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This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u > 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , a a , μ > 0 \mu \gt 0 , N ≥ 2 N\ge 2 , and 2 < p < 2 s ∗ 2\lt p\lt {2}_{s}^{\ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) \left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) \left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.

In this paper we first establish a new critical point theorem which may produce a relationship between the sign-changing critical point, the Morse index, the upper bound of the critical value, the value of the type inf k ≤ dim ⁡ Y ∞ Y ⊂ E , ⁡ sup Y ⁡ G and the number of nodal domains. An unified approach is introduced for these composite properties. As one application of the abstract theorem, we study the existence of infinitely many sign-changing bound states for the following nonlinear stationary Schrodinger equation: − Δ u + a ( x ) u = | u | p − 2 u , u ∈ H 1 ( R N ) , where N ≥ 2 , p > 2 and p 2 N / ( N − 2 ) : = 2 ⁎ when N > 2 ; the potential a ( x ) ∈ C ( R N , R ) is a nonnegative function verifying suitable decay assumptions, but being not symmetric or periodic. We obtain the information on the precise estimates of the energies, the Morse indices and the number of nodal domains for these sign-changing bound states. We believe that the abstract theorem in this article will have more applications.

Note: Please see pdf for full abstract with equations. We consider the existence and asymptotic behavior on mass of positive solutions to the following system −Δu + λ1u = μ1u3 + α1|u|p−2u + βv2u in R4, −Δv + λ2v = μ2v3 + α2|v|p−2v + βu2v in R4, under the normalized mass constraints ∫R4u2 = a21 and ∫R4v2 = a22, where a1, a2 are prescribed, μ1, μ2, β > 0, α1, α2 ∈ R, p ∈ (2, 4) and λ1, λ2 ∈ R appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., αi < 0(i = 1, 2). Then turning to the case of αi > 0(i = 1, 2), if 2 < p < 3, we show that the problem admits a ground state and an excited state. Moreover, we give a precise asymptotic behavior of these two solutions as (a1, a2) → (0, 0) and a1 ~ a2. This seems to be the first contribution regarding the multiplicity as well as synchronized mass collapse behavior of normalized solutions to Schrödinger systems with Sobolev critical exponent. When 3 ≤ p < 4, we prove an existence as well as a non-existence (p = 3) results of ground states. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively. 2010 Mathematics Subject Classification : 35J50, 35B33, 35B09, 35B40.

In this paper, we consider polyharmonic Lane–Emden equations $$ \begin{equation*} (-\Delta )^m u=|u|^{p-1}u, \ \ \ \mbox{in} \ \ \ {\mathbb R}^n, \end{equation*}$$where $m\geq 3$. We classify the stable or stable outside a compact set solutions when $m=3$ or $4$ for any dimensions and when $m\geq 5$ for large dimensions. In the process, we exhibit the general Joseph–Lundgren exponent (including both local and nonlocal cases) in a concise form and prove related properties. The key ingredient of the proof of the classification is a monotonicity formula for general polyharmonic equations, which may have application in regularity theory for higher-order elliptic equations.

In this paper, we will study the following PDE in R N involving multiple Hardy-Sobolev critical exponents: { Δ u + ∑ i = 1 l λ i u 2 ⁎ ( s i ) − 1 | x | s i + u 2 ⁎ − 1 = 0 in R N , u ∈ D 0 1 , 2 ( R N ) , where 0 s 1 s 2 ⋯ s l 2 , 2 ⁎ : = 2 N N − 2 , 2 ⁎ ( s ) : = 2 ( N − s ) N − 2 and there exists some k ∈ { 1 , ⋯ , l } such that λ i > 0 for 1 ≤ i ≤ k ; λ i 0 for k + 1 ≤ i ≤ l . We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence of the positive ground state solution. The regularity of the least-energy solution is also investigated.

