作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
In this paper, we study the existence and multiplicity of solutions with a prescribed L-2-norm for a class of nonlinear fractional Choquard equations in Double-struck capital R-N: (-Delta)su-lambda u=(kappa a*|u|p-2u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${( - \Delta )<^>s}u - \lambda u = ({\kappa _a}*|u{|<^>{p - {2_u}}})$$\end{document} where N > 3, s is an element of (0, 1), alpha is an element of (0, N), p is an element of(max{1+a+2sN,2}N+aN-2s)and kappa a(x)=|x|a-N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (max\{ 1 + \frac{{a + 2s}}{N},2\} \frac{{N + a}}{{N - 2s}})and{\kappa _a}(x) = |x{|<^>{a - N}}$$\end{document} considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c > 0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that, we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover, by using a minimax procedure, we prove that for any c > 0, there are infinitely many radial critical points of I restricted on S(c).
作者机构：
[Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
In this paper, we consider the nonlocal Kirchhoff problem -(c(2)a + cb integral(R3 )vertical bar del u vertical bar(2)) Delta u + V(x)u = u(p), u > 0, u is an element of H1(R-3) where a, b > 0, 1 < p < 5 are constants, c > 0 is a parameter. Under some assumptions on V(x), we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
existence of a weak solution;fractional Laplacian;double critical exponents;Hardy term;weighted Morrey space;improved Sobolev inequality
摘要：
In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:
0.1
$${( - \Delta )^s}u - \gamma {u \over {{{\left| x \right|}^{2s}}}} = {{{{\left| u \right|}^{2_s^ * (\beta ) - 2}}u} \over {{{\left| x \right|}^\beta }}} + \left[ {{I_\mu } * {F_\alpha }( \cdot ,u)} \right](x){f_\alpha }(x,u),\;\;\;\;u \in {\dot H^{^s}}({\mathbb{R}^n}),$$
where
$$s \in (0,1),0 \le \alpha ,\beta < 2c < n,\mu \in (0,n),\gamma < {\gamma _H},{I_\mu }(x) = {\left| x \right|^{ - \mu }},{F_\alpha }(x,u) = {{{{\left| {u(x)} \right|}^{2_\mu ^\# (\alpha )}}} \over {{{\left| x \right|}^{{\delta _\mu }(\alpha )}}}},\;{f_\alpha }(x,u) = {{{{\left| {u(x)} \right|}^{2_\mu ^\# (\alpha ) - 2}}u(x)} \over {{{\left| x \right|}^{{\delta _\mu }(\alpha )}}}},2_\mu ^\# (\alpha ) = (1 - {\textstyle{\mu \over {2n}}}) \cdot 2_s^ * (\alpha ),\;{\delta _\mu }(\alpha ) = (1 - {\textstyle{\mu \over {2n}}})\alpha ,\;2_s^ * (\alpha ) = {{2(n - \alpha )} \over {n - 2s}}$$
and
$${\gamma _H} = {4^s}{{{\Gamma ^2}({\textstyle{{n + 2s} \over 4}})} \over {{\Gamma ^2}({\textstyle{{n - 2s} \over 4}})}}.$$
. We show that problem (0.1) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings
0.2
$${\dot H^s}({\mathbb{R}^n})\alpha {L^{2_s^ * (\alpha )}}({\mathbb{R}^n},{\left| y \right|^{ - \alpha }})\alpha {L^{p,{\textstyle{{n - 2s} \over 2}}p + pr}}({\mathbb{R}^n},{\left| y \right|^{ - pr}}),$$
where s ∈ (0, 1), 0 < α < 2s < n, p ∈ [1, 2s*(α)) and
$$r = {\textstyle{\alpha \over {2_s^ * (\alpha )}}}.$$
. We also establish an improved Sobolev inequality,
0.3
$${\left( {\int_{{\mathbb{R}^n}} {{{{{\left| {u(y)} \right|}^{2_s^ * (\alpha )}}} \over {{{\left| y \right|}^\alpha }}}} {\rm{d}}y} \right)^{{\textstyle{1 \over {2_s^ * (\alpha )}}}}} \le C\left\| u \right\|_{{{\dot H}^s}({\mathbb{R}^n})}^\theta \left\| u \right\|_{{L^{p,{\textstyle{{n - 2s} \over 2}}p + pr}}({\mathbb{R}^n},{{\left| y \right|}^{ - pr}})}^{1 - \theta },\;\;\;\;\;\;\forall u \in {\dot H^s}({\mathbb{R}^n}),$$
where
$$s \in (0,1),\;0 \le \alpha < 2s < n,p \in [1,2_s^ * (\alpha )),\;r = {\alpha \over {2_s^ * (\alpha )}},\;0 < \max {\rm{\{ }}{2 \over {2_s^ * (\alpha )}}{\rm{,}}{{2_s^ * - 1} \over {2_s^ * (\alpha )}}{\rm{\} }} < \theta < 1,\;2_s^ * = {{2n} \over {n - 2s}}$$
and C= C(n, s, α) 0 is a constant. Inequality (0.3) is a more general form of Theorem 1 in Palatucci, Pisante [1]. By using the mountain pass lemma along with (0.2) and (0.3), we obtain a nontrivial weak solution to problem (0.1) in a direct way. It is worth pointing out that (0.2) and (0.3) could be applied to simplify the proof of the existence results in [2] and [3].
