On the extremal solutions of superlinear Helmholtz problems

In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

The Existence of Strong Solution for Generalized Navier-Stokes Equations with p x -Power Law under Dirichlet Boundary Conditions

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling by p x -power law with Dirichlet boundary condition under the restriction 3 n / n + 2 n + 2 < p x < 2 n + 1 / n − 1 . In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction 2 n + 1 / n + 3 < p x < 2 n + 1 / n − 1 , which arises from the nonflatness of domain.

Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type

This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent Δ p ( x ) 2 u − M ( ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x ) Δ p ( x ) u + | u | p ( x ) − 2 u = λ f ( x , u ) + μ g ( x , u ) in Ω , u = Δ u = 0 on ∂ Ω , $$\begin{array}{} \left\{\begin{array}{} \Delta^2_{p(x)}u-M\bigg(\displaystyle\int\limits_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,\text{d} x \bigg)\Delta_{p(x)} u + |u|^{p(x)-2}u = \lambda f(x,u)+\mu g(x,u) \quad \text{ in }\Omega,\\ u=\Delta u = 0 \quad \text{ on } \partial\Omega, \end{array}\right. \end{array}$$ where p − := inf x ∈ Ω ¯ p ( x ) > max 1 , N 2 , λ > 0 $\begin{array}{} \displaystyle p^{-}:=\inf_{x \in \overline{\Omega}} p(x) \gt \max\left\{1, \frac{N}{2}\right\}, \lambda \gt 0 \end{array}$ and μ ≥ 0 are real numbers, Ω ⊂ ℝ N (N ≥ 1) is a smooth bounded domain, Δ p ( x ) 2 u = Δ ( | Δ u | p ( x ) − 2 Δ u ) $\begin{array}{} \displaystyle \Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u) \end{array}$ is the operator of fourth order called the p(x)-biharmonic operator, Δ p(x) u = div(|∇u| p(x)–2∇u) is the p(x)-Laplacian, p : Ω → ℝ is a log-Hölder continuous function, M : [0, +∞) → ℝ is a continuous function and f, g : Ω × ℝ → ℝ are two L 1-Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.

Bifurcation analysis for a modified quasilinear equation with negative exponent

In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent

Given a smooth bounded domain [Formula: see text] in [Formula: see text] with [Formula: see text], we study the existence and the profile of positive solutions for the following elliptic Neumann problem: [Formula: see text] where [Formula: see text] is a large exponent and [Formula: see text] denotes the outer unit normal vector to the boundary [Formula: see text]. For suitable domains [Formula: see text], by a constructive way we prove that, for any non-negative integers [Formula: see text], [Formula: see text] with [Formula: see text], if [Formula: see text] is large enough, such a problem has a family of positive solutions with [Formula: see text] boundary layers and [Formula: see text] interior layers which concentrate along [Formula: see text] distinct [Formula: see text]-dimensional minimal submanifolds of [Formula: see text], or collapse to the same [Formula: see text]-dimensional minimal submanifold of [Formula: see text] as [Formula: see text].

Blow-Up Results for a Class of Quasilinear Parabolic Equation with Power Nonlinearity and Nonlocal Source

This paper deals with a class of quasilinear parabolic equation with power nonlinearity and nonlocal source under homogeneous Dirichlet boundary condition in a smooth bounded domain; we obtain the blow-up condition and blow-up results under the condition of nonpositive initial energy.

Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

Qualitative properties of singular solutions to fractional elliptic equations

In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} where $0<\alpha <1$ , $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$ , and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$ , rather than for every $t>0$ . Our main result is that the solutions satisfy the estimate \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} This estimate is new even for $\Gamma =\{0\}$ . As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.

Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces

We consider the following fractional reaction-diffusion equation u t ( t ) + ∂ t ∫ 0 t g α ( s ) A u ( t − s ) d s = t γ f ( u ) , $$ u_t(t) + \partial_t \int\nolimits_{0}^{t} g_{\alpha}(s) \mathcal{A} u(t-s) ds = t^{\gamma} f(u),$$ where g α (t) = t α−1/Γ(α) (0 < α < 1), f ∈ C([0, ∞)) is a non-decreasing function, γ > −1, and A $\mathcal{A}$ is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in L r (Ω), for non-negative initial data u 0 ∈ L r (Ω), when r > 1 and Ω ⊂ ℝ N is either a smooth bounded domain or the whole space ℝ N . The case r = 1 is also studied.

Maximum Principles and ABP Estimates to Nonlocal Lane–Emden Systems and Some Consequences

This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system involving fractional Laplace operators: { ( - Δ ) s ⁢ u = λ ⁢ ρ ⁢ ( x ) ⁢ | v | α - 1 ⁢ v in ⁢ Ω , ( - Δ ) t ⁢ v = μ ⁢ τ ⁢ ( x ) ⁢ | u | β - 1 ⁢ u in ⁢ Ω , u = v = 0 in ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x% )\lvert v\rvert^{\alpha-1}v&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta)^{t}v&\displaystyle=\mu\tau(x)\lvert u\rvert^{\beta-1}u&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=v=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{n}\setminus\Omega,\end{aligned}\right. where s , t ∈ ( 0 , 1 ) {s,t\in(0,1)} , α , β > 0 {\alpha,\beta>0} satisfy α ⁢ β = 1 {\alpha\beta=1} , Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} , n ≥ 1 {n\geq 1} , and ρ and τ are continuous functions on Ω ¯ {\overline{\Omega}} and positive in Ω. We establish some maximum principles depending on Ω. In particular, we explicitly characterize the measure of Ω for which the maximum principles corresponding to this problem hold in Ω. For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Ω. Aleksandrov–Bakelman–Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small | Ω | {\lvert\Omega\rvert} has to be to ensure the positivity of the obtained solutions.