In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth:−Δu+λ1u=μ1u3+βuv2+2qpyu2qp−1v2inΩ,−Δv+λ2v=μ2v3+βu2v+2yu2qpvinΩ,u,v>0inΩ,u,v=0on∂Ω,$$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$whereΩ⊂ ℝ4is a smooth bounded domain with smooth boundary∂Ω;p,qare positive coprime integers with 1 <2qp$\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$< 2;μi> 0 andλi∈ ℝ are fixed constants,i= 1, 2;β> 0,y> 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

In this paper we study the nonhomogeneous semilinear fractional Schrodinger equation with critical growth $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u + u = {u^{2_s^* - 1}} + \lambda (f(x,u)) + h(x)),}&{x \in {\mathbb{R}^N},} \\ {u \in {H^s}({\mathbb{R}^N}),\;\;\;u(x) > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in {\mathbb{R}^N},} \end{array}} \right.$$ where s ∈ (0, 1), N > 4s, and λ > 0 is a parameter, $$2_s^* = {{2N} \over {N - 2s}}$$ is the fractional critical Sobolev exponent, f and h are some given functions. We show that there exists 0 λ*, a unique solution (λ*,uλ*) if λ = λ*, which shows that (λ*,uλ*) is a turning point in HS(ℝN) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schr\"odinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-\delta u+\lambda_1 u=|u|^{2^*-2}u+{\nu\alpha} |u|^{\alpha-2}|v|^\beta u,\quad \text{in }\mathbb{R}^N, &-\delta v+\lambda_2 v=|v|^{2^*-2}v+{\nu\beta} |u|^\alpha |v|^{\beta-2}v,\quad \text{in }\mathbb{R}^N, &\int u^2=a^2,\;\;\; \int v^2=b^2, \end{aligned} \right. \end{equation*} where $N=3,4$, $\alpha,\beta>1$, $20$ and $\alpha+\beta\le 2+\frac{4}{N}$, we show that the system has a normalized ground state solution for $02+\frac{4}{N}$ we prove the existence of a threshold $\nu_1\ge 0$ such that a normalized ground state solution exists for $\nu>\nu_1$, and does not exist for $\nu

In this paper, we study the following $k$-coupled nonlinear Schr\"odinger system with Sobolev critical exponent: \begin{equation*} \left\{ \begin{aligned} -\Delta u_i & +\lambda_iu_i =\mu_i u_i^{2^*-1}+\sum_{j=1,j\ne i}^{k} \beta_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;\Omega,\newline u_i&>0 \quad \hbox{in}\; \Omega \quad \hbox{and}\quad u_i=0 \quad \hbox{on}\;\partial\Omega, \quad i=1,2,\cdots, k. \end{aligned} \right. \end{equation*} Here $\Omega\subset \mathbb{R}^N $ is a smooth bounded domain, $2^{*}=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\lambda_1(\Omega)0$ and $ \beta_{ij}=\beta_{ji}\ne 0$, where $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. We characterize the positive least energy solution of the $k$-coupled system for the purely cooperative case $\beta_{ij}>0$, in higher dimension $N\ge 5$. Since the $k$-coupled case is much more delicated, we shall introduce the idea of induction. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the $k$-coupled system decreases as $k$ grows. Moreover, we establish the existence of positive least energy solution of the limit system in $\mathbb{R}^N$, as well as classification results., Comment: arXiv admin note: text overlap with arXiv:1209.2522

In this paper, we study equations involving fully nonlinear nonlocal operators \begin{document}$ \mathcal {F}_{\alpha}(u(x)) = C_{n, \alpha}P.V.\int_{ \mathbb R^n}\frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}}dz = f(u(x)), \; \; \; x\in \mathbb R^n. $\end{document} We shall establish a maximum principle for anti-symmetric functions on any half space, and obtain key ingredients for proving the symmetry and monotonicity for positive solutions to the fully nonlinear nonlocal equations. Especially, a Liouville theorem is derived, which will be useful in carrying out the method of moving planes on unbounded domains for a variety of problems with fully nonlinear nonlocal operators.

