Solutions For Kirchhoff Type Problems Research Articles

Unilateral global interval bifurcation and one-sign solutions for Kirchhoff type problems

<abstract><p>In this paper, we study the following Kirchhoff type problems:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{l} -(\int_{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u^{3}+g(u, \lambda), \, \, \, \, \, \, \, \, \mathrm{in}\, \, \Omega,\\ u = 0, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \mathrm{on}\, \, \partial\Omega, \end{array} \right. $\end{document} </tex-math></disp-formula></p> <p>where $ \lambda $ is a parameter. Under some natural hypotheses on $ g $ and $ \Omega $, we establish a unilateral global bifurcation result from interval for the above problem. By applying the above result, under some suitable assumptions on nonlinearity, we shall investigate the existence of one-sign solutions for a class of Kirchhoff type problems.</p></abstract>

The third solution for a Kirchhoff-type problem with a critical exponent

The aim of this paper is to analyze the multiplicity of solutions for a nonhomogeneous Kirchhoff-type problem on R4 with a critical exponent. The first two positive solutions are deduced by using the variational method, and the third solution is found with the help of Brezis-Lieb's lemma and Mazur's lemma. Moreover, we obtain a new compactness condition and improve on the existing results in the literature.

Existence of nonnegative nontrivial solutions for Kirchhoff type problems with variable exponent

This paper is devoted to study a class of Kirchhoff type problems with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of nonnegative nontrivial solutions to this problem is obtained.

Multiplicity of solutions for Kirchhoff type problem involving eigenvalue

This paper is concerned with the existence and multiplicity of weak solutions for a p(x)-Kirchhoff problem by using variational method and genus theory. We prove the simplicity and boundedness of the principal eigenvalue.

Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki–Lions type conditions

Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki–Lions type conditions

Existence of ground state solutions for Kirchhoff type problem without the Ambrosetti–Rabinowitz condition

In this paper, we prove the existence of a positive ground state solution for Kirchhoff type problem with critical exponential growth without the Ambrosetti–Rabinowitz condition in dimension two.

Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

In this paper, we consider the following Kirchhoff type problem{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2uinR3,u∈H1(R3), where a>0 is a constant, b and λ are positive parameters, and 2<p<6. Suppose that the nonnegative continuous potential V represents a potential well with the bottom V−1(0), the equation has been extensively studied in the case 4≤p<6. In contrast, no existence result of solutions is available for the case 2<p<4 due to the presence of the term (∫R3|∇u|2dx)Δu. By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2<p<4. Moreover, we also explore the decay rate of the positive solutions as |x|→∞ as well as their asymptotic behavior as b→0 and λ→∞.

Semi-classical solutions for Kirchhoff type problem with a critical frequency

AbstractIn the present paper, we consider the following Kirchhoff type problem$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$whereb&gt; 0,p∈ (4, 6), the potential$V\in C(\mathbb R^3,\mathbb R)$andɛis a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e.,$\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities

In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature.

Two Positive Solutions for Kirchhoff Type Problems with Hardy-Sobolev Critical Exponent and Singular Nonlinearities

We consider the following singular Kirchhoff type equation with Hardy-Sobolev critical exponent \[ \begin{cases} \displaystyle -\left( a + b \int_{\Omega} |\nabla u|^2 \, dx \right) \Delta u = \frac{u^{3}}{|x|} + \frac{\lambda}{|x|^{\beta} u^{\gamma}}, & x \in \Omega, u > 0, & x \in \Omega, u = 0, & x \in \partial \Omega, \end{cases} \] where $\Omega \subset \mathbb{R}^{3}$ is a bounded domain with smooth boundary $\partial \Omega$, $0 \in \Omega$, $a,b,\lambda \gt 0$, $0 \lt \gamma \lt 1$, and $0 \leq \beta \lt (5+\gamma)/2$. Combining with the variational method and perturbation method, two positive solutions of the equation are obtained.

Multiplicity of positive solutions for Kirchhoff type problems with nonlinear boundary condition

In this paper, we study the existence of multiple positive solutions to problem \[\left \{\begin{aligned} &\bigg (a+b \int _\Omega (|\nabla u|^2+|u|^2)\,dx\bigg )(-\Delta u+u)=|u|^{4}u &&\mbox {in } \Omega, \\ &\frac {\partial u}{\partial \nu }=\lambda |u|^{q-2}u &&\mbox {on } \partial \Omega,\end{aligned} \right . \] where $\Omega \subset \mathbb {R}^{3}$ is a smooth bounded domain, $a, b \gt 0$, $\lambda \gt 0$ and $1\lt q\lt 2$. Based on the Nehari manifold and variational methods, we prove that the problem has at least two positive solutions, and one of the solutions is a positive ground state solution.