摘要：
<p style='text-indent:20px;'>In this article, we deal with the existence, qualitative and symmetry properties of normalized solutions to the following nonlinear Schrödinger system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+(x_{1}^{2}+x_{2}^{2})u = \lambda_{1}u+\mu_{1}u^{3}+\beta uv^{2}, &\quad x\in \mathbb{R}^3,\\ -\Delta v+(x_{1}^{2}+x_{2}^{2})v = \lambda_{2}v+\mu_{2}v^{3}+\beta u^{2}v, &\quad x\in \mathbb{R}^3, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \mu_{i}>0 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ i $\end{document}</tex-math></inline-formula> = 1, 2), <inline-formula><tex-math id="M4">\begin{document}$ \beta>0 $\end{document}</tex-math></inline-formula>, and the frequencies <inline-formula><tex-math id="M5">\begin{document}$ \lambda_{1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \lambda_{2} $\end{document}</tex-math></inline-formula> are unknown and appear as Lagrange multipliers. In addition, we study the stability of the corresponding standing waves for the related time-dependent Schrödinger systems. We mainly extend the results in J. Bellazzini et al. (Commun. Math. Phys. 2017), which dealt with mass-supercritical nonlinear Schrödinger equation with partial confinement, to cubic nonlinear Schrödinger systems with partial confinement.</p>
摘要：
In this paper, we revisit the singularly perturbation problem -(epsilon(2)a + epsilon b integral(R3) vertical bar del u vertical bar(2)) Delta u + V(x)u = vertical bar u vertical bar(p-1)u in R-3, (0.1) where a, b, epsilon > 0, 1 < p < 5 are constants and V is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem - (a + b integral(R3) vertical bar del u vertical bar(2)) Delta u + u = vertical bar u vertical bar(p-1)u in R-3. Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for epsilon > 0 sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity. (C) 2019 Elsevier Inc. All rights reserved.
期刊：
Journal of Differential Equations,2019年266(11):7101-7123 ISSN：0022-0396
通讯作者：
Li, Gongbao
作者机构：
[Li, Gongbao; Ye, Hongyu] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Li, Gongbao; Ye, Hongyu] Wuhan Univ Sci & Technol, Coll Sci, Wuhan 430065, Hubei, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Kirchhoff equation;Mass concentration;Constrained minimization;Normalized solutions;Sharp existence
摘要：
In this paper, we study the existence and concentration behavior of minimizers for i(V)(c) = inf (u is an element of Sc) I-V(u), here S-c ={u is an element of H-1(R-N)vertical bar integral(RN) V(x)vertical bar u vertical bar(2) < +infinity, vertical bar u vertical bar(2) = c > 0} and IV(u) =1/2 integral(RN) (a vertical bar del u vertical bar(2)+ V(x)vertical bar u vertical bar(2)) + b/4 (integral(RN)vertical bar del u vertical bar(2))(2) - 1/p integral(RN)vertical bar u vertical bar(p), where N = 1, 2, 3 and a, b > 0 are constants. By the Gagliardo-Nirenberg inequality, we get the sharp existence of global constraint minimizers of i(V) (c) for 2 < p < 2* when V(x) >= 0, V(x) is an element of L-loc(infinity)(R-N) and lim(vertical bar x vertical bar ->+infinity) V(x) = +infinity. For the case p is an element of(2, 2N+8/N)\{4}, we prove that the global constraint minimizers u(c) of iV(c) behave like u(c)(x) approximate to c/vertical bar Q(p)vertical bar(2) (m(c)/c)(n/2) Q(p) (mc/c x - z(c)) , for some z(c) is an element of R-N when c is large, where Q(p) is, up to translations, thw unique positive solution of -N(p-2)/4 Delta Q(p) + 2N-p(N-2)/4 Q(p) = vertical bar Q(p)vertical bar(p-2) Q(p) in R-N and m(c) = (root a(2)D(1)(2)-4bD(2)i(0)(c)+aD(1)/2bD(2))1/2, D-1 = Np-2N-4/2N(p-2) and D-2 = 2N+8-Np/4N(p-2) . (C) 2018 Elsevier Inc. All rights reserved.