We study the following coupled system of quasilinear equations: − Δ p u + μ | u | p − 2 u = | u | q − 2 u + α λ | u | α − 2 u | v | β , x ∈ R N , − Δ p v + ν | v | p − 2 v = | v | p ∗ − 2 v + β λ | u | α | v | β − 2 v , x ∈ R N , where N ≥ 3 , μ , ν , λ > 0 , α , β ≥ 1 , 2 p q p ∗ and α + β = p . We establish some results about the existence and regularity of ground state solutions for the p-Laplacian systems by using variational methods. We also study the asymptotic behavior of solutions as the coupling parameter λ tends to zero.

In this paper, we study the solutions of the following k-coupled elliptic system − Δ u i = ∑ j = 1 k β i j | u i | q − 1 u i | u j | q + 1 in R N , i = 1 , … , k , where q ≥ 1 , N ≥ 2 , k is a fixed positive integer and B = ( β i j ) i , j = 1 k is a real symmetric matrix. We analyze the impact of the properties of the matrix B on the solutions. In particular, when the matrix B is strictly copositive, we give a complete classification for stable and finite Morse index solutions (whether positive or sign-change) with 3 ≤ 2 q + 1 p c ( N ) , where p c ( N ) is the Joseph–Lundgren exponent. The results can be applied to subcritical, critical and supercritical values of the exponent 2 q + 1 .

In this paper, we study the symmetry of components of positive solutions for the fractional elliptic systems in the whole space R N . As an application, we prove the uniqueness of positive solutions for fractional Schrodinger type systems with critical exponents. We also prove the symmetry of components and some Liouville type theorems for a class of fractional noncooperative systems, which contains the systems arising in Bose–Einstein condensation and in biological models of Lotka–Volterra type.

This paper concerns an elliptic system with critical exponents: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_j-\frac{\lambda _j}{|x|^2}u_j=u_j^{2^*-1}+\sum \limits _{k\ne j}\beta _{jk}\alpha _{jk}u_j^{\alpha _{jk}-1}u_k^{\alpha _{kj}},\;\;x\in {{\mathbb {R}}}^N,\\ u_j\in D^{1,2}({{\mathbb {R}}}^N),\quad u_j>0 \;\; \hbox {in} \quad {{\mathbb {R}}}^N\setminus \{0\},\quad j=1, \ldots ,r,\end{array}\right. } \end{aligned}$$ where $$N\ge 3, r\ge 2, 2^*=\frac{2N}{N-2}, \lambda _j\in (0, \frac{(N-2)^2}{4})$$ for all $$ j=1, \ldots ,r $$ ; $$\beta _{jk}=\beta _{kj}$$ ; $$\alpha _{jk}>1, \alpha _{kj}>1,$$ and $$\alpha _{jk}+\alpha _{kj}=2^* $$ for all $$k\ne j$$ . Note that the nonlinearities $$u_j^{2^*-1}$$ and the coupling terms are all critical in arbitrary dimension $$N\ge 3 $$ . The signs of the coupling constants $$\beta _{ij}$$ are decisive for the existence of the ground-state solutions. We show that the critical system with $$r\ge 3$$ has a positive ground-state solution for all $$\beta _{jk}>0$$ with some constraint on $$\lambda _j$$ . However, there is no ground-state solution when all $$\beta _{jk}$$ are negative. It is also proved that the positive solution of the system is radially symmetric. Furthermore, we obtain an uniqueness theorem for the case $$r\ge 3$$ with $$N=4$$ and an existence theorem for the case $$r=2$$ with general coupling exponents.

