Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

This paper concerns the existence and concentration of positive solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity. The proof relies on variational method, truncated methods, and nonsmooth critical points theory. Some related results are improved.     

Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Using variational methods, we establish existence of multi-bump solutions for the following class of problems

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2 u +(\lambda V(x)+1)u = f(u), \quad \text{ in } \quad \mathbb {R}^{N},\\ u \in H^{2}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$

where \(N \ge 1\), \(\Delta ^2\) is the biharmonic operator, f is a continuous function with subcritical growth, \(V : \mathbb {R}^N \rightarrow \mathbb {R}\) is a continuous function verifying some conditions and \(\lambda >0\) is a real constant large enough.

     

Ground state and nodal solutions for a class of biharmonic equations with singular potentials

In this paper, we are concerned with a class of fourth order elliptic equations of Kirchhoff type with singular potentials in $\mathbb{R}^{N}.$ The existence of ground state and nodal solutions are obtained by using variational methods and properties of Hessian matric. Furthermore, the "energy doubling" property of nodal solutions is also explored.     

Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin

In this paper, we investigate the existence of multiple solutions for a class of biharmonic equations where the nonlinearity involves a concave term at the origin. The solutions are obtained from the versions of mountain pass lemma and linking theorem.     

Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation ${(\mathcal E)_\lambda}$ in ${\mathbb{R}^N}$ , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that ${(\mathcal E)_\lambda}$ admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when ${\lambda > \overline{\lambda} \ge \lambda^*}$ , the existence of a second independent nontrivial non-negative entire solution of ${(\mathcal{E})_\lambda}$ is proved under a further natural assumption on A.     

一类非线性Schrodinger-Maxwell方程组半经典解的存在性

本文将研究如下非线性Schrodinger—Maxwell方程组问题 {-ε^2△u＋V（x）u＋K（x）Фu=｜u｜^p-2u, x∈R^3, -△Ф=4πK（x）u^2, x∈R^3. 当势函数V（x）和电量函数K（x）满足一定假设条件时,作者利用变分法证明了ε充分小时,该方程组半经典解的存在性．     

Existence and concentration of positive solutions for a class of gradient systems

Some gradient systems with two competing potential functions are considered. Bound states (solutions with finite energy) are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of positive solutions are related to a suitable ground energy function.     

Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Existence and stability of oscillating solutions for a class of delay differential equations

In this paper we study a Hopf bifurcation for a general class of simple delay differential equations with a term that can describe a treatment. Conditions guaranteeing appearance of the Hopf bifurcation as well as stability of arising periodic orbits are proved. Applications of proved theorems to a general version of the logistic equation are presented.     

Existence of three nontrivial solutions for a class of superlinear elliptic equations

The existence of three nontrivial solutions for a class of superlinear elliptic equations is obtained by using variational theorems of mixed type due to Marino and Saccon and Linking Theorem.     

Existence of homoclinic solutions for a class of neutral functional differential equations

By means of Mawhin’s continuation theorem and some analysis methods, the existence of 2kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above periodic solutions set.     

Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),p∈(max{1 +(α+2s)/N,2},(N+α)/(N-2s)) and κα(x)=|x|α-N. To get such solutions,we look for critical points of the energy functional I(u) =1/2∫RN|(-△)s/2u|2-1/(2p)∫RN(κα*|u|p)|u|p on the constraints S(c)={u∈Hs(RN):‖u‖L2(RN)2=c},c >0.For the value p∈(max{1+(α+2s)/N,2},(N+α)/(N-2s)) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c>0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that,we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover,by using a minimax procedure, we prove that for any c>0, there are infinitely many radial critical points of I restricted on S(c).     

Radial entire solutions for supercritical biharmonic equations

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.