Infinitely many bound state solutions for Kirchhoff type problems

This paper is concerned with the following Kirchhoff type problems (0.1)−(a+b∫Ω|∇u|2dx)Δu=λ|u|q−2u+|u|p−2u,x∈Ω,u=0,x∈∂Ω,where constants a,b>0, the parameter λ≥0, 1<q<2<p<6 and Ω is a smooth bounded domain in R3. We consider two cases: the concave–convex nonlinearities (λ>0) and the superlinear nonlinearities (λ=0). By using the genus type min–max argument, we prove that (0.1) admits infinitely many bound state solutions for the above two cases.

Infinitely many sign-changing solutions for Kirchhoff type problems in [formula omitted

In this paper, we consider the following nonlinear Kirchhoff type problem: −a+b∫R3|∇u|2Δu+V(x)u=f(u),inR3,u∈H1(R3),where a,b>0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik–Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity |u|p−2u with p∈(2,4].

Uniqueness, existence and concentration of positive ground state solutions for Kirchhoff type problems in [formula omitted

In this paper, we first prove the uniqueness of positive ground state solution for the following Kirchhoff equation in R3 with constant coefficients{−(a+b∫R3|∇u|2)Δu+cu=d|u|p−1uinR3,u>0,u∈H1(R3), where a, b, c, d>0 are positive constants, 3<p<5. Then we use the uniqueness result to obtain the existence and concentration theorems of positive ground state solutions to the following Kirchhoff equation with competing potential functions−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=K(x)|u|p−1uinR3, for a sufficiently small positive parameter ε.

Nehari-type ground state solutions for Kirchhoff type problems in ℝN

ABSTRACTIn the present paper, by employing Non-Nehari manifold method, we make use of a weak condition on the existence of Nehari-type ground state solutions for the following Kirchhoff-type problem (0.1) which weakens both the variant Ambrosetti–Rabinowitz condition and the variant Nehari-type condition related to Problem (0.1). Furthermore, we allow to be sign-changing, which is different from the previous literature.

Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb R^3$

We are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle - \bigg(\epsilon ^2a + \epsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u + u = Q(x)|u|^{p-2}u,&amp; x\in \mathbb{R}^3, \hfill \\ u \in H^1(\mathbb{R}^3), \quad u &gt; 0, &amp; x\in \mathbb{R}^3 , \end{cases} \end{equation*} where $\epsilon&gt; 0 $ is a parameter, $a, b&gt; 0$ are constants, $p\in (2, 6)$, and $Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number of positive solutions when $\epsilon $ is sufficiently small.

Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities

In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, by the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues

We study the following Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ccc} -\left(a+b\int_{\Omega}|\nabla u|^2dx \right) \Delta u=f(x,u), &\mbox{in} \Omega, \\ u=0, &\text{on} \partial \Omega. \end{array} \right. \end{equation*} Note that $F(x,t)=\int_0^1 f(x,s)ds$ is the primitive function of $f$. In the first result, we prove the existence of solutions by applying the $G-$Linking Theorem when the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_k$ and $\mu_{k+1}$ allowing for resonance with $\mu_{k+1}$ at infinity. In the second result, for the case that the quotient $\frac{4F(x,t)}{bt^4}$ stays between $\mu_1$ and $\mu'_{2}$ allowing for resonance with $\mu'_{2}$ at infinity, we find a nontrivial solution by using the classical Linking Theorem and argument of the characterization of $\mu'_2$. Meanwhile, similar results are obtained for degenerate problem.

Ground state sign-changing solutions for Kirchhoff type problems in bounded domains

In the present paper, we consider the existence of ground state sign-changing solutions for a class of Kirchhoff-type problems(0.1){−(a+b∫Ω|∇u|2dx)△u=f(u),x∈Ω;u=0,x∈∂Ω, where Ω⊂RN is a bounded domain with a smooth boundary ∂Ω, N=1,2,3, a>0, b>0 and f∈C(R,R). Under some weak assumptions on f, with the aid of some new analytical skills and Non-Nehari manifold method, we prove that (0.1) possesses one ground state sign-changing solution ub, and its energy is strictly larger than twice that of the ground state solutions of Nehari-type. Furthermore, we establish the convergence property of ub as the parameter b↘0. Our results improve and generalize some results obtained by W. Shuai (2015) [34].

Multiple solutions for Kirchhoff type problems involving super-linear and sub-linear terms

Multiple solutions for Kirchhoff type problems involving super-linear and sub-linear terms