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With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation φ(x)=∫DE(x,y)G(y,φ(y))dy. This result is applied to branching d...

This paper establishes global bifurcation and eigenvalue results for the following complex $ k $-Hessian equation $ \begin{equation} \left\{ \begin{array}{ll} S_k\left(u_{i\overline{j}}\right) = \lambda^k f(-u)\,\, &\text{in}\,\, B,\\ u = 0 &\text{on}\,\, \partial B. \end{array} \right.\nonumber \end{equation} $ The existence/nonexistence, uniqueness and multiplicity of radially symmetric negative solutions are investigated. Moreover, a-priori bound of radially symmetric negative solutions is also obtained.

We establish the unilateral global bifurcation result for the following nonlinear operator equation $$u = L\left( \lambda \right)u + H\left( {\lambda,u} \right),\,\left( {\lambda,u} \right) \in \mathbb{R}{^m} \times X$$ where m is a positive integer, X is a Banach space, L(·) is a positively homogeneous completely continuous operator and H: ℝm·X → X is completely continuous with H = o (||u||) near u = 0 uniformly on bounded λ sets.

By variational method, we obtain various existence and multiplicity results for the following double phase problem { − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω . In particular, we find a sign-changing ground state solution. Some properties of double phase operator are also obtained.

We establish a unilateral global bifurcation result for the following problem where Ω is a bounded domain in . Based on this global bifurcation result, we also studied the global structure of one-sign solutions according to different asymptotic behaviors (sublinear/linear/superlinear/jumping) nonlinearity near zero.

We study the global structure of admissible solutions for the following k -Hessian equation S k D 2 u = λ k f ( − u ) . By bifurcation and topological methods, we determine the interval of λ for the existence of admissible solution for this problem.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.

In this paper, we investigate the global regularity of Leray–Hopf weak solutions to the 3D Navier–Stokes equations in terms of pressure. In particular, we establish three Serrin-type regularity criteria in framework of the anisotropic Lebesgue spaces. Our result can be understood as a generalization of the notable works of Y. Zhou (2006) [23] , C. Cao and E. Titi (2008) [5] , and Y. Zhou and M. Pokorný (2010) [24] .

The uniqueness obtained in [1] may not right if f ( s ) / s k is only decreasing for s > 0 . We correct this error by requiring f ( s ) / s k is “strictly” decreasing for s > 0 .

This paper is concerned with the following semilinear elliptic problem { − Δ u = λ m ( x ) f ( u ) in R N , u → 0 as | x | → + ∞ , where λ is a real parameter and m is a weight function which may be sign-changing. For the linear case, i.e., f ( u ) = u , we investigate the spectral structure. For the semilinear case, we study the existence and asymptotic behavior of one-sign and nodal solutions by bifurcation method.

We study the following eigenvalue problem of k-Hessian equation { S k ( D 2 u ) = λ k f ( − u ) in B , u = 0 on ∂ B . Global bifurcation result is established for this problem. As applications of the bifurcation result, we determine the intervals of λ for the existence, nonexistence, uniqueness and multiplicity of radially symmetric k-admissible solutions for this problem.

We study the existence/nonexistence and multiplicity of radial solutions of the following mean curvature problem in Minkowski spacetime $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\quad \text {in}\,\, \mathbb {R}^N,\\ u\rightarrow 0&{}\quad \text {as}\,\, \vert x\vert \rightarrow +\infty . \end{array} \right. \end{aligned}$$ Moreover, we also obtain the rate of decay of solutions at \(\infty \).

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

In this paper, we establish a unilateral global bifurcation result from intervals for a class of semilinear problems on the unit ball. By applying the above result, we obtain the spectrum of a class of half-linear problems on the unit ball. Our results partially give a positive answer to the open problem proposed by Berestycki.

This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded regular domain with \begin{document}$ N\geq3 $\end{document}, \begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document} are general Hamilton-Jacobi-Bellman operators, \begin{document}$ \mu $\end{document} is a real parameter. By using bifurcation theory, we determine the range of parameter \begin{document}$ \mu $\end{document} of the above problem which has one or multiple constant sign solutions according to the behaviors of \begin{document}$ f $\end{document} at \begin{document}$ 0 $\end{document} and \begin{document}$ \infty $\end{document}, and whether \begin{document}$ f $\end{document} satisfies the signum condition \begin{document}$ f(s)s>0 $\end{document} for \begin{document}$ s\neq0 $\end{document}.

