Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity

In the present paper, by applying variant mountain pass theorem and Ekeland variational principle we study the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity $$ \left\{\begin{array}{ll} -(a + b \int\nolimits_{\Omega} |\nabla{u}|^{2})\triangle{u} = \alpha(x)|u|^{q-2}u + f(x, u),\quad{\rm in}\;\Omega,\\ u = 0,\;\quad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\rm on}\;\partial\Omega, \end{array} \right. $$ A new existence theorem and an interesting corollary of four nontrivial solutions are obtained.     

Global W2,p (2<=p

This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.     

Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein– Gordon–Maxwell equations on R 3

In this paper we study the following nonhomogeneous Schrödinger–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+V(x)u+ \phi u=f(x,u)+h(x),} \quad {\rm in}\,\,\,{\mathbf{R}}^3,\\ {-\triangle \phi=u^2, \qquad\qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,{\mathbf{R}}^3,} \end{array} \right.$ where f satisfies the Ambrosetti–Rabinowitz type condition. Under appropriate assumptions on V, f and h, the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory. Similar results for the nonhomogeneous Klein–Gordon–Maxwell equations $\left\{\begin{array}{ll} {-\triangle u+[m^2-(\omega+\phi)^2]u=|u|^{q-2}u+h(x), \quad {\rm in} \,\,\,{\mathbf{R}}^3,}\\ {-\triangle \phi+ \phi u^2=-\omega u^2, \qquad\qquad\qquad\qquad\qquad\,\,\, {\rm in} \,\,\,{\mathbf{R}}^3,} \end{array} \right.$ are also obtained when 2 < q < 6.     

Positive Solution of a Semilinear Elliptic Equation on RN

In this paper, we obtain the existence of positive solution of {-Δu = b(x)(u - λ)^p_+,\qquad x ∈ R^N λ > 0, |∇ u| ∈ L² (R^N),\qquad u ∈ L\frac{2N}{N-2} (R^N) under the assumptions that 1 < p < \frac{N+2}{N-2}, N ≥ 3, b(x) satisfies b(x) ∈ C(R^N), b(x) > 0 in R^N b(x) →_{|x|→∞}b^∞ and b(x) > \frac{4}{p+3}b^∞ for x ∈ R^N     

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.     

Fujita exponents for evolution problems with nonlocal diffusion

We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem

一类三点边值问题正解的存在性

在与线性问题第一特征值相关的条件下,通过应用不动点指数理论讨论了三点边值问题u″ 9(t)f(u)=0,t∈(0,1),u′(0)=0,u(1)=αu(η)正解的存在性,这里η∈(0,1),α∈R且0<α<1.本文结果推广和改进了文献[1]的主要结论.     

A Variational Approach to a Kirchhoff-type Problem Involving Two Parameters

In this paper, the authors establish the existence of at least three weak solutions for the Kirchhoff-type problem $$\left\{\begin{array}{ll}-K \left( \int_{\Omega}| \nabla u(x)|^{2}dx \right) \Delta u(x)= \lambda f(x,u)+\mu g(x,u),\quad {\rm in}\; \Omega,\\u=0, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad {\rm on}\; \partial \Omega, \end{array} \right.$$ under appropriate hypotheses. The proofs are based on variational methods.     

On Stable Solutions to a Weighted Degenerate Elliptic Equation with Advection Terms

Mathematical Notes - In this paper, we study the elliptic equations $$-G_\alpha u+c(\text {x} )\cdot\nabla_\alpha u=h(\text {x} )e^{u}, \qquad \text {x} = (x,y) \in \mathbb R^{N_{1}}\times \mathbb...     

Some Generic Properties of Non Degeneracy for Critical Points of Functionals and Applications

We give some generic properties of non degeneracy for critical points of functionals. We apply these results, obtaining some theorems of multiplicity of solutions for the equation

$ \left\{\begin{array}{ll} -\varepsilon^2\Delta_g u + u = |u|^{p-2}u \quad {\rm in}\ M \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad, \\ u \in H_g^1(M) \end{array}\right. $

where M is a compact Riemannian manifold of dimension n and \({2 < p < \frac{2n}{n\,-\,2}}\).

     

Existence of C1-solutions to Certain Non-uniformly Degenerate Elliptic Equations

We are concerned with the Dirichlet problem of {div A(x, Du) + B(z) = 0 \qquad in Ω u= u_0 \qquad \qquad on ∂ Ω Here Ω ⊂ R^N is a bounded domain, A(x, p) = (A¹ (x, p), ... >A^N (x, p}) satisfies min{|p|^{1+α}, |p|^{1+β}} ≤ A(x, p) ⋅ p ≤ α_0(|p|^{1+α}+|p|^{1+β}) with 0 < α ≤ β. We show that if A is Lipschitz, B and u_0 are bounded and β < max {\frac{N+2}{N}α + \frac{2}{N},α + 2}, then there exists a C¹-weak solution of (0.1).     

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

Classification and existence of positive solutions of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations of the form

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C ¹-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.     

On the Existence and Stability of Positive Solutions for Some Pairs of Differential Equations

In this paper, we are concerned with the existence and stability of the positive solutions of a semilinear elliptic system - Δu(x) = a(x)v^6(x) + e(x) - Δv(x) = b(x}u^μ(x) + m(x) \qquad in Ω u = v = 0 \qquad\qquad on ∂Ω where Ω ⊂ R^N is a bounded domain with smooth boundary ∂Ω. It is shown that under the suitable conditions on \delta, μ, there exist a stable and an unstable positive solutions for this system if e and m are sufficiently small in L^∞.     

Geometry of phase space and solutions of semilinear elliptic equations in a ball

We consider the problem

where denotes the unit ball in , , and . Merle and Peletier showed that for there is a unique value such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionally

then for close to , a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if . We establish a similar assertion for the problem

where , , and satisfies the same condition as above.

     

Some remarks on quasilinear parabolic problems with singular potential and a reaction term

In this paper we deal with the following quasilinear parabolic problem $$\left\{\begin{array}{l@{\quad}l} (u^\theta)_t - \Delta_p {u} = \lambda \frac{u^{p - 1}}{|x|^{p}} + u^q + f,\,\, u \geq 0 \quad {\rm in} \;\;\Omega \times (0, T),\\ u(x, t) = 0 \quad\qquad\qquad\qquad\qquad\qquad\qquad {\rm on}\; \partial \Omega \times(0, T),\\ u(x, 0) = u_0(x), \,\,\, \qquad\qquad\qquad\qquad\qquad x \in\; \Omega,\end{array}\right.$$ where θ is either 1 or (p ? 1), \({N \geq 3, \,\Omega \subset \mathcal{IR}^N}\) is either a bounded regular domain containing the origin or \({\Omega \equiv \mathcal{IR}^N}\) , 1 < p < N, q > 0 and u ₀ ≥  0, f ≥  0 with suitable hypotheses. The aim of this work is to get natural conditions to show the existence or the nonexistence of nonnegative solutions. In the case of nonexistence result, we analyze blow-up phenomena for approximated problems in connection with the classical Harnack inequality, in the Moser sense, more precisely in connection with a strong maximum principle. We also study when finite time extinction (1 < p < 2) and finite speed propagation (p > 2) occur related to the reaction power.     

Existence and Comparison Results for Non-Uniformly Parabolic Problems

In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem

Hölder regularity for a Kolmogorov equation

We study the interior regularity properties of the solutions to the degenerate parabolic equation,

which arises in mathematical finance and in the theory of diffusion processes.