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We extend the results in Kloeden-Simsen [CPAA 2014] to p(x, t)- Laplacian problems on time-dependent Lebesgue spaces with variable exponents. We study the equation ∂uλ/∂t (t) − div (Dλ(t, x)|∇uλ(t)|p(x,t)−2∇2uλ(t) ) + |uλ(t)| p(x,t)−2uλ(t) = B(t, uλ(t)) on a bounded smooth domain Ω in Rn, n ≥ 1, with a homogeneous Neumann boundary condition, where the exponent p(·) ∈ C(Ω¯ × [τ, T], R+) satisfies min p(x, t) > 2, and λ ∈ [0, ∞) is a parameter. We establish the existence and upper semicontinuity of pullback attractors for this equation under the assumption, amongst others, that B is globally Lipschitz in its second variable and Dλ ∈ L∞([τ, T] × Ω, R+) is bounded from above and below, monotonically nonincreasing in time and continuous in the parameter λ. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of radially symmetric solutions to the subcritical semilinear elliptic problem $$\displaylines{ -\Delta u = u^{\frac{N+2}{N-2}}/[\log(e+u)]^{\alpha}\quad \text{in } \Omega=B_R(0)\subset\mathbb{R}^N,\cr u>0,\quad \text{in } \Omega,\cr u=0,\quad \text{on } \partial \Omega, }$$ as $\alpha\to 0^+$. Using asymptotic estimates, we prove that there exists an explicitly defined constant L(N,R)>0, only depending on N and R, such that $$ \limsup_{\alpha\to0^+} \frac{\alpha u_\alpha (0)^2} {[\log(e+u_\alpha (0))]^{1+\frac{\alpha(N+2)}{2}}} \leq L(N,R) \le 2^*\liminf_{\alpha\to0^+}\frac{\alpha u_\alpha (0)^2} {[\log(e+u_\alpha (0))]^{\frac{\alpha(N-4)}2}}. $$

We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a H¨older nonlinear σ(u). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a "strange" nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on L2 (0, T), independently of the regularity of σ. [ABSTRACT FROM AUTHOR]

In this article we study the behavior of the solutions of non-autonomous damped wave equations when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood collapses toward the boundary as a parameter approaches zero. We prove the continuity of the set of equilibria for these equations. Moreover, if an equilibrium solution of the limit problem is hyperbolic, then we show that the perturbed equation has only one equilibrium solution nearby.

We prove an asymptotic monotonicity formula for bounded, globally minimizing solutions (in the sense of Morse) to a class of semilinear elliptic systems of the form Δu = Wu(u), x ∈ ℝn, n≥2, with W: ℝm → ℝ, m≥1, nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution u). As an application, we can prove a Liouville type theorem under various assumptions. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of radially symmetric solutions to the subcritical semilinear elliptic problem ... as ff ! 0+. Using asymptotic estimates, we prove that there exists an explicitly defined constant L(N;R) > 0, only depending on N and R, such that... . [ABSTRACT FROM AUTHOR]

Consider the KPP-type equation of the form $\Delta u+f(u)=0$, where $f:[0,1] \to \mathbb R_{+}$ is a concave function. We prove for arbitrary dimensions that there is no solution bounded in $(0,1)$. The significance of this result from the point of view of probability theory is also discussed.

In this article we study the behavior of the solutions of nonautonomous damped wave equations when some reaction terms are concentrated in a neighborhood of the boundary and this neighborhood collapses toward the boundary as a parameter approaches zero. We prove the continuity of the set of equilibria for these equations. Moreover, if an equilibrium solution of the limit problem is hyperbolic, then we show that the perturbed equation has only one equilibrium solution nearby. [ABSTRACT FROM AUTHOR]

The uniqueness of positive solution of a semilinear cooperative Hamiltonian elliptic system with two equations is proved for the case of sublinear and superlinear nonlinearities. Implicit function theorem, bifurcation theory, and ordinary differential equation techniques are used in the proof.

