Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension

In this paper, we study the following Kirchhoff‐type elliptic problem where is a bounded domain with smooth boundary ?Ω, a,b,λ,μ > 0 and 1 < q < 2^?=2N/(N ? 2). When N = 4, we obtain that there is a ground state solution to the problem for q∈(2,4) by using a variational methods constrained on the Nehari manifold and also show the problem possesses infinitely many negative energy solutions for q∈(1,2) by applying usual Krasnoselskii genus theory. In addition, we admit that there is a positive solution to the equations for N≥5 under some suitable conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiplicity of solutions for critical Kirchhoff type equations

We prove the existence of multiple solutions for critical Kirchhoff equations involving positive operators in closed manifolds.     

Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents

In this paper, we study the Dirichlet problem for a class of semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents and get the existence of infinitely many solutions in both case.     

Positive ground state solutions for fractional Kirchhoff type equations with critical growth

We study the existence of positive ground state solutions for the following fractional Kirchhoff type equation where a,b > 0 are constants, μ is a positive parameter, with and s ∈ (0,1). Under suitable assumptions on V(x), by using a monotonicity trick and a global compactness principle, we prove that the equation admits a positive ground state solution if and μ > 0 large enough.     

Low regularity global solutions for nonlinear evolution equations of Kirchhoff type

We study the global solvability of the Cauchy-Dirichlet problem for two second order in time nonlinear integro-differential equations:

1)

the extensible beam/plate equation

     

Kirchhoff type problems involving $p$-biharmonic operators and critical exponents

In this paper, we study the existence of solutions for a class of Kirchhoff type problems involving $p$-biharmonic operators and critical exponents. The proof is essentially based on the mountain pass theorem due to Ambrosetti and Rabinowitz [2] and the Concentration Compactness Principle due to Lions [18,19].     

Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents

We formulate the concentration-compactness principle at infinity for both subcritical and critical case. We show some applications to the existence theory of semilinear elliptic equations involving critical and subcritical Sobolev exponents.     

Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents

In this paper, we study multiplicity of positive solutions for a class of Kirchhoff type of equations with the nonlinearity containing both singularity and critical exponents. We obtain two positive solutions via the variational and perturbation methods.     

Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent

In this paper we investigate the following Kirchhoff type elliptic boundary value problem involving a critical nonlinearity: $$\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=\mu g(x,u)+u^5, u>0& \text{in }\Omega,\\ u=0& \text{on }\partial \Omega,\end{array}\right. {\rm {(K1)}}$$ here \({\Omega \subset \mathbb{R}^3}\) is a bounded domain with smooth boundary \({\partial \Omega, a,b \geq 0}\) and a + b > 0. Under several conditions on \({g \in C(\overline{\Omega} \times \mathbb{R}, \mathbb{R})}\) and \({\mu \in \mathbb{R}}\) , we prove the existence and nonexistence of solutions of (K1). This is some extension of a part of Brezis–Nirenberg’s result in 1983.     

Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu ^p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa _ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have \(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=u ^p possesses a positive solution inH ₀ ¹ (Ω). We also prove the existence of nonradial solutions to the problem ?Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ?Ω, Ω=B(0,1). for a class off(r, u).     

Existence of positive solutions for nonlinear Kirchhoff type problems in with critical Sobolev exponent

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: where a, b > 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods. Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem where ϵ > 0 is a parameter, V (x) is a positive continuous potential, and with 4 < p < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.     

POSITIVE SOLUTIONS AND BIFURCATION OF FULLY NONLINEAR ELLIPTIC EQUATIONS INVOLVING SUPER－CRITICAL SOBOLEV EXPONENTS

ＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＡＮＤＢＩＦＵＲＣＡＴＩＯＮＯＦＦＵＬＬＹＮＯＮＬＩＮＥＡＲＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳＩＮＶＯＬＶＩＮＧＳＵＰＥＲ－ＣＲＩＴＩＣＡＬＳＯＢＯＬＥＶＥＸＰＯＮＥＮＴＳ￥ＱＵＣＨＡＮＧＺＨＥＮＧ（屈长征）（Ｉｎｓ...     

On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents

Letp=2N/(N –2),N 3 be the limiting Sobolev exponent and ^N a bounded smooth domain. We show that for H ^–1(),f satisfies some conditions then–u=c ₁ u ^p–1 +f(x,u) + admits at least two positive solutions.     

One-bump positive solutions for the Kirchhoff equations with perturbation in

In this paper, we study the Kirchhoff equations with perturbation in . Applying the finite reduction method, we prove that the equation has one-bump positive solutions under some suitable conditions that are given in Section 1.     

Existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities

In this paper we study the nonlinear Kirchhoff equations on the whole space. We show the existence, non-existence, and multiplicity of solutions to this problem with asymptotically linear nonlinearities. This result can be regarded as an extension of the result in Li et al. (2012).