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ -\Delta v+\lambda _2v=\mu _2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad \int _{{{\mathbb {R}}^N}}v^2=a_2^2, \end{array}\right. } \end{aligned}$$ where $$p,r_1+r_2

In this paper we deal with a nonlinear Schrodinger–Poisson equation involving critical Sobolev exponent. Under a local condition imposed on the potential, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values. In the proofs we apply Szulkin and Weth’s generalized Nehari manifold methods, penalization techniques and Ljusternik–Schnirelmann theory.

We study the multiplicity of concentrating solutions to the nonlinear fractional Kirchhoff equation $$\begin{aligned} \left( \varepsilon ^{2s}a+\varepsilon ^{4s-3}b\int _{\mathbb R^3}|(-\Delta )^{\frac{s}{2}}u|^2dx\right) (-\Delta )^s u+V(x)u=f(u)~~\text{ in }~~\mathbb R^3, \end{aligned}$$ where $$\varepsilon >0$$ is a positive parameter, $$(-\Delta )^s$$ is the fractional laplacian with $$s\in (\frac{3}{4},1), a,b$$ are positive constants, and V is a positive potential such that $$\inf _{\partial \Lambda }V>\inf _{\Lambda }V$$ for some open bounded subset $$\Lambda \subset \mathbb R^3.$$ We relate the number of positive solutions with the topology of the set where V attains its minimum in $$\Lambda $$ . The proof is based on the Ljusternik–Schnirelmann theory.

Consider the following Schrodinger system: where either or is a smooth bounded domain. Note that the cubic nonlinearities and the coupling terms are of critical growth whenever dimension . We give a characterization of the least energy solutions when or is a smooth bounded domain of , if the coupling matrix is positively or negatively definite with and exists and satisfies for . Furthermore, when and , we obtain a nonexistence theorem about the least energy solutions provided attraction and repulsion coexist, i.e. some of are positive but some others are negative.

In this paper, we study the normalized solutions to the following system { − Δ u + ( V 1 ( x ) + λ 1 ) u = μ 1 | u | p − 2 u + β v in R N , − Δ v + ( V 2 ( x ) + λ 2 ) v = μ 2 | v | q − 2 v + β u in R N , ∫ R N u 2 = a , ∫ R N v 2 = b , with the mass-subcritical condition 2 p , q 2 + 4 N , where μ 1 , μ 2 , a , b > 0 , β ∈ R ∖ { 0 } are prescribed; λ 1 , λ 2 ∈ R are to be determined. We prove the existence of a solution with prescribed L 2 -norm under some various conditions on the potential V 1 , V 2 : R N → R . The proof is based on the refined energy estimates.

In this paper, by constructing an appropriate Byeon-Wang type penalization function, we consider the critical Choquard equation − ϵ 2 Δ u + V ( x ) u = 1 ϵ α ( I α ⁎ | u | N + α N − 2 ) | u | N + α N − 2 − 2 u + λ g ( u ) in R N , where N ≥ 3 , ( N − 4 ) + α N , ϵ , λ > 0 are parameters, and I α is the Riesz potential. We obtained the localized semiclassical states concentrating at local minimizers of potential. It is the first time to study Choquard equations using this method as far as we know.

Consider the following non-local critical system (0.1) { ( − Δ ) s u − λ 1 u = μ 1 | u | 2 ⁎ − 2 u + α γ 2 ⁎ | u | α − 2 u | v | β in Ω , ( − Δ ) s v − λ 2 v = μ 2 | v | 2 ⁎ − 2 v + β γ 2 ⁎ | u | α | v | β − 2 v in Ω , u = 0 , v = 0 in R N ∖ Ω , where ( − Δ ) s is fractional Laplacian, 0 s 1 and all λ 1 , λ 2 , μ 1 , μ 2 , γ > 0 , 2 ⁎ : = 2 N N − 2 s is a fractional Sobolev critical exponent, N > 2 s , α , β > 1 , α + β = 2 ⁎ , and Ω is an open bounded domain in R N with Lipschitz boundary. Under proper conditions, we establish the existence result of the ground state solution to system (0.1) .