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
关键词：
Schrodinger-Kirchhoff type equation;variational methods;multiple positive solutions;concentrating solution;penalization method
摘要：
In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrodinger-Kirchhoff type -epsilon M-p(epsilon(p-N)integral(RN)vertical bar del u vertical bar(p))Delta(p)u + V(x)vertical bar u vertical bar(p-2)u = f(u) in R-N, where Delta(p) is the p-Laplacian operator, 1 < p < N, M : R+ -> R+ and V : R-N -> R+ are continuous functions, epsilon is a positive parameter, and f is a continuous function with subcritical growth. We assume that. V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and LyusternikSchnirelrnann theory, we prove the existence, multiplicity-, arid concentration of solutions for the above equation.
作者机构：
[Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Xiang, Chang-Lin] Yangtze Univ, Sch Informat & Math, Jingzhou 434023, Peoples R China.
期刊：
ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA,2018年43:991-1021 ISSN：1239-629X
通讯作者：
Li, Gongbao
作者机构：
[Jia, Huifang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Li, Gongbao] Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Li, Gongbao] N;Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
关键词：
Concentration behavior;Ground state solution;Schrödinger-Kirchhoff type equations;Variational methods
摘要：
In this paper, we study the following fractional Schrodinger Kirchhoff type problem (Q(epsilon)) {L(epsilon)(s)u = K (x)f (u) in R-3, u is an element of H-s (R-3), where L-epsilon(s) is a nonlocal operator defined by L(epsilon)(s )u = M (1/epsilon(3-2s )integral integral R3x R3 vertical bar u(x)-u(y)vertical bar(2)/vertical bar x-y vertical bar(3+2s)dx dy +1/epsilon(3)integral R3 V(x)u(2)dx)[epsilon(2s) (-Delta)(s) u + V(x ) u], epsilon is a small positive parameter, 3/4 < s < 1 is a fixed constant, the operator (-Delta)(s )is the fractional Laplacian of order s, M, V, K and f are continuous functions. Under proper assumptions on M, V, K and f, we prove the existence and concentration phenomena of solutions of the problem (Q(epsilon)). With minimax theorems and the Ljusternik-Schnirelmann theory, we also obtain multiple solutions of problem (Q(epsilon)) by employing the topology of the set where the potentials V(x) attains its minimum and K(x) attains its maximum.
期刊：
ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA,2017年42(1):405-428 ISSN：1239-629X
通讯作者：
Li, Gongbao
作者机构：
[Li, Gongbao; Luo, Xiao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan, Peoples R China.;[Li, Gongbao; Luo, Xiao] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
摘要：
In this paper, we study the existence and multiplicity of solutions with a prescribed L-2-norm for a class of nonlinear Chern Simons Schrodinger equations in R-2 where To get such solutions we look for critical points of the energy functional on the constraints When p = 4, we prove a sufficient condition for the nonexistence of constrain critical points of I on S-r(c) for certain c and get infinitely many minimizers of I on Sr(8 pi). For the value p epsilon (4, +infinity) considered, the functional I is unbounded from below on Sr(c). By using the constrained minimization method on a suitable submanifold of S-r(c), we prove that for certain c > 0, I has a critical point on Sr(c). After that, we get an H-1-bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on S-r(c) for any c epsilon (0, 4 pi/root p-3).