In this article we prove the existence of an infinite number of radial solutions to Δu + K(r)f(u) = 0 with a nonlinear boundary condition on the exterior of the ball of radius R centered at the origin in ℝN such that limr → ∞ u(r) = 0 with any given number of zeros where f : ℝ → ℝ is odd and there exists a β > 0 with f < 0 on (0, β), f > 0 on (β, ∞) with f superlinear for large u, and K(r) ~ r-α with 0 < α < 2(N - 1). [ABSTRACT FROM AUTHOR]

This article studies the existence and nonexistence of global solutions to the initial boundary value problems for semilinear wave and heat equation, and for Cauchy problem of nonlinear Schrödinger equation. This is done under three possible initial energy levels, except the NLS as it does not have comparison principle. The most important feature in this article is a new hypothesis on the nonlinear source terms which can include at least eight important and popular power-type nonlinearities as special cases. This article also finds some kinds of divisions for the initial data to guarantee the global existence or finite time blowup of the solution of the above three problems. [ABSTRACT FROM AUTHOR]

This article concerns the existence and multiplicity of solutions to the superlinear elliptic problems. We introduce a new superlinear condition which is proved to be weaker than the Ambrosetti-Rabinowitz condition, the nonquadratic condition, the monotonicity conditions. As an application, positive solution and infinitely many solutions to semilinear elliptic equation with general subcritical growth are obtained, which generalize some known results. [ABSTRACT FROM AUTHOR]

In this article, we study the elliptic boundary value problem $$\displaylines{ -\Delta u+a(x)u=g(x, u) \quad \text{in } \Omega,\cr u = 0\quad \text{on } \partial \Omega, }$$ where $ \Omega\subset \mathbb{R}^N$ $(N\geq3)$ is a bounded domain with smooth boundary $ \partial\Omega$ and the potential $a(x)$ is allowed to be sign-changing. We establish the existence of infinitely many nontrivial solutions by variant fountain theorem developed by Zou for sublinear nonlinearity.

This article concerns the existence and multiplicity of weak solutions of the nonlinear fractional elliptic problem. We extend some well known results of semilinear Laplacian equations to the nonlocal fractional setting. Using the variational methods based on the critical point theory, sub-supersolutions methods and Morse theory, we show that the problem has at least 6 nontrivial solutions. [ABSTRACT FROM AUTHOR]

This article focuses on a one-dimensional nonlinear wave equation which is the Euler-Lagrange equation of a variational principle whose Lagrangian density involves linear terms and zero term as well as quadratic terms in derivatives of the field. We establish the global existence of weak solutions to its Cauchy problem by the method of energy-dependent coordinates which allows us to rewrite the equation as a semilinear system and resolve all singularities by introducing a new set of variables related to the energy. [ABSTRACT FROM AUTHOR]

In this paper we study the Cauchy problem for a system of semi-linear hyperbolic equations. We prove a theorem on the nonexistence of global solutions with positive initial energy. [ABSTRACT FROM AUTHOR]

In this article, we study the approximate controllability and homegenization results of a semi-linear elliptic problem with Robin boundary condition in a periodically perforated domain. We prove the existence of minimal norm control using Lions constructive approach, which is based on Fenchel-Rockafeller duality theory, and by means of Zuazua's fixed point arguments. Then, as the homogenization parameter goes to zero, we link the limit of the optimal controls (the limit of fixed point of the controllability problems) with the optimal control of the corresponding homogenized problem. [ABSTRACT FROM AUTHOR]

In this article, we analyze the boundary behavior of solutions to the boundary blow-up elliptic problem $$ Delta u =b(x)f(u), quad ugeq 0,; xinOmega,; u|_{partial Omega}=infty, $$ where $Omega$ is a bounded domain with smooth boundary in $mathbb{R}^N$, $f(u)$ grows slower than any $u^p$ ($p > 1$) at infinity, and $b in C^{alpha}(ar{Omega})$ which is non-negative in Omega and positive near $partialOmega$, may be vanishing on the boundary.

We use a method based on Picone-type identities to find oscillation conditions for the equation $$ sum_{i j =1}^n frac{partial}{partial x_i} Big( a_{ij}(x) frac{partial}{partial x_j} Big)u + f(x,u, abla u) + c(x) u =0,, $$ with Dirichlet boundary conditions on bounded and unbounded domains. In this article, the above method substitudes the traditional Riccati techniques [3,8] used for unbounded domains.

By means of a Picone's type identity, we prove uniqueness and oscillation of solutions to an elliptic semilinear equation with Dirichlet boundary conditions.

In this paper, we study the multiplicity of nontrivial nonnegative solutions for a semilinear elliptic equation involving nonlinear boundary condition and sign-changing potential. With the help of the Nehari manifold, we prove that the semilinear elliptic equation: $$displaylines{ -Delta u+u=lambda f(x)|u|^{q-2}u quad hbox{in }Omega , cr frac{partial u}{partial u }=g(x)|u| ^{p-2}u quad hbox{on }partial Omega , }$$has at least two nontrivial nonnegative solutions for $lambda $is sufficiently small.

