Search

In this paper we study the following nonlinear Schrödinger system: { − Δ u + α u = | u | p − 1 u + 2 q + 1 λ | u | p − 3 2 u | v | q + 1 2 , x ∈ R 3 , − Δ v + β v = | v | q − 1 v + 2 p + 1 λ | u | p + 1 2 | v | q − 3 2 v , x ∈ R 3 , u (x) → 0 , v (x) → 0 , as | x | → ∞ , where 3 ≤ p , q < 5 , α, β are positive parameters. We show that there exists λ k > 0 such that the equation has at least k radially symmetric sign-changing solutions and at least k seminodal solutions for each k ∈ N and λ ∈ (0 , λ k) . Moreover, we show the existence of a least energy radially symmetric sign-changing solution for each λ ∈ (0 , λ 0) where λ 0 ∈ (0 , λ 1 ] . [ABSTRACT FROM AUTHOR]

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f (⋅) to satisfy ∫ − ∞ t e λ s ∥ f (s) ∥ L 2 2 d s < ∞ for all t ∈ R , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H 1 topology and the usual (L 2 , L 2) pullback D λ -attractor indeed can attract in the H 1 -norm, provided that ∫ − ∞ t e λ s ∥ f (s) ∥ H − 1 (O s) 2 d s < ∞ and f ∈ L loc 2 (R , L 2 (O s)) . [ABSTRACT FROM AUTHOR]

The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K ∞ (R + d) that was defined in (Bachar et al. 2002:41, 2002) for d = 2 and in (Bachar and Mâagli 9(2):153–192, 2005) for d ≥ 3 and adapted to study some nonlinear elliptic problems in the half space. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants λ and μ to the following system: Δ u = λ f (x , u , v) , Δ v = μ g (x , u , v) in R + d , lim x → (ξ , 0) u (x) = a 1 ϕ 1 (ξ) , lim x → (ξ , 0) v (x) = a 2 ϕ 2 (ξ) for all ξ ∈ R d − 1 , lim x d → ∞ u (x) x d = b 1 , lim x d → ∞ v (x) x d = b 2 , where ϕ 1 and ϕ 2 are nontrivial nonnegative continuous functions on ∂ R + d = R d − 1 × { 0 } , a 1 , a 2 , b 1 , b 2 are nonnegative constants such that (a 1 + b 1) (a 2 + b 2) > 0 . The functions f and g are nonnegative and belong to a class of functions containing in particular all functions of the type f (x , u , v) = p (x) u α g 1 (v) and g (x , u , v) = q (x) g 2 (u) v β with α ≥ 1 , β ≥ 1 , g 1 , g 2 are continuous on [ 0 , ∞) , and p , q are nonnegative functions in K ∞ (R + d) . [ABSTRACT FROM AUTHOR]

We reprove a result by Bartsch, Weth, and Willem (Calc. Var. Partial Differ. Equ. 18(3):253–268, 2003) concerning the nondegeneracy of bubble solutions for a critical semilinear elliptic equation involving the polyharmonic operator. The merit of our proof is that it does not rely on the comparison theorem. The argument of our proof mainly uses the stereographic projection with the Funk–Hecke formula, which works for general critical semilinear elliptic equations. [ABSTRACT FROM AUTHOR]

We consider the following singular semilinear problem { Δ u (x) + p (x) u γ = 0 , x ∈ D (in the distributional sense) , u > 0 , in D , lim | x | → 0 | x | n − 2 u (x) = 0 , lim | x | → ∞ u (x) = 0 , where γ < 1 , D = R n ∖ { 0 } (n ≥ 3 ) and p is a positive continuous function in D, which may be singular at x = 0 . Under sufficient conditions for the weighted function p (x) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the following elliptic boundary value problem: { − Δ u + V ( x ) u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , $$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x, u),\quad x\in \Omega , \\ u=0, \quad x \in \partial \Omega , \end{cases} $$ where Ω ⊂ R N $\Omega \subset {\mathbb {R}}^{N}$ is a bounded domain with smooth boundary ∂Ω, and f is allowed to be sign-changing and is of sublinear growth near infinity in u. For both cases that V ∈ L N / 2 ( Ω ) $V\in L^{N/2}(\Omega )$ with N ≥ 3 $N\geq 3$ and that V ∈ C ( Ω , R ) $V\in C(\Omega , \mathbb {R})$ with inf Ω V ( x ) > − ∞ $\inf_{\Omega }V(x)>-\infty $ , we establish a sequence of nontrivial solutions converging to zero for above equation via a new critical point theorem.

