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The existence and spatio-temporal patterns of $$2\pi $$ -periodic solutions to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer $$O(2) \times \Gamma \times \mathbb Z_2$$ -equivariant degree theory. The solutions are supposed to take their values in a prescribed symmetric domain D, while O(2) is related to the reversal symmetry combined with the autonomous form of the system. The group $$\Gamma $$ reflects symmetries of D and/or possible coupling in the corresponding network of identical oscillaltors, and $$\mathbb Z_2$$ is related to the oddness of the right-hand side. Abstract results, based on the use of Gauss curvature of $$\partial D$$ , Hartman–Nagumo type a priori bounds and Brouwer equivariant degree techniques, are supported by a concrete example with $$\Gamma = D_8$$ —the dihedral group of order 16.

We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.

We have proved that all the closed connected sets of solutions of the complex Ginzburg–Landau equation {−Δu(x)+2i〈A(x),∇u(x)〉+‖A(x)‖2u(x)=λ(1−|u(x)|2)u(x)in Ω,u=0on ∂Ω, bifurcating from the set of normal solutions {0}×(0,+∞)⊂H01(Ω,C)×(0,+∞) are unbounded, where Ω⊂R2 is an open, bounded domain with smooth boundary, A(x1,x2)=(−x2,x1) and ‖⋅‖ is the usual norm in R2.

Let N ≥ 3 , 2 ∗ = 2 N / ( N − 2 ) and p ∈ ( 2 , 2 ∗ ) . Our purpose in this paper is to consider the multiple existence of solutions of problem − e 2 Δ u + u = | u | p − 2 u + λ f u ∈ H 1 ( R N ) , where e , λ > 0 , f ∈ L 2 ( R N ) ∩ C 2 ( R N ) , f ≥ 0 and f ≢ 0 . We will show the effect of the shape of f to the number of solutions.

In this paper we present a general framework for applications of the twisted equivariant degree (with one free parameter) to an autonomous $\Gamma$-symmetric system of functional differential equations in order to detect and classify (according to their symmetric properties) its periodic solutions. As an example we establish the existence of multiple non-constant periodic solutions of delay Lotka-Volterra equations with $\Gamma$-symmetries. We also include some computational examples for several finite groups $\Gamma$.

In this paper, we consider the multiple existence of sign changing solutions of coupled nonlinear Schrodinger equations (P) { − Δ u + μ 1 u = λ 1 u 3 + β v 2 u in R 3 − Δ v + μ 2 v = λ 2 v 3 + β u 2 v in R 3 where μ 1 , μ 2 , λ 1 , λ 2 are positive and β 0 .

Let ( M , g ) be a noncompact, connected, orientable smooth N -dimensional Riemannian manifold without boundary. We consider the existence of solutions of problem (P) Δ g u + a u log u = 0 on M , where Δ g stands for the Laplacian in M and a > 0 .

Let N > 2 and Ω C ℝ N be a bounded domain with boundary ∂Ω. Let Γ C ∂Ω be closed. Our purpose in this paper is to consider the existence of stable solutions u ∈ H 1 (Ω, C) of the Ginzburg-Landau equation {―Δu(x) = λ(w 2 0 (x) ― |u| 2 )u in Ω, u = g on ∂Ω\Γ, ∂u ∂v = 0 on Γ where A > 0, w 0 ∈ C 2 (Ω,ℝ + ) and g ∈ C 2 (∂Ω\Γ) such that |g(x)| = wo(x) on ∂Ω\Γ.

Let N ⩾ 3 , 2 * = 2 N / ( N − 2 ) and Ω ⊂ R N be a bounded domain with a smooth boundary ∂Ω and 0 ∈ Ω . Our purpose in this paper is to consider the existence of solutions of Henon equation: { − Δ u ( x ) = | x | α | u ( x ) | 2 * − 1 in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where α > 0 .

In this paper, we consider the multiple existence of nonradial positive solutions of coupled nonlinear Schrodinger system $$ \left \{\begin{array}{ll} - \Delta{u} + \mu_{1}u = u^{3} + \beta {v}^{2}u \quad {\rm in} \quad {\mathbb{R}}^{3} \\ {-\Delta{v} + \mu_{2}v = v^{3} + \beta u^{2}v \quad {\rm in} \quad {\mathbb{R}^3}}\end{array} \right. {\quad \quad (P)}$$ where μ1, μ2 > 0 with \(\mu_1 \neq \mu_2\) and β < 0.

