Volume 7, Issue 4

The paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case Ω=ℝN{\Omega=\mathbb{R}^{N}}, we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.

In this paper, we present the proof of superlensing an arbitrary object using complementary media and we study reflecting complementary media for electromagnetic waves. The analysis is based on the reflecting technique and new results on the compactness, existence, and stability for the Maxwell equations with low regularity data.

In this paper, we are concerned with the nonlinear elliptic systems in divergence form under controllable growth condition. We prove that the weak solution u is locally Hölder continuous besides a singular set by using the direct method and classical Morrey-type estimates. Here the Hausdorff dimension of the singular set is less than n-p{n-p}. This result not only holds in the interior, but also holds up to the boundary.

In this paper, we deal with fractional p -Laplacian equations of the form {(-Δ)ps⁢u=λ⁢f⁢(x,u),x∈Ω,u⁢(x)=0,x∈ℝN∖Ω,\left\{\begin{aligned} \displaystyle(-\Delta)_{p}^{s}u&\displaystyle=\lambda f% (x,u),&&\displaystyle x\in\Omega,\\ \displaystyle u(x)&\displaystyle=0,&&\displaystyle x\in\mathbb{R}^{N}\setminus% \Omega,\end{aligned}\right. where λ∈(0,+∞){\lambda\in(0,+\infty)}, 0<s<1<p<+∞{0<s<1<p<+\infty} and Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}}, N⩾2{N\geqslant 2}, is a bounded domain with smooth boundary. With assumptions on f⁢(x,t){f(x,t)} just in Ω×(-δ,δ){\Omega\times(-\delta,\delta)}, where δ>0{\delta>0} is small, existence and multiplicity of nontrivial solutions are obtained via variational methods.

Local asymptotic stability analysis is conducted for an initial-boundary-value problem of a Korteweg–de Vries equation posed on a finite interval [0,2⁢π⁢7/3]{[0,2\pi\sqrt{7/3}]}. The equation comes with a Dirichlet boundary condition at the left end-point and both the Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the associated linearized equation around the origin is not asymptotically stable. In this paper, the nonlinear Korteweg–de Vries equation is proved to be locally asymptotically stable around the origin through the center manifold method. In particular, the existence of a two-dimensional local center manifold is presented, which is locally exponentially attractive. Analyzing the Korteweg–de Vries equation restricted on the local center manifold, we obtain a polynomial decay rate of the solution.

In this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

In this paper, we concern ourselves with the following Kirchhoff-type equations: {-(a+b⁢∫ℝ3|∇⁡u|2⁢𝑑x)⁢△⁢u+V⁢u=f⁢(u) in ⁢ℝ3,u∈H1⁢(ℝ3),\left\{\begin{aligned} \displaystyle-\biggl{(}a+b\int_{\mathbb{R}^{3}}\lvert% \nabla u\rvert^{2}\,dx\biggr{)}\triangle u+Vu&\displaystyle=f(u)\quad\text{in % }\mathbb{R}^{3},\\ \displaystyle u&\displaystyle\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right. where a , b and V are positive constants and f has critical growth. We use variational methods to prove the existence of ground state solutions. In particular, we do not use the classical Ambrosetti–Rabinowitz condition. Some recent results are extended.

In this paper, we consider the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms: {ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+β⁢θx=0,(x,t)∈(0,1)×(0,∞),ρ3⁢θt⁢t-δ⁢θx⁢x+γ⁢ψt⁢t⁢x+∫0tg⁢(t-s)⁢θx⁢x⁢(s)⁢ds+μ1⁢θt⁢(x,t)+μ2⁢θt⁢(x,t-τ)=0,(x,t)∈(0,1)×(0,∞),\left\{\begin{aligned} &\displaystyle\rho_{1}\varphi_{tt}-K(\varphi_{x}+\psi)_% {x}=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{2}\psi_{tt}-b\psi_{xx}+K(\varphi_{x}+\psi)+\beta\theta_{x}% =0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{3}\theta_{tt}-\delta\theta_{xx}+\gamma\psi_{ttx}+\int_{0}^% {t}g(t-s)\theta_{xx}(s)\,\mathrm{d}s+\mu_{1}\theta_{t}(x,t)+\mu_{2}\theta_{t}(% x,t-\tau)=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\end{aligned}\right. together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ1,μ2{\mu_{1},\mu_{2}} are positive constants. Under a hypothesis between the weight of the delay term and the weight of the friction damping term, we prove the global existence of solutions by using the Faedo–Galerkin approximations together with some energy estimates. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the above two weights, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.

The aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations for it. Then we show the existence of solutions and meaningful properties such as measurability and upper semicontinuity for the solution set of the mixed variational quasi hemivariational inequality associated to the partial differential hemivariational inequality. Relying, on these properties we are able to prove the existence of mild solutions for partial differential hemivariational inequalities. Furthermore, we show the compactness of the set of the corresponding mild trajectories.

Using concentration-compactness arguments, we prove a variant of the Brézis–Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.