期刊：
Journal of Mathematical Analysis and Applications,2017年455(2):1559-1578 ISSN：0022-247X
通讯作者：
Li, Gongbao
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
关键词：
Gauged Schr?dinger equation;Least energy sign-changing solutions;Asymptotic behavior
摘要：
<![CDATA[<ce:abstract xmlns:ce="" xmlns="" id="ab0010" view="all" class="author"> <ce:section-title id="st0010">Abstract</ce:section-title> <ce:abstract-sec id="as0010" view="all"> <ce:simple-para id="sp0010" view="all">We study the existence and asymptotic behavior of the least energy sign-changing solutions to a gauged nonlinear Schr?dinger equation<ce:display> <ce:formula id="fm0010"> <mml:math xmlns:mml="" altimg="si1.gif" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="true">{</mml:mo> <mml:mtable> <mml:mtr> <mml:mtd columnalign="left"> <mml:mo>?</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mo stretchy="true" maxsize="3.8ex" minsize="3.8ex">(</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup>
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2016年36(2):731-762 ISSN：1078-0947
通讯作者：
He, Yi
作者机构：
[He, Yi] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[He, Yi] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
会议名称：
2013 Workshop on Variational Problems and Evolution Equations
摘要：
We study the existence, concentration and multiplicity of weak solutions to the
quasilinear Schrödinger equation with critical Sobolev growth
\begin{equation*}
\left\{ \begin{gathered}
- {\varepsilon ^2}\Delta u + V(x)u - {\varepsilon ^2}\Delta (u^2)u = W(x){u^{q - 1}} + {u^{2\cdot{2^*} - 1}} {\text{ in }}{\mathbb{R}^N},\\
u > 0{\text{ in }}{\mathbb{R}^N},\\
\end{gathered} \right.
\end{equation*}
where $\varepsilon $ is a small positive parameter, $N \ge 3$, ${2^ * } = \frac{{2N}}
{{N - 2}}$, $4 < q < 2 \cdot {2^ * }$, $\min V > 0$ and $\inf W > 0$. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials $V(x)$ attains its minimum and $W(x)$ attains its maximum.
期刊：
MATHEMATICAL CONTROL AND RELATED FIELDS,2016年6(4):551-593 ISSN：2156-8472
通讯作者：
Li, Gongbao
作者机构：
[He, Yi] South Cent Univ Nationalities, Sch Math & Stat, Wuhan 430074, Peoples R China.;[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
We are concerned with a class of singularly perturbed quasilinear Schrödinger equations of the following form: \[ - {\varepsilon ^2}\Delta u - {\varepsilon ^2}\Delta ({u^2})u + V(x)u = h(u),{\text{ }}u > 0{\text{ in }}{\mathbb{R}^N}, \] where $\varepsilon $ is a small positive parameter, $N \ge 3$ and the nonlinearity $h$ is of critical growth. We construct a family of positive solutions ${u_\varepsilon } \in {H^1}({\mathbb{R}^N})$ of the above problem which concentrates around local minima of $V$ as $\varepsilon \to 0$ under certain assumptions on $h$. Our result especially solves the above problem in the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(2 < q \le 4,{\text{ }}\lambda > 0)$ and completes the study made in some recent works in the sense that, in those papers only the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(4 < q < 2 \cdot {2^ * },{\text{ }}\lambda > 0)$ was considered. Moreover, our main results extend also the arguments used in Byeon and Jeanjean [14], which deal with Schrödinger equations with subcritical nonlinearities, to the quasilinear Schrödinger equations with critical nonlinearities.
作者机构：
[Li, Gong-bao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gong-bao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
In this paper, we study the existence of multiple solutions to the following nonlinear elliptic boundary value problem of p-Laplacian type: {-Delta(p)u = f(x,u), x is an element of Omega, u = 0, x is an element of partial derivative Omega, where 1 < p < infinity, Omega subset of R-N is a bounded smooth domain, Delta(p)u = div (vertical bar Du vertical bar(p-2)Du) is the p-Laplacian of u and f: Omega x R -> R satisfies lim(vertical bar t vertical bar ->infinity) f(x, t)/vertical bar t vertical bar(p-2)t = l uniformly with respect to x is an element of Omega, and l is not an eigenvalue of -Delta(p) in W-0(1,p)(Omega) but f(x, t) dose not satisfy the Ambrosetti-Rabinowitz condition. Under suitable assumptions on f (x, t), we have proved that (*) has at least four nontrivitial solutions in W-0(1,p) (Omega) by using Nonsmooth Mountain-Pass Theorem under (C)(c) condition. Our main result generalizes a result by N. S. Papageorgiou, E. M. Rocha and V. Staicu in 2008 (Calculus of Variations and Partial Differential Equations, 33: 199-230(2008)) and a result by G. B. Li and H. S. Zhou in 2002 (Journal of the London Mathematical Society, 65: 123-138(2002)).
