We are concerned with the following mean curvature problem in Minkowski space {[ -div (∇ v/√(1-|∇ v|^2) )=λ m(|x|)f(v) in ℝ^N,; v(|x|)→ 0 as |x|→ +∞ , ]. where N≥ 3 , λ >0 is a parameter, m∈ C_loc^α (ℝ^N, ℝ) for some α∈ (0, 1) is a weighted function and f∈ C(ℝ, ℝ) . Depending on the behavior of f near 0 and infinity, we investigate the existence and multiplicity of one-sign or sign-changing radial solutions to the problem. Moreover, we also obtain the rate of decay of solutions at ∞ . The proof of the main results is based upon the bifurcation technique.

We show the existence and multiplicity of solutions for the fourth-order periodic boundary value problem u””(t)-λ u(t)=f(t,u(t))-h(t), t∈ [0,1], u(0)=u(1), u'(0)=u'(1), u”(0)=u”(1), u”'(0)=u”'(1), where λ∈ℝ is a parameter, h∈ L^1(0,1) , and f:[0,1]×ℝ→ℝ is an L^1 -Carathéodory function. Moreover, f is sublinear at +∞ and nondecreasing with respect to the second variable. We obtain that if λ is sufficiently close to 0 from the left or right, then the problem has at least one or two solutions, respectively. The proof of main results is based on bifurcation theory and the method of lower and upper solutions.

We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator −u′1−u′2′=λa(x)f(u,u′),x∈R, $-{\left(\frac{{u}^{\prime }}{\sqrt{1-{u}^{\prime 2}}}\right)}^{\prime }=\lambda a\left(x\right)f\left(u,{u}^{\prime }\right), x\in \mathbb{R},$ where λ > 0 is a parameter, a∈C(R,R) $a\in C\left(\mathbb{R},\mathbb{R}\right)$ is a 2π-periodic sign-changing function with ∫02πa(x)dx<0 ${\int }_{0}^{2\pi }a\left(x\right)\mathrm{d}x{< }0$ , f∈C(R×R,R) $f\in C\left(\mathbb{R}{\times}\mathbb{R},\mathbb{R}\right)$ satisfies a generalized regular-oscillation condition. Moreover, for the special case that f does not contain derivative term, we also investigate the global structure of 2π-periodic odd/even sign-changing solutions set under some parity conditions. The proof of our main results are based upon bifurcation techniques.

We are concerned with the linear problem {[ -Δ u+κ/|x|^2 x·∇ u =λ K(|x|) u, x∈ℝ^N,; u(x)>0, x∈ℝ^N,; u(x)→ 0, |x|→∞ , ]. where λ is a positive parameter, κ∈ [0,N-2) , N> 2 and K:ℝ^N → (0,∞ ) is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue λ _1^rad and the corresponding positive eigenfunction φ _1 satisfies lim _|x|→∞φ _1(|x|)=c/|x|^N-2-κ for some c>0 . As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.

We are concerned with the Neumann problem in some FLRW spacetimes(P){div(∇uf(u)f(u)2−|∇u|2)+f′(u)f(u)2−|∇u|2(N+|∇u|2f(u)2)=λNg(|x|,u)inB(R),|∇u|0 is a parameter. We show that (P) has infinitely many radially symmetric sign-changing solutions under some appropriate conditions. The proof of our main result is based upon bifurcation techniques.

Let E = {u ∈ C1[0, 1]: u(0) = u(1) = 0}. Let S with v = {+, −} denote the set of functions u ∈ E which have exactly k − 1 interior nodal zeros in (0, 1) and vu be positive near 0. We show the existence of S-shaped connected component of S -solutions of the problem $$\left\{ {\begin{array}{*{20}{c}}{\begin{array}{*{20}{c}}{{{\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)}^\prime } + \lambda a(x)f(u) = 0,}&{x \in (0,1)}\end{array}} \\{u(0) = u(1) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{array}} \right.$$ where λ > 0 is a parameter, a ∈ C([0, 1], (0, ∞)). We determine the intervals of parameter λ in which the above problem has one, two or three S -solutions. The proofs of the main results are based upon the bifurcation technique.

We are concerned with the Neumann problem with Minkowski-curvature operator {div (del u(x)/root 1 - vertical bar del u(x)vertical bar(2)) = g(u(x)) in Omega, (P) u(x) > 0 in Omega, u -> 0 as vertical bar x vertical bar -> infinity, partial derivative u/partial derivative v = -sigma on partial derivative Omega, where Omega is an exterior domain in R-N and partial derivative/partial derivative v denotes the normal interior derivative on partial derivative Omega. We have shown the existence of radial positive solutions of (P) when g satisfies some suitable assumptions. The proof of our main result is based upon the shooting method.

We show the global structure of positive solutions for the Neumann problem involving mean curvature operator P $$\begin{aligned} \left\{ \begin{array}{ll} -(\frac{u'}{\sqrt{1-u'^2}})'=\lambda a(r)f(u), &{}\quad r\in (0,R), \\ u'(0)=u'(R)=0,&{} \\ \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$a:[0,R]\rightarrow {\mathbb {R}}$$ is an $$L^1$$ -function which is allowed to change sign and $$f:[0,\infty )\rightarrow [0,\infty )$$ is continuous. Depending on the behavior of f near 0 and $$\infty $$ , we obtain that there exists $$0<\lambda _*\le \lambda ^*$$ such that for any $$\lambda >\lambda ^*$$ , problem (P) possesses at least two positive solutions, while it has no solution for $$\lambda \in (0,\lambda _*)$$ . The proof of the main results is based upon bifurcation method.

In this paper, we are concerned with elliptic problems -Δ u= f(u)+ g( | x | ,u,x/| x |·∇ u), x∈Ω , u|_∂Ω=0, where Ω ={x∈ℝ^N:R_1<|x|2 , 0< R_10 , 0<β < 1/4(R_2-R_1)^2 . We obtain infinitely many radial solutions with prescribed nodal properties using bifurcation techniques.

This paper is concerned with sublinear perturbations of resonant linear polyharmonic problems. We establish some {\it a priori} bounds and use these together with Leray-Schauder continuation and bifurcation arguments to obtain extensions of some known results of Mawhin and Schmitt on the multiplicity of solutions of nonlinear elliptic eigenvalue problems with the parameter near resonance.

We consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ where $\lambda ,\mu \geq 0$, $\mathbb{T}=\{2,\ldots ,N-1\}$ with $N>3$, $\phi (s)=s/\sqrt{1-s^{2}}$. The function $f:=\lambda a(t,s)+\mu b(t,s)$ is either sublinear, or superlinear, or sub-superlinear near $s=0$. Applying the topological method, we prove the existence of either one or two, or three positive solutions.