Existence of Ground States for Kirchhoff-Type Problems with General Potentials
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Existence of Ground States for Kirchhoff-Type Problems with General Potentials Fuli He1 · Dongdong Qin1 · Xianhua Tang1 Received: 11 August 2020 / Accepted: 13 October 2020 © Mathematica Josephina, Inc. 2020
Abstract In this paper, we consider the following Kirchhoff-type problem:
− a + b R3 |∇u|2 dx u + V (x)u = f (u), x ∈ R3 ; u ∈ H 1 (R3 ),
where a, b > 0, V ∈ C(R3 , R) and f ∈ C(R, R). Using variational method and some new analytical techniques, we show the existence of ground state solutions for the above problem. Assumptions imposed on the potential V and the nonlinearity f are general, and they are satisfied by several functions. Our results generalize and improve the ones obtained recently in [Li and Ye, J. Differential Equations (2014)], [Tang and Chen, Calc. Var. Partial Differential Equations (2017)], [Guo, J. Differential Equations (2015)] and some other related literature. Keywords Kirchhoff-type problem · Ground states · General potentials · Variational method Mathematics Subject Classification 35J20 · 35J65
B
Dongdong Qin [email protected] Fuli He [email protected] Xianhua Tang [email protected]
1
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, People’s Republic of China
123
F. He et al.
1 Introduction In this paper, we study the existence of ground state solutions for the following Kirchhoff-type problem:
− a + b R3 |∇u|2 dx u + V (x)u = f (u), x ∈ R3 ; u ∈ H 1 (R3 ),
(1.1)
where a, b > 0, V : R3 → R and f : R → R satisfy the following basic assumptions: (V1) V ∈ C(R3 , R) and V := lim inf |y|→∞ V (y) ≥ V (x) for all x ∈ R3 ; ∞ R3
(V2) inf u∈H 1 (R3 )\{0}
a|∇u|2 +V (x)u 2 dx 2 R3 u dx
:= ζ0 > 0;
(F1) f ∈ C(R, R) and there exists a constant C0 > 0 such that | f (t)| ≤ C0 1 + |t|5 ,
∀ t ∈ R;
(F2) f (t) = o(t) as t → 0 and | f (t)| = o |t|5 as |t| → +∞. The ground state solution means that a nontrivial solution has minimal energy among the energy of all nontrivial solutions of (1.1), that is the least energy solution. As well known, under the above assumptions, a weak solution to (1.1) corresponds to a critical point of the energy functional I(u) =
1 2
−
2 b a|∇u|2 + V (x)u 2 dx + |∇u|2 dx 4 R3 R3
R3
F(u)dx, u ∈ H 1 (R3 ),
(1.2)
t here and in the sequel, F(t) := 0 f (s)ds. Under various assumptions on the potential V and the nonlinearity f , problem like or similar to (1.1) has been wildly investigated in the literature via variational methods, see, for instance, the existence of positive solutions [1,9,11,22,24,28], ground state solutions [2,3,8,14,22,33], multiple or infinitely many solutions [6,16,21,37,39, 40], sing-changing solutions [10,30,32,34], semiclassical solutions and asymptotic behavior [16,26,37] and some other related results [4,5,17–19,23,27,29,31,35]. When the following 4-superlinear condition is satisfied, (SF) lim|t|→∞
F(t) t4
= ∞.
The mountain-pass theorem and Nehari manifold method were well applied in the literature in order to find ground state solutions of (1.1),
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