Let $Omega$ be a bounded domain in $mathbb{R}^{N}$, $Ngeq 3$, and $p=frac{2N}{N-2}$ the limiting Sobolev exponent. We show that for $fin H^1_0(Omega)^ast$, satisfying suitable conditions, the nonlinear elliptic problem $$displaylines{ -Delta u =|u |^{ p-2 }u +f quad hbox{in } Omega cr u=0 quad hbox{on } partialOmega }$$ has at least three solutions in $H_{0}^{1}(Omega)$.

We show the exact asymptotic behaviour near the boundary for the classical solution to the Dirichler problem $$ -Delta =k(x)g(u)+lambda |abla u|^q, quad u>0,; xin Omega,quad uig|_{partial{Omega}}=0, $$ where $Omega$ is a bounded domain with smooth boundary in $mathbb R^N$. We use the Karamata regular varying theory, a perturbed argument, and constructing comparison functions.

We consider the nonlinear eigenvalue problem u"(t) + λ(u(t)p = u(t)q) = 0, u(t) > 0, -1 < t < 1, u(1) = u(-1) = 0, where 1 < p < q are constants and λ > 0 is a parameter. It is known in [13] that the bifurcation curve λ(α) consists of two branches, which are denoted by λ±(α). Here, α = ∥uλ∥∞. We establish the asymptotic behavior of the turning point αp of λ(α), namely, the point which satisfies dλ(αp)=dα = 0 as p → q and p → 1. We also establish the asymptotic formulas for λ+(α) and λ-(α) as α → 1 and α → 0, respectively. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and infinitely many solutions for the elliptic boundary-value problem Our main tools are the local linking and symmetric mountain pass theorem in critical point theory. [ABSTRACT FROM AUTHOR]

By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions to the semilinear elliptic problem $$ Delta u -|abla u|^q=b(x)g(u),quad u>0 quadext{in }Omega, quad uig|_{partial Omega}=+infty, $$ where $Omega$ is a bounded domain in $mathbb{R}^N$ with smooth boundary, $qin (1, 2]$, $g in C[0,infty)cap C^1(0, infty)$, $g(0)=0$, $g$ is increasing on $[0, infty)$, and $b$ is non-negative non-trivial in $Omega$, which may be singular or vanishing on the boundary.

We consider the boundary blow-up nonlinear elliptic problems $Delta upmlambda | abla u|^q=k(x)g(u)$ in a bounded domain with boundary condition $u|_{partial Omega}=+infty$, where $qin [0, 2]$ and $lambdageq0$. Under suitable growth assumptions on $k$ near the boundary and on $g$ both at zero and at infinity, we show the existence of at least one solution in $C^2(Omega)$. Our proof is based on the method of explosive sub-supersolutions, which permits positive weights $k(x)$ which are unbounded and/or oscillatory near the boundary. Also, we show the global optimal asymptotic behaviour of the solution in some special cases.

In this article, we consider a semilinear elliptic equations of the form $Delta u+f(u)=0$, where $f$ is a concave function. We prove for arbitrary dimensions that there is no solution bounded in $(0,1)$. The significance of this result in probability theory is also discussed.

We generalize previous uniqueness results on a semilinear elliptic equation with zero Dirichlet boundary condition and superlinear, subcritical nonlinearity. Our proof is based on a bifurcation approach and a Pohozaev type integral identity, which greatly simplifies the previous arguments.

In this article, we study the second-order forced Liínard equation x" + f(x)x' + g(x) = e(t). By using the topological degree theory, we prove that the equation has at least one positive periodic solution when g admits a repulsive singularity near the origin and satisfies some semilinear growth conditions near infinity. Recent results in the literature are generalized and complemented. [ABSTRACT FROM AUTHOR]

In our previous article [1] we estimated the Lp-norm (p ≥ 1) of the solution to damped fractional wave equation. In this article, we prove other Lp estimates, with some emphasis on requiring less regularity of the initial data. We also study the Strichartz type estimate of this equation. Finally we present some application of these estimates, for proving existence of global solutions to semilinear damped fractional wave equations. [ABSTRACT FROM AUTHOR]

We investigate the exact number of positive solutions of the semilinear Dirichlet boundary value problem $Delta u+f(u) = 0$ on a ball in ${mathbb R}^n$ where $f$ is a strictly convex $C^2$ function on $[0,infty)$. For the one-dimensional case we classify all strictly convex $C^2$ functions according to the shape of the bifurcation diagram. The exact number of positive solutions may be 2, 1, or 0, depending on the radius. This full classification is due to our main lemma, which implies that the time-map cannot have a minimum. For the case $n>1$ we prove that for sublinear functions there exists a unique solution for all $R$. For other convex functions estimates are given for the number of positive solutions depending on $R$. The proof of our results relies on the characterization of the shape of the time-map.