期刊：
ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA,2015年40(2):729-766 ISSN：1239-629X
通讯作者：
Li, Gongbao
作者机构：
[He, Yi; Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[He, Yi; Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
We are concerned with the following Schrodinger-Poisson equation with critical nonlinearity: {-epsilon(2)Delta u + V(x)u + psi u = lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(4)u in R-3, -epsilon(2)Delta psi = u(2) in R-3, u > 0, u is an element of H-1(R-3), where epsilon > 0 is a small positive parameter, lambda > 0, 3 < p <= 4. Under certain assumptions on the potential V, we construct a family of positive solutions u epsilon is an element of H-1 (R-3) which concentrates around a local minimum of V as epsilon --> 0. Subcritical growth Schrodinger-Poisson equation {-epsilon(2)Delta u + V(x)u + psi u = f(u) in R-3, -epsilon(2)Delta psi = u(2) in R-3, u > 0, u is an element of H-1(R-3), has been studied extensively, where the assumption for f(u) is that f(u) similar to vertical bar u vertical bar(p-2)u with 4 < p < 6 and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. The more difficult critical case is studied in this paper. As g(u) := lambda vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(4)u with 3 < p < 4 does not satisfy the Ambrosetti-Rabinowitz condition (there exists mu > 4, 0 < mu integral(u)(0) g(s) ds <= g(u)u), the boundedness of Palais-Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function g(s)/s(3) is not increasing for s > 0 prevents us from using the Nehari manifold directly as usual. The main result we obtained in this paper is new.
期刊：
Calculus of Variations and Partial Differential Equations,2015年54(3):3067-3106 ISSN：0944-2669
通讯作者：
Li, Gongbao
作者机构：
[He, Yi; Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[He, Yi; Li, Gongbao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
35J60;35J92;Primary 35J20
摘要：
We are concerned with the following Kirchhoff type equation with critical nonlinearity:
$$\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{| {\nabla u} |}^2}} } \Bigr )\Delta u + V(x)u = \lambda {| u |^{p - 2}}u + {| u |^4}u{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}), \\ \end{array} \right. \end{aligned}$$
where
$$\varepsilon $$
is a small positive parameter,
$$a,b>0$$
,
$$\lambda > 0$$
,
$$2 < p \le 4$$
. Under certain assumptions on the potential V, we construct a family of positive solutions
$${u_\varepsilon } \in {H^1}({\mathbb {R}^3})$$
which concentrates around a local minimum of V as
$$\varepsilon \rightarrow 0$$
. Although, critical growth Kirchhoff type problem
$$\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{| {\nabla u} |}^2}} } \Bigr )\Delta u + V(x)u = f(u)+{u^5}{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}) \\ \end{array} \right. \end{aligned}$$
has been studied in e.g. He et al. [18], where the assumption for f(u) is that
$$f(u) \sim |u{|^{p - 2}}u$$
with
$$4 < p < 6$$
and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. As
$$g(u): = \lambda {| u |^{p - 2}}u + {| u |^4u}$$
with
$$2<p\le 4$$
does not satisfy the Ambrosetti-Rabinowitz condition (
$$\exists \mu > 4, 0 < \mu \int _0^u {g(s)ds \le g(u)u}$$
), the boundedness of Palais–Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function
$$g(s)/{s^3}$$
is not increasing for
$$s > 0$$
prevents us from using the Nehari manifold directly as usual. Our result extends the main result in He et al. [18] concerning the existence and concentration of positive solutions to the case where
$$f(u) \sim |u{|^{p - 2}}u$$
with
$$4 < p < 6$$
.
期刊：
Journal of Differential Equations,2014年257(2):566-600 ISSN：0022-0396
通讯作者：
Li, Gongbao
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Gongbao] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
Kirchhoff equation;Ground state solutions;Pohoz̆aev type identity;Variational methods
摘要：
In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: {-(a + b integral(R3) vertical bar Du vertical bar(2)) Delta u + V(x)u = vertical bar u vertical bar(p-1)u, x is an element of R-3, (0.1) u is an element of H-1 (R-3), u > 0, x is an element of R-3, where a, b > 0 are constants, 2 < p < 5 and V : R-3 -> R. Under certain assumptions on V. we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results especially solve problem (0.1) in the case where p is an element of (2, 3], which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in [14] concerning the existence of positive solutions to the nonlinear Kirchhoff problem {-(epsilon(2)a + epsilon b integral(R3) vertical bar Du vertical bar(2)) Delta u + V(x)u = f(u), x is an element of R-3, u is an element of H-1 (R-3), u > 0, x is an element of R-3, where epsilon > 0 is a parameter, V(x) is a positive continuous potential and f (u) similar to vertical bar u vertical bar(p-1)u with 3 < p < 5 and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [7,33], which deal with Schrodinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem. (C) 2014 Elsevier Inc. All rights reserved.
