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.

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.

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>

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>

Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis

This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = Δ p − ρ ∇ · ( p ∇ w ) + λ p ( 1 − p ) , w t = − γ p w β in a bounded smooth domain Ω ⊂ R N ( N = 1 , 2 ) , where ρ , λ , γ > 0 and β ≥ 1 . More precisely, it is shown that the corresponding solution ( p , w ) converges to ( 1 , 0 ) with an explicit exponential rate if β = 1 , and polynomial rate if β > 1 as t → ∞ , respectively, in L ∞ -norm.

Properties of solutions to some weighted p-Laplacian equation

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,\] on smooth domain and for varying nonlinearity \(f\).

Sign-changing tower of bubbles to an elliptic subcritical equation

This paper is concerned with the following nonlinear elliptic problem involving nearly critical exponent [Formula: see text]: [Formula: see text] in [Formula: see text], [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded smooth domain in [Formula: see text], [Formula: see text], [Formula: see text] is a small positive parameter. As [Formula: see text] goes to zero, we construct a solution with the shape of a tower of sign changing bubbles.