Let ( M , g ) be a compact, connected, orientable smooth N -dimensional Riemannian manifold without boundary. Let p ∈ ( 2 , 2 ∗ ) , where 2 ∗ = 2 N / ( N − 2 ) . We consider the multiple existence of solutions of the problem (P) − e 2 Δ g u + u = | u | p − 2 u on M , where Δ g stands for the Laplacian in M .

We study Brezis–Nirenberg type theorems for the equation − Δ u + g ( x , u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N , g ( x , ⋅ ) is increasing and f is a dissipative nonlinearity. We apply such theorems for studying existence and multiplicity of positive solutions for the equation − Δ u = u − q + λ u p in Ω , u = 0 on ∂ Ω , where q > 0 , p > 1 and λ > 0 .

In this paper, we consider the multiple existence of solutions of the coupled nonlinear Schrodinger equations (P) { − Δ u + μ 1 u = u 3 + β v 2 u in R 3 − Δ v + μ 2 v = v 3 + β u 2 v in R 3 where μ 1 , μ 2 and β > 0 .

Let N ≥ 3 , 2 p 2 ∗ = 2 N / ( N − 2 ) , e > 0 and Ω be a bounded domain with a smooth boundary ∂ Ω . Our purpose in this paper is to consider the multiple existence of sign changing solutions of the problem − e 2 Δ u + u = | u | p − 2 u , u ∈ H 0 1 ( Ω ) .

Let N ⩾ 3 , 1 p ( N + 2 ) / ( N − 2 ) and 0 λ ( N − 2 ) 2 / 4 . We study the existence of multiple sign-changing solutions of u ∈ H 1 ( R N ) , − Δ u + u − λ u / | x | 2 = | u | p − 1 u in R N , which are not radially symmetric.

Let N ⩾ 3 , 2 * = 2 N N − 2 and p ∈ ( 2 , 2 * ) . Our purpose in this paper is to consider multiple existence of solutions of problem − Δ u + u = | u | p − 2 u + f , u ∈ H 1 ( R N ) , where f ∈ L 2 ( R N ) , f ⩾ 0 and f ≢ 0 .

We show that the problem has multiple solutions including sign-changing ones, where μ > 0, N ≥ 3, 1 < p < (N + 2)/(N − 2) and Q : ℝN → ℝ is some continuous function. We obtain signchanging solutions which are not only local minimum types in N* but also mountain pass types in N*, where N* = {u = u+ +u− ∈ H1(ℝN) : u+ ≠ 0, u− ≠ 0, ∫ℝN (|∇u±|2 + μ|u±|2) dx = ∫ℝN Q(x)|u±|p+1 dx}.

Let N ≥ 3 , 2 ∗ = 2 N / ( N − 2 ) . Our purpose in this paper is to consider the existence and multiplicity of solutions of the problem { − Δ u = | u | 2 ∗ − 2 u + f on R N u ∈ H 1 ( R N ) where f ∈ L 2 ∗ / ( 2 ∗ − 1 ) ( R N ) ∩ L ∞ ( R N ) , f ≥ 0 and f ≢ 0 .

Let N ≥ 4 N\geq 4 , let 2 ∗ = 2 N / ( N − 2 ) 2^{\ast }=2N/(N-2) and let Ω ⊂ R N \Omega \subset \mathbb {R}^{N} be a bounded domain with a smooth boundary ∂ Ω \partial \Omega . Our purpose in this paper is to consider the existence of solutions of the problem: \[ { − Δ u − μ u | x | 2 a m p ; = | u | 2 ∗ − 1 a m p ; a m p ; in Ω , u a m p ; > 0 a m p ; a m p ; in Ω , u a m p ; = 0 a m p ; a m p ; on ∂ Ω , \left \{ \begin {aligned} -\Delta u - \mu \frac {u}{\vert x\vert ^2} &= \vert u\vert ^{2^\ast -1} && \text {in $\Omega $}, \\ u & > 0 && \text {in $\Omega $},\\ u & = 0 && \text {on $\partial \Omega $}, \end {aligned} \right . \] where 0 > μ > ( N − 2 2 ) 2 . 0>\mu >(\frac {N-2}{2})^{2}.

Let N ⩾ 7 , 2 * = 2 N / ( N - 2 ) and Ω ⊂ R N be a bounded domain with a smooth boundary ∂ Ω and a : Ω ¯ ⟶ R is a continuous mapping. In this paper we consider the existence and multiplicity of positive solutions of Dirichlet boundary value problem of the form - ▽ ( a ( y ) ▽ u ) = | u | 2 * - 2 u , u ∈ H 0 1 ( Ω ) .