Solutions of the stationary semilinear Cahn-Hilliard-type equation which are exponentially decaying at infinity, are studied. Using the mounting pass lemma allows us to determinate the existence of a radially symmetric solution. On the other hand, the application of Lusternik-Schnirel'man (L-S) category theory shows the existence of, at least, a countable family of solutions. However, through numerical methods it is shown that the whole set of solutions, even in 1D, is much wider. This suggests that, actually, there exists, at least, a countable set of countable families of solutions, in which only the first one can be obtained by the L-S min-max approach. [ABSTRACT FROM AUTHOR]

Let $q(t)$ be a continuous 2 π-periodic function with $\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt>0$. We propose a new approach to establish the existence of Aubry-Mather sets and quasi-periodic solutions for the following time-periodic parameters semilinear Duffing-type equation: where $f(t,x)$ is a continuous function, 2 π-periodic in the first argument and continuously differentiable in the second one. Under some assumptions on the functions q and f, we prove that there are infinitely many generalized quasi-periodic solutions via a version of the Aubry-Mather theorem given by Pei. Especially, an advantage of our approach is that it does not require any high smoothness assumptions on the functions $q(t)$ and $f(t,x)$. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following semilinear elliptic systems: where $V(x)$, $H(x)$ are nonnegative continuous functions. Under some appropriate assumptions on $V(x)$, $H(x)$, and $F(x, u, v)$, we prove the existence of infinitely many small negative-energy solutions by using the fountain theorem established by Zou. Recent results from the literature are extended. [ABSTRACT FROM AUTHOR]

This paper concerns the asymptotic behavior of solutions to the homogeneous Neumann exterior problems of a class of semilinear parabolic equations with convection and reaction terms. The critical Fujita exponents theorems are established. It is shown that the global existence and blow-up of solutions depends on the reaction term, the convection term and the spatial dimension. [ABSTRACT FROM AUTHOR]

In this paper, we investigate residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary elliptic optimal control problems. By using the hp finite element approximation for both the state and the co-state and the hp discontinuous Galerkin finite element approximation for the control, we derive a posteriori error bounds in $L^{2}$- $H^{1}$ norms for the Neumann boundary optimal control problems governed by semilinear elliptic equations. We also give $L^{2}$- $L^{2}$ a posteriori error estimates for the optimal control problems. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximations for the semilinear Neumann boundary optimal control problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of non-negative solutions to the semilinear equation $-\Delta_{H} u = K(\xi)\vert u\vert ^{2^{\sharp}-2}u + \mu \vert \xi \vert _{H}^{\alpha}u$ in a bounded domain $\Omega\subset\mathbb{H}^{N}$ with Dirichlet boundary conditions. Here $\mathbb{H}^{N}$ is the Heisenberg group and $2^{\sharp}= 2q/(q-2)$ is the critical exponent of the Sobolev embedding on the Heisenberg group. The function $K(\xi)$ may be sign changing on Ω. Using the variational method, we prove that this problem has at least two non-negative solutions provided μ, α, and $K(\xi)$ satisfy some conditions. [ABSTRACT FROM AUTHOR]

In this paper we study a class of semilinear degenerate parabolic equations arising in mathematical finance and in the theory of diffusion processes. We show that blow-up of spatial derivatives of smooth solutions in finite time occurs to initial boundary value problems for a class of degenerate parabolic equations. Furthermore, nonexistence of nontrivial global weak solutions to initial value problems is studied by choosing a special test function. Finally, the phenomenon of blow-up is verified by a numerical experiment. [ABSTRACT FROM AUTHOR]

