle analyses are introduced.
In this paper, we study the following singularly perturbed Kirchhoff type equation in subcritical casewhereUnder some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of V in the semiclassical limit by applying some new techniques and subtle analyses. Moreover, we use Moser iteration to conquer the difficulty caused by the lack of Sobolev inequality.
In this paper, we dedicate to studying the following semilinear Schrödinger system equation*-Δu+V1(x)u=Fu(x,u,v)amp;mboxin~RN,r-Δv+V2(x)v=Fv(x,u,v)amp;mboxin~RN,ru,v∈H1(RN),endequation* where the potential Vi are periodic in x,i=1,2, the nonlinearity F is allowed super-quadratic at some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.
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