The main purpose of this paper is to construct families of positive and changing-sign solutions for both the slightly subcritical and slightly supercritical equations $-\Delta u+V(x)u=N(N-2)|u|^{\frac{4}{N-2}\pm\varepsilon}u$ in $\mathbb R^N,$ which blow-up and concentrate at different points of $\mathbb R^N$ as $\varepsilon$ goes to 0, under certain conditions on the potential $V.$

In this paper, we consider the existence of periodic solutions of reaction diffusion systems by using S 1 -degree theory due to Dylawerski et al., see Jodel et al. (Ann. Pol. Math. 41 (1991) 243).

In the case ofK≠D(A)¯, we study Cauchy problems and periodic problems for nonlinear evolution equationu(t)∈K,u′(t)+Au(t)∋f(t,u(t)),0≤t≤T, whereAisa maximal monotone operator on a Hilbert spaceH,Kis a closed, convex subset ofH,Vis a subspace ofH, andf:[0,T]×(K∩V)→His of Carathéodory type.

In this paper, we consider the existence of limit cycles of coupled van der Pol equations by using S1-degree theory due to Dylawerski et al. (see Ann. Polon. Math. 62 (1991) 243).

Let N⩾3, p>2∗=2N/(N−2) and Ω ⊂RN be a bounded domain with a boundary ∂Ω. Our purpose in this paper is to consider the existence of solutions of problem: (P)−Δu=up−1onΩ,u>0onΩ,u=0on∂Ω.

In this paper, we establish a multiple existence result of periodic solutions for a nonlinear parabolic equations with singular nonlinearity at zero. To establish our result, We use an approximating process by smooth nonlinear functions to avoid a singularity in nonlinear term. And we adapt topological argument based on the variational structure of functionals corresponding to approximating equations.

Let N⩾2 and Ω⊂ R N be a bounded domain. In the present paper, we show the existence of infinity many solutions of nonlinear Dirichlet boundary value problem −Δu=g(x,u), u∈H 0 1 (Ω), where g :Ω× R → R is a continuous function satisfying lim|t|→0g(x,t)/t→∞ for all x∈Ω .

Let N⩾3 and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and consider the semilinear boundary value problem of the form −Δu+λu=∣u∣2*−2uin Ω,u>0in Ω,u=0on ∂Ω, (P) where λ∈R and 2*=2N/(N−2). We show the effect of topology and geometry of the domain Ω on the existence and multiplicity of solutions of this problem.

In this paper, we show the existence of stable and unstable $T-$periodic solutions for a semilinear parabolic equation $\frac{\partial u}{\partial t} - \Delta u = g(x,u) + h( t, x ),\quad \text{in} \quad (0,T) \times \Omega$ $u=0 ,\quad \text{on}\quad (0,T) \times \partial \Omega$ $u(0) = u(T),\quad \text{in} \quad \overline \Omega$ where $\Omega \subset R^N$ is a bounded domain with a smooth boundary, $g:\overline{\Omega} \times R \rightarrow R$ is a continuous function such that $g(x,\cdot )$ has a superlinear growth for each $x \in \overline{\Omega} $ and $h:(0,T) \times \Omega \to R$ is a continuous function.

In this paper, we consider the solvability of nonlinear equations of the form Au+Cu 3 p where A is an m-accretive operator on a Banach space X, C is a mapping on X and p ∈ X.

In this paper, we show the existence of stable and unstable periodic solutions for a semilinear parabolic equation $\qquad\qquad \frac{\partial u}{\partial t}-\Delta_x u -\lambda_1 u +g(u) =s \phi + h$ in $ R\times \Omega$ $\qquad\qquad u(t,x) =0 $ on $R\times \partial \Omega$ $\qquad\qquad u(0,x)=u(2\pi, x)$ on $\Omega$ where $g$ is a continuous function on $R$, $\phi$ denotes the positve normalized eigenfunction corresponding to the first eigenvalue $\lambda_1$ of problem (L), $s \in R$, and $h \in C([0,2\pi],C^1_0(\overline{\Omega})).$

