Multiplicity of solutions for a class of quasilinear elliptic equation involving the critical Sobolev and Hardy exponents

In this paper, by using concentration-compactness principle and a new version of the symmetric mountain-pass lemma due to Kajikiya (J Funct Anal 225:352–370, 2005), infinitely many small solutions are obtained for a class of quasilinear elliptic equation with singular potential $$- \Delta_p u - \mu \frac{|u|^{p-2}u}{|x|^p} =\frac{|u|^{p^\ast(s)-2}u}{|x|^s} + \lambda f(x, u),\quad u\in H_0^{1,p}(\Omega).$$      

Variational Inclusion Systems in $\mathbb{R}^N$

The authors study the existence of nontrivial solutions to p-Laplacian variational inclusion systems

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn

We establish sufficient conditions under which the quasilinear equation $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.     

Existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for a class of doubly nonlinear parabolic equations

The class of equations of the type (1) $$\partial u/\partial t - div\overrightarrow a (u,\nabla u) = f,$$ such that (2) $$\begin{array}{l} \overrightarrow a (u,p) \cdot p \ge v_0 |u|^l |p|^m - \Phi _0 (u), \\ |\overrightarrow a (u,p)| \le \mu _1 |u|^l |p|^{m - 1} + \Phi _1 (u) \\ \end{array}$$ with some m ∈ (1,2), l≥0, and Φ _i (u)≥0 is studied. Similar equations arise in the study of turbulent filtration of gas or liquid through porous media. Existence and uniqueness in some class of Hölder continuous generalized solutions of the Cauchy-Dirichlet problem for equations of the type (1), (2), is proved. Bibliography: 9 titles.     

Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri

In this paper we deal with the existence of weak solutions for the following Neumann problem¶¶$ \left\{{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) $ \left\{\begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) &; $ \mbox{in $ \mbox{in \Omega$}\\ {\partial u \over \partial \nu} = 0 $}\\ {\partial u \over \partial \nu} = 0 &; $ \mbox{on $ \mbox{on \partial \Omega$} \right. $}\end{array} \right. ¶¶ where $ \nu $ \nu is the outward unit normal to the boundary $ \partial\Omega $ \partial\Omega of the bounded open set _boxclose^N \Omega \subset \mathbb{R}^N . The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.     

一类奇异拟线性椭圆方程正解的多重性

研究奇异拟线性椭圆型方程{-div(|x|~(-ap)|▽u|~(p-2)▽u) + f(x)|u|~(p-2) = g(x)\u|~(q-2)u + λh(x)|u|~(r-2),x R~N,u(x) 0,x∈ R~N,其中λ0是参数,1pN(N3),1rpgp*=0a(N—p)/p,p*=Np/{N~pd),aa+l,d=a+l-60,权函数f(x),g(x),h(x)满足一定的条件.利用山路引理和Ekeland变分原理证明了问题至少有两个非平凡的弱解.     

一类具奇异势的拟线性椭圆方程的无穷多解

本文用变分法和集中紧性原理获得了一类具奇异势的拟线性椭圆方程-Δ_pu=μ(|μ|~(P~*(s)-2)u)/(|x|~s) λf(x,u),u∈H_0~(1,p)(Ω)的无穷多解．     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.     

A note on regularity of solutions for degenerate singular elliptic boundary value problems

We prove the \(C^{1,\beta }\)-boundary regularity and a comparison principle for weak solutions of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^{N},N>1\ \)with smooth boundary \(\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ \)and f is allowed to be unbounded.

     

Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.     

对带混合非线性项的基尔霍夫方程规范解问题的补充

在本文中,我们研究了带有质量约束条件的基尔霍夫方程规范解的存在性问题, $$-\Big(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} \text{d} x\Big)\Delta u=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u,~~x\in \mathbb{R}^{3},$$ 其中质量约束条件为$$S_{c}:=\Big\{u \in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}|u|^{2} \text{d} x=c\Big\},$$ 这里$a$, $b$, $c>0$, $\mu\in \mathbb{R}$, $20$, 当$(p, q)$ 属于$\mathbb{R}^{2}$中的某个域时. 我们通过使用约束最小化、集中紧性原理和Minimax方法证明了规范解的存在性.我们部分地扩展了已经被研究的结果.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

Infinitely many solutions for a zero mass Schodinger-Poisson-Slater problem with critical growth

In this paper, we are concerned with the following Schr\"{o}dinger-Poisson-Slater problem with critical growth: $$ -\Delta u+(u^{2}\star \frac{1}{|4\pi x|})u=\mu k(x)|u|^{p-2}u+|u|^{4}u\,\,\mbox{in}\,\,\R^{3}. $$ We use a measure representation concentration-compactness principle of Lions to prove that the $(PS)_{c}$ condition holds locally. Via a truncation technique and Krasnoselskii genus theory, we further obtain infinitely many solutions for $\mu\in(0,\mu^{\ast})$ with some $\mu^{\ast}>0$.     

具Hardy-Sobolev临界指数的奇异椭圆方程多解的存在性

运用变分方法研究了下面问题-Δpu=μupx(s)s-2u f(x,u),x∈Ω,u=0,x∈Ω,多重解的存在性,其中Ω是一个具有光滑边界的有界区域.     

L ∞-solutions for some nonlinear degenerate elliptic and parabolic equations

In this paper we prove the existence of bounded solutions for equations whose prototype is:

Regular solutions of initial-boundary value problems for linear and nonlinear wave-equations I

This paper deals with the question of the existence of classical solutions for the equations $$\frac{{\partial ^{2} u}{\partial t^{2} }} + \sum_{\begin{subarray}{l} |\alpha| \leqslant m \\ | \beta | \leqslant m \end{subarray}} D^{\alpha} (A_{\alpha \beta } (x,t) D^{\beta} u) = f (t,x,u)$$ on [0,T] × G. G is a bounded or unbounded domain; the differential operator in the space variables is elliptic; the initial values of u are prescribed and D^αu (t,x) vanishes for (t,x) ∈ [0,T] × ?G, |α|≤ m?1. First we develop a method for solving regularly linear wave equations. In contrast to the usual compatibility conditions, our method requires less differentiability in t but imposes some boundary conditions on f(t). It allows some applications to nonlinear problems which will be treated in the second part of this paper and which e.g. enable us to solve ?² u/?t²?A(t)u+u³=f.     

Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions

In this paper, the authors consider the positive solutions of the system of the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rmdiv}(| ∇u |^{p−2} ∇u) + f(u, v), & (x, t) ∈ Ω × (0, T ), & \\ v_t = {\rmdiv}(| ∇v |^{p−2} ∇v) + g(u, v), &(x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η}= h(u, v), \frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in$\boldsymbol{R}^n$with smooth boundary $∂Ω, p > 2$, $h(· , ·)$ and $s(· , ·)$ are positive $C^1$ functions, nondecreasing in each variable. The authors find conditions on the functions $f, g, h, s$ that prove the global existence or finite time blow-up of positive solutions for every $(u_0, v_0)$.