A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

Monotonicity of the reflected Bessel transition density on the diagonal

For α∈R$\alpha \in \mathbb{R}$, let p^R(t,x,x)$p^R(t,x,x)$ denote the diagonal of the transition density of the α$\alpha$-Bessel process in (0,1]$(0,1]$, killed at 0 and reflected at 1. As a function of x$x$, if either α≥3$\alpha \ge 3$ or α=1$\alpha =1$, then for t>0$t>0$, the diagonal is nondecreasing. This monotonicity property fails if 1≠α<3$1\ne \alpha <3$.     

Multiple solutions for a class of semilinear elliptic equations with general potentials

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations −△u+a(x)u=g(x,u)$-△u+a\left(x\right)u=g(x,u)$ in a bounded smooth domain of R^N(N≥3)${\mathbb{R}}^N(N\ge 3)$ with the Dirichlet boundary value, where the primitive of the nonlinearity g$g$ is of superquadratic growth near infinity in u$u$ and the potential a$a$ is allowed to be sign-changing. Recent results in the literature are generalized and significantly improved.     

Boundary value problems in spaces of distributions on smooth and polygonal domains

We study boundary value problems of the form -Δu=f$-\mathrm{\Delta }u=f$ on Ω$\Omega$ and Bu=g$\mathit{Bu}=g$ on the boundary ∂Ω$\mathrm{\partial }\Omega$, with either Dirichlet or Neumann boundary conditions, where Ω$\Omega$ is a smooth bounded domain in Rⁿ${\mathbb{R}}^n$ and the data f,g$f,g$ are distributions  . This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution u$u$ in terms of f and g  . For f=0$f=0$, taking advantage of the fact that u$u$ is harmonic on Ω$\Omega$, we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f≠0$f\ne 0$ and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case.     

Existence and multiplicity of nonnegative solutions for semilinear elliptic problems involving nonlinearities indefinite in sign

Let r,s∈]1,2[$r,s\in ]1,2[$ and λ,μ∈]0,+∞[$\lambda ,\mu \in ]0,+\infty [$. In this paper, we deal with the existence and multiplicity of nonnegative and nonzero solutions of the Dirichlet problem with 0$0$ boundary data for the semilinear elliptic equation −Δu=λu^s−1−u^r−1$-\Delta u=\lambda u^{s-1}-u^{r-1}$ in Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N≥2$N\ge 2$. We prove that there exists a positive constant Λ$\Lambda$ such that the above problem has at least two solutions, at least one solution or no solution according to whether λ>Λ$\lambda >\Lambda$, λ=Λ$\lambda =\Lambda$ or λ<Λ$\lambda <\Lambda$. In particular, a result by Hernandéz, Macebo and Vega is improved and, for the semilinear case, a result by Díaz and Hernandéz is partially extended to higher dimensions. Finally, an answer to a conjecture, recently stated by the author, is also given.     

Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators D$D$ with 0∈kerD$0\in kerD$ is shown for which the fact that a non-zero t?u(t)$t?u\left(t\right)$ belongs to a cone and −Du$-Du$ to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0$Du=0$ are given.