In this article, we study a class of indefinite semilinear elliptic problems in ℝN. By using the fibering maps and studying some properties of the Nehari manifold, we obtain the existence and multiplicity of positive solutions. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the existence and uniqueness of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The discussions are based on analytic semigroup theory and fixed point theorems. An example illustrates the main results. [ABSTRACT FROM AUTHOR]

In this article we consider elliptic equations whose nonlinear term depends on the gradient of the unknown. We assume that the nonlinearity has a asymptotically linear growth at zero and at infinity with respect to the second variable. By applying Morse theory and an iterative method, we prove the existence of nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this article, we prove theorems on existence, uniqueness, and nonexistence of solutions for nonlocal problems of a semilinear wave equations in one space variable. [ABSTRACT FROM AUTHOR]

We consider the elliptic problem with nonlinear boundary conditions: -Δu + bu = f(x, u) in Ω, -∂vu = |u|q-1u - g(u) on ∂Ω, where Ω is a bounded domain in Rn. Proving the existence of solutions of this problem relies essentially on a variational argument. However, since Lq+1(∂Ω) ⊂ H¹(Ω) does not hold for large q, the standard variational method can not be applied directly. To overcome this difficulty, we use approximation methods and uniform a priori estimates for solutions of approximate equations. [ABSTRACT FROM AUTHOR]

Let be G a closed subgroup of O(N). We consider the semilinear elliptic problem Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. where Ω ⊂ RN is a smooth bounded domain, N ≥ 4. We establish the R multiplicity of symmetric positive solutions, nodal solutions, and solutions which are Λ invariant but are not Γ̃ invariant, where Γ̃ O(N). [ABSTRACT FROM AUTHOR]

Under the Keller-Osserman condition on f + g, we show the existence and nonexistence of entire solutions for the semilinear elliptic system Δu = p(x)f(v), Δv = q(x)g(u), x ∈ ℝN, where p, q : ℝN → [0,∞) are continuous functions. [ABSTRACT FROM AUTHOR]

We prove some existence of positive solutions to the semilinear elliptic system Δu = λp(x)g(v) Δv = µq(x)f(u) in the half space Rn+, n ≥ 2, subject to some Dirichlet conditions, where λ and µ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the Kato class K∞(Rn+). [ABSTRACT FROM AUTHOR]

For a given semilinear parabolic equation with polynomial nonlinearity, many solutions blow up in finite time. For a certain class of these equations, we show that some of the solutions which do not blow up actually tend to equilibria. The characterizing property of such solutions is a finite energy constraint, which comes about from the fact that this class of equations can be written as the flow of the L² gradient of a certain functional. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the asymptotic behavior of the classical solution to the semilinear boundary-value problem Δu + a(x)uσ = 0 In Rn, u > 0, lim|x|→∞u(x) = 0, where σ < 1. The special feature is to consider the function a in Cαloc(Rn), 0 < α < 1, such that there exists c > 0 satisfying 1/c L(|x|+1)/(1+|x|)λ ≤ a (x) ≤ c L(|x|+1)/(1+|x|)λ, where L(t) := exp (∫t1z(s)/sds) , with z ∈ C ([1, ∞)) such that limt→∞ z(t) = 0. The comparable asymptotic rate of a(x) determines the asymptotic behavior of the solution. [ABSTRACT FROM AUTHOR]

In this work we consider a class of impulsive fractional-order semilinear evolution equations with a nonlocal initial condition. By means of solution operator and application of fixed point theorems we established the existence and uniqueness of a mild solution. [ABSTRACT FROM AUTHOR]

We use a method based on Picone-type identities to find oscillation conditions for the equation n ∑ ij=1 ∂/∂xi (aij(x)∂/∂xj)u + f(x, u, ∇u) + c(x)u = 0, with Dirichlet boundary conditions on bounded and unbounded domains. In this article, the above method substitudes the traditional Riccati techniques used for unbounded domains. [ABSTRACT FROM AUTHOR]

In this article, we consider a semilinear elliptic equations of the form Δu + f(u) = 0, where f is a concave function. We prove for arbitrary dimensions that there is no solution bounded in (0, 1). The significance of this result in probability theory is also discussed. [ABSTRACT FROM AUTHOR]