We study the following Monge-Ampere equation { ( det ⁡ ( D 2 u ) ) 1 N = λ f ( − u ) in Ω , u = 0 on ∂ Ω by bifurcation technique. We establish some results about the existence, nonexistence, uniqueness and multiplicity of convex solutions for this problem. Our results generalize and improve many important known results from previous literature.

We consider a double phase problem in RN with a q−1-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational methods based on the topological degree and critical point theory together with the Nehari manifold method and deformation lemma, we find three nontrivial solutions: among them, the first one is the positive ground state solution, the second one is the negative ground state solution, and the third one is the least energy sign-changing solution, which changes sign only once. We also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions.

We study the global structure of positive solutions of the following mean curvature equation in the Minkowski space \[ -\div \bigg (\frac {\nabla u}{\sqrt {1-\vert \nabla u\vert ^2}}\bigg )= \lambda f(x,u), \] on an annular domain with the Robin boundary condition. According to the behavior of $f$ near $0$, we obtain the existence and multiplicity of positive solutions for this problem.

In this paper, we investigate the structure of the nontrivial solution set for the following nonlinear operator equation u = L ( λ ) u + H ( λ , u ) , ( λ , u ) ∈ R m × X , where m is a positive integer, X is a Banach space, L ( ⋅ ) : X → X is a (positively) homogeneous compact operator and H : R m × X → X is compact with H = o ( ‖ u ‖ ) near u = 0 uniformly on bounded λ sets. We obtain some results involving (unilateral) global bifurcation. Two examples of p-Laplacian problem with jumping nonlinearity and nonlocal boundary value problem are given to demonstrate how the theory can be applied.

We establish the existence of nontrivial nonnegative solution for the following 0-Dirichlet problem with mean curvature operator in the Minkowski space $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla u}{\sqrt{1-\vert \nabla u\vert ^2}}\right) = \lambda f(x,u)\,\, &{}\text {in}\,\, \Omega ,\\ u=0&{}\text {on}\,\, \partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a general bounded domain of $$\mathbb {R}^N$$ . By bifurcation and topological methods, we determine the interval of parameter $$\lambda $$ in which the above problem has nontrivial nonnegative solution according to sublinear or linear nonlinearity at zero.

We focus on the structure of the solution set for the nonlinear equation [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous operators. Under certain hypotheses on [Formula: see text] and [Formula: see text], unilateral global bifurcations for eigenvalue problems are presented. Some applications are illustrated for nonlinear ordinary and partial differential equations. In particular, the existence and multiplicity of one-sign solutions for Monge–Ampère equation is discussed.

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge–Ampere equation $$\begin{aligned} \left\{ \begin{array}{lll} \det \left( D^2u\right) =\lambda ^N a(x)f(-u)\,\, &{}\quad \text {in}\,\, \Omega ,\\ u=0~~~~~~~\,\,&{}\quad \text {on}\,\, \partial \Omega . \end{array} \right. \end{aligned}$$ We establish global bifurcation results for the problem. We find intervals of \(\lambda \) for the existence, multiplicity and nonexistence of strictly convex solutions for this problem.

Using bifurcation method, we investigate the existence, nonexistence and multiplicity of positive solutions for the following Dirichlet problem involving mean curvature operator in Minkowski space $$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla v}{\sqrt{1-\vert \nabla v\vert ^2}}\right) = \lambda f(\vert x\vert ,v) &{}\quad \text {in}\,\, B_R(0),\\ v=0&{}\quad \text {on}\,\, \partial B_R(0). \end{array} \right. \end{aligned}$$ We managed to determine the intervals of the parameter \(\lambda \) in which the above problem has zero, one or two positive radial solutions corresponding to sublinear, linear, and superlinear nonlinearities f at zero respectively. We also studied the asymptotic behaviors of positive radial solutions as \(\lambda \rightarrow +\infty \).

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle -\bigg(a+b\int_\Omega \vert \nabla u\vert^2dx\bigg)\Delta u=\lambda u+h(x,u,\lambda) &\text{in } \Omega,\\ u=0 &\text{on }\Omega. \end{cases} \end{equation*} Under some natural hypotheses on $h$, we show that $(a\lambda_1,0)$ is a bifurcation point of the above problem. As an application of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we provide a positive answer to an open problem involving the case $a=0$.

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.