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.

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.

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 .

A semilinear problem with a gradient term in the nonlinearity

<p style='text-indent:20px;'>We consider the following semilinear problem with a gradient term in the nonlinearity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u&gt;0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> be a bounded, smooth domain in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>. We prove that when <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a unit ball and <inline-formula><tex-math id="M5">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ q\in (0,q^*(N)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ q^*(N)\in (1,2) $\end{document}</tex-math></inline-formula>, we have infinitely many radial solutions for <inline-formula><tex-math id="M8">\begin{document}$ 2\leq N&lt;2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda = \tilde \lambda $\end{document}</tex-math></inline-formula>. On the other hand, for <inline-formula><tex-math id="M10">\begin{document}$ N&gt;2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> there exists a unique radial solution for <inline-formula><tex-math id="M11">\begin{document}$ 0&lt;\lambda&lt;\tilde \lambda $\end{document}</tex-math></inline-formula>.</p>

Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

<p style='text-indent:20px;'>In this paper, we consider the following Schrödinger-Poisson equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\triangle u + u + \phi u = u^{5}+\lambda g(u), &amp;\hbox{in}\ \ \Omega, \\\ &amp; -\triangle \phi = u^{2}, &amp; \hbox{in}\ \ \Omega, \\\ &amp; u, \phi = 0, &amp; \hbox{on}\ \ \partial\Omega.\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{3} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> and the nonlinear growth of <inline-formula><tex-math id="M4">\begin{document}$ u^{5} $\end{document}</tex-math></inline-formula> reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.</p>

Bounds for subcritical best Sobolev constants in W1, p

<p style='text-indent:20px;'>This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega),\quad 1\leq q&lt; \begin{cases} \frac{Np}{N-p},&amp; 1\leq p&lt;N\\ \infty,&amp; p = N \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ N\geq p\geq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> or the whole space. The Sobolev limiting case <inline-formula><tex-math id="M5">\begin{document}$ p = N $\end{document}</tex-math></inline-formula> is also covered by means of a limiting procedure.</p>

Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation- div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),where Ω is a bounded smooth domain of 𝕉N.

Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.