Existence and global behavior of positive solutions of semilinear fractional Dirichlet problems in exterior domains

We prove the existence of multipeak positive solutions for an elliptic Neumann problem in exterior domains involving nearly critical Sobolev exponent.

The existence of a positive solution for a class of asymptotically lin- ear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function.

Semilinear mixed problems in exterior domains for σ-evolution equations with friction and coefficients depending on spatial variables

We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$: (*) \begin{align} \left\{ \begin{array}{ll} -\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\ \\ u=0, & x\in\partial \Omega,\\ \end{array}\right. \end{align}where a > 0, $b\geq0$, and λ > 0 are constants, $\partial\Omega\neq\emptyset$, $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in H_{0}^{1}(\Omega)$, and $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if \begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}has only finite number of positive solutions in $H^1(\mathbb R^3)$ and the diameter of the hole $\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.

Multiple positive solutions for an inhomogeneous semilinear problem in exterior domains

We state a result concerning the limit of a class of minimization problems. This result is applied to describe the asymptotic behaviour of the solutions of an elliptic Dirichlet problem in exterior domains $\Omega$ of $\mathbb{R}^N$, when $\mathbb{R}^N$ \ $Omega$ becomes larger and larger.

We study the variety of solutions of the inhomogeneous div--curl problem in exterior domains in dependence on the decay conditions on div and curl. Here we consider the Neumann as well as the Dirichlet boundary value prescription where in the first case the topological impact is decisive. In the second case the integrability conditions on div, curl and the boundary values are more difficult. Finally we present Hölder estimates for the solution of the Dirichlet or Neumann problem where it is unique.

In this note, we consider the boundary value problem in exterior domains for the \(p\)-Laplacian system, \(p\in (1,2)\). For suitable \(p\) and \(L^r\)-spaces, \(r>n\), we furnish existence of a high-regular solution that is a solution whose second derivatives belong to \(L^r(\varOmega )\). Hence, in particular we get \(\lambda \)-Holder continuity of the gradient of the solution, with \(\lambda =1-\frac{n}{r}\). Further, we improve previous results on \(W^{2,2}\)-regularity in a bounded domain.

The paper concerns with the existence of positive solutions for the following Kirchhoff problem { − ( a + b ∫ Ω | ∇u | 2 d x ) Δu + u = | u | p − 2 u + ε | u | 4 u in Ω , u ∈ H 0 1 ( Ω ) , where a, b>0, 4<p<6, 0 $ ]]> ε > 0 is a parameter and Ω ⊂ R 3 is an exterior domain, that is, Ω is an unbounded domain with R 3 ∖Ω nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when R 3 ∖Ω is contained in a small ball and 0 $ ]]> ε > 0 is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309–330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation.

Strongly singular problems in exterior domains

In this paper we study the existence and multiplicity of solutions for the semilinear elliptic equation $-\Delta u = Q(x)f'(u)$ in an exterior domain with Neumann boundary conditions. We prove the existence of a positive ground state as well as a sign-changing solution under a double power growth condition on the nonlinearity.

We prove the existence of a sign changing solution to the semilinear elliptic problem \(\ -\Delta u+u=\mid u\mid^{p-2}u, \ u\in H_{0}^{1}(\Omega)\), in an exterior domain Ω having finite symmetries.

Global solutions for a nonlinear absorption parabolic problem in exterior domains

Review for "Kinetic Monte Carlo methods for three-dimensional diffusive capture problems in exterior domains"

Review for "Kinetic Monte Carlo methods for three-dimensional diffusive capture problems in exterior domains"