In this paper we consider the semilinear parabolic equation Σmi,j=1 aijXiXju - ∂tu + Vup = 0 with a general class of potentials V = V(ξ, t), where A = {aij}i,j is a positive definite symmetric matrix and the Xi's denotes a system of left-invariant vector fields on a Carnot group G. Based on a fixed point argument and by establishing some new estimates involving the heat kernel, we study the existence and large-time behavior of global positive solutions to the preceding equation. [ABSTRACT FROM AUTHOR]

This work is concerned with a mixed boundary value problem for a semilinear parabolic equation with a memory term. Under suitable conditions, we prove that the energy functional decays to zero as the time tends to infinity by the method of perturbation energy, in which the usual exponential and polynomial decay results are only special cases. [ABSTRACT FROM AUTHOR]

We establish existence and multiplicity results for weak solutions of a degenerate quasilinear elliptic system. By using Ekeland’s variational principle, the mountain pass theorem and the saddle point theorem in critical point theory, we obtain the existence of one or three solutions for an elliptic system with Dirichlet boundary conditions under some restriction on λ. This kind of results was firstly obtained by Mawhin and Schmitt (Ann. Pol. Math. 51:241-248, 1990) for a semilinear two-point boundary value problem. [ABSTRACT FROM AUTHOR]

The existence of at least one solution to the second-order nonlocal boundary value problems on a half line is investigated by using Mawhin's continuation theorem. [ABSTRACT FROM AUTHOR]

A class of semilinear elliptic equations involving strong resonance or non-resonance is reconsidered here. The multiplicity of solutions is investigated by using the variational method, and the results complement earlier ones. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonlinear equation involving the fractional Laplacian: (-Δ)5v(x) = b(x)f (v(x)), x ∈ R, where s ∈ (0, 1), b : R→R is a periodic, positive, even function and -f is the derivative of a double-well potential G. That is, G ∈ C2,γ (0 < γ < 1), G(1) = G(-1) < G(τ ) ∀ τ ∈ (-1, 1), G'(-1) = G'(1) = 0. We show the existence of layer solutions of the equation for s ≥ 12 and for some odd nonlinearities by variational methods, which is a bounded solution having the limits ±1 at ±∞. Asymptotic estimates for layer solutions as |x|→+∞and the asymptotic behavior of them as s ↑ 1 are also obtained. [ABSTRACT FROM AUTHOR]

We study the following system of equations: 0.1 { Δ u 1 = p 1 (| x |) f 1 (u 1 , u 2) in R N , Δ u 2 = p 2 (| x |) f 2 (u 1 , u 2) in R N. Here f 1 , f 2 are continuous nonlinear functions that satisfy Keller–Osserman-type conditions, and p 1 and p 2 are continuous weight functions. We establish the existence of radial solutions for this system under various boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we consider the Cauchy problem of a generalization of time-fractional diffusion equation with variable coefficients in R+n+1, where the time derivative is replaced by a regularized hyper-Bessel operator. The explicit solution of the inhomogeneous linear equation for any n∈Z+ and its uniqueness in a weighted Sobolev space are established. The key tools are Mittag-Leffler functions, M-Wright functions and Mikhlin multiplier theorem. At last, we obtain the existence of solution of the semilinear equation for n=1 by using a fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the nonlinear boundary problem for first-order impulsive semilinear differential inclusions. We establish existence results by using Martelli’s fixed point theorem with upper and lower solutions method. We find that by giving different definitions of lower and upper solutions we can get all existence results. We also present an example. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of classical and strong solutions of a fractional semilinear evolution equation using the method of a semigroup and the Banach fixed theorem. [ABSTRACT FROM AUTHOR]

We establish the existence, nonexistence, and multiplicity of positive solutions to semilinear elliptic systems with integral boundary conditions when positive multiparameters vary on the boundary. We prove the results by using sub- and supersolution argument and fixed point index theory. [ABSTRACT FROM AUTHOR]