High perturbations of a new Kirchhoff problem involving the p-Laplace operator

In the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

A three solutions theorem for Pucci's extremal operator and its application

In this article we prove a three solution type theorem for the following boundary value problem: \begin{equation*} \label{abs} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u) =f(u)& \text{in }\Omega,\\ u =0& \text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ and $f\colon [0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function. This is motivated by the work of Amann \cite{aman} and Shivaji \cite{shivaji1987remark}, where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when $f$ has a sublinear growth and $f(0)=0.$

Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation det ( D 2 u ) = b ( x ) g ( − u ) , u < 0 in Ω and u = 0 on ∂ Ω , $$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$ where Ω is a bounded, smooth and strictly convex domain in ℝ N (N ≥ 2), b ∈ C∞(Ω) is positive and may be singular (including critical singular) or vanish on the boundary, g ∈ C 1((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $ and g is normalized regularly varying at zero with index −γ(γ>1). Our results reveal the refined influence of the highest and the lowest values of the (N − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.

Long-time dynamics of a class of nonlocal extensible beams with time delay

This work is devoted to the following nonlocal extensible beam equation with time delay: [Formula: see text] on a bounded smooth domain [Formula: see text]. The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

Statistical Solutions for a Nonautonomous Modified Swift-Hohenberg Equation

We consider the nonautonomous modified Swift-Hohenberg equation $$u_t+\Delta^2u+2\Delta u+au+b|\nabla u|^2+u^3=g(t,x)$$ on a bounded smooth domain $\Omega\subset\R^n$ with $n\leqslant 3$. We show that, if $|b|<4$ and the external force $g$ satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space $H_0^2(\Omega)$. Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order $$2m\ge 4$$ 2 m ≥ 4 have been shown so far only when the principal part of L is the polyharmonic operator $$(-\Delta )^m$$ ( - Δ ) m . In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.

Positive solutions of the prescribed mean curvature equation with exponential critical growth

In this paper, we are concerned with the problem $$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ - div ∇ u 1 + | ∇ u | 2 = f ( u ) in Ω , u = 0 on ∂ Ω , where $$\Omega \subset {\mathbb {R}}^{2}$$ Ω ⊂ R 2 is a bounded smooth domain and $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → R is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.

Nonnegative solutions for the fractional Laplacian involving a nonlinearity with zeros

We study the nonlocal nonlinear problem $$\begin{aligned} \left\{ \begin{array}[c]{lll} (-\Delta )^s u = \lambda f(u) &{} \text{ in } \Omega , \\ u=0&{}\text{ on } \mathbb {R}^N{\setminus }\Omega , \quad (P_{\lambda }) \end{array} \right. \end{aligned}$$ ( - Δ ) s u = λ f ( u ) in Ω , u = 0 on R N \ Ω , ( P λ ) where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^N$$ R N , $$N>2s$$ N > 2 s , $$0<s<1$$ 0 < s < 1 ; $$f:\mathbb {R}\rightarrow [0,\infty )$$ f : R → [ 0 , ∞ ) is a nonlinear continuous function such that $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 and $$f(t)\sim |t|^{p-1}t$$ f ( t ) ∼ | t | p - 1 t as $$t\rightarrow 0^+$$ t → 0 + , with $$2<p+1<2^*_s$$ 2 < p + 1 < 2 s ∗ ; and $$\lambda $$ λ is a positive parameter. We prove the existence of two nontrivial solutions $$u_{\lambda }$$ u λ and $$v_{\lambda }$$ v λ to ($$P_{\lambda }$$ P λ ) such that $$0\le u_{\lambda }< v_{\lambda }\le 1$$ 0 ≤ u λ < v λ ≤ 1 for all sufficiently large $$\lambda $$ λ . The first solution $$u_{\lambda }$$ u λ is obtained by applying the Mountain Pass Theorem, whereas the second, $$v_{\lambda }$$ v λ , via the sub- and super-solution method. We point out that our results hold regardless of the behavior of the nonlinearity f at infinity. In addition, we obtain that these solutions belong to $$L^{\infty }(\Omega )$$ L ∞ ( Ω ) .

Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in Ω , − Δ ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , where Ω is a bounded smooth domain of R 3 , a > 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 < p < 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 < p < 12 5 .