In this paper, we are concerned with a superlinear parabolic equation \[ { ∂ u ∂ t − Δ u = u p + h ( t , x ) , a m p ; ( t , x ) ∈ R + × Ω , u = 0 , a m p ; ( t , x ) ∈ R + × ∂ Ω , u > 0 , a m p ; ( t , x ) ∈ R + × ∂ Ω , \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}}{{\partial t}} - \Delta u = {u^p} + h(t,x),} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \Omega ,} \hfill \\ {{u = 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ {{u > 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ \end {array} } \right . \] where Ω ⊂ R N \Omega \subset {{\mathbf {R}}^N} is a bounded domain with smooth boundary ∂ Ω \partial \Omega , h is T-periodic with respect to the first variable, and 1 > p > N + 2 N − 2 1 > p > \frac {{N + 2}}{{N - 2}} if N ≥ 3 N \geq 3 and 1 > p > + ∞ 1 > p > + \infty if N ≤ 2 N \leq 2 . It is shown that there exist a stable and an unstable positive T-periodic solution for this problem if h is sufficiently small in L ∞ {L^\infty } .

In this paper, we consider the existence and multiplicity of periodic solutions of the problem u ′ + A u ∋ g ( t , u ) u’ + Au \ni g(t,u) where A A is a subdifferential of a convex function defined in a Hilbert space H H and g : R × H → H g:R \times H \to H is a Carathéodory function periodic with respect to the first variable.

1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 Mathematisches Institut, University Giessen, Arndtstr. 2, 35392 Giessen, Germany 3 Graduate School of Environment and Information Science, Yokohama National University, Tokiwadai, Yokohama 240-8501, Japan 4 Mathematics Department, University of California, Irvine, CA 92697-3875, USA 5 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-4205, USA

In this paper, we show that if $\mu>0$ is small enough, the problem \begin{equation*} \left\{ \begin{aligned} -\Delta u -\mu\frac{u}{|x|^2} & =|u|^{2^\ast-2}u & & \text{in $\Omega$,}\\ u & =0 & & \text{on $\partial\Omega$} \end{aligned} \right. \end{equation*} has at least cat $\Omega -1$ positive solutions, where $\Omega$ is a noncontractible, bounded domain in $\mathbb R^N (N\geq 4)$ such that its boundary is smooth and $0 \in \Omega$.

In this paper, we show the multiple existence of positive solutions of semilinear elliptic problems of the form $$ -\Delta u=\vert u\vert ^{2^{*}-2}u+f, \quad u\in H_{0}^{1}(\Omega), $$ where $\Omega\subset{\mathbb R}^{N}$ is a bounded domain, $2^{*}$ is the Sobolev critical exponent and $f\in L^{2}(\Omega)$.

We show existence and multiple existence results of subharmonic solutions for the system of second order differential equation $\ddot{u}(t)+ G'(u(t))=f(t)$ for $t\in \mathbb R$, where $N \in \mathbb N$, $f \in C(\mathbb R, \mathbb R^N)$ is a $T$-periodic function, $G\in C^{2}(\mathbb R^{N},\mathbb R)$ is a nonconvex functional.

We study the existence of positive solutions for singular Dirichlet problems. By the variational method, we show that if $\lambda_1$ is contained in some interval, then a singular Dirichlet problem has a radially symmetric, positive solution on an annulus $\Omega$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with homogeneous Dirichlet boundary condition. Since we consider a problem with singularity, we have difficulties that either the functional $I$ associated with the problem is not differentiable even in the sense of Gâteaux or the strong maximum principle is not applicable. But we show that a function which attains some minimax value for $I$ is a solution. We also treat the case that $\Omega$ is a ball.

In this paper we prove the global existence of a solution to a boundary initial value problem for a forced quasilinear hyperbolic Volterra equation under the assump- tion that the forcing term remains small and can be decom- posed into a time-periodic part and a part that decays to zero as t →∞ . We also show that the solution converges to a time- periodic function as t →∞ ; the latter is a periodic solution of a related history value problem.

Let $N \geq 3$ and $\Omega \subset \mathbb R^{N} $ be a bounded domain with a smooth boundary $\partial \Omega $. We %In this paper we consider a semilinear boundary value problem of the form %existence and multiplicity of solutions of problem $$ \cases -\Delta u = \vert u\vert ^{2^*-2} u +f &\text{\rm in } \Omega,\\ u> 0 & \text{\rm in } \Omega, \\ u= 0 & \text{\rm on } \partial \Omega, \endcases \leqno \text{\rm (P)} $$ where $f \in C(\overline \Omega )$ and $2^* = 2N/(N-2)$. We show the effect of topology of $\Omega $ on the multiple existence of solutions.