In this paper we study the following quasilinear nonlocal differential equations with convolution coefficients -M ((a * u^γ )(1) ) (φ _p(u'))' = f(λ , x, u), x ∈ (0,1), where γ >0 , φ _p(u)=|u|^p-2u with p>1 is the p-Laplacian operator, and the reaction term f(λ , x, u) includes linear, eigenvalue, sublinear, and logistic cases. We establish existence results for positive solutions and describe the global structure of the solution set in the degenerate case. The proofs of the main results are based upon fixed point arguments.

We are concerned with the problem with Minkowski-curvature operator on an exterior domain {[ -div (ϕ _N(∇ u(x)) )=λ a(|x|)f(u(x)), in B^c,; ∂ u/∂ n| _∂ B^c=0, lim _|x|→∞u(x)=0, ]. where ∂ u/∂ n denotes the outward normal derivative of u, ϕ _N(y)=y/√(1-|y|^2), y∈ℝ^N, B^c={x∈ℝ^N:|x|>R} is an exterior domain in ℝ^N, N≥ 3, R>0, a∈ L^1_loc((0,1],ℝ) may change sign, λ is a nonnegative real parameter, f∈ C([0,∞ ),ℝ) with f(0)>0 . The proof of the main result is based on the Leray-Schauder fixed point theorem.

We are concerned with the existence of nodal solutions for the n-th order boundary value problem {[ u^(n)(t)+a(t) f(u(t))=0, t∈ (0,π ),; u^(i)(0)=0, i∈{i_1,...,i_n-m},; u^(j)(π )=0, j∈{j_1,...,j_m}, ]. where n≥ 2 , 1≤ m≤ n-1, {i_1,...,i_n-m}, {j_1,...,j_m} are subsets of the set of integers in {0,..., n-1} ; a:[0,π ]→ (-1)^mℝ^- is continuous and does not vanish identically on any subinterval of [0, π ] ; f: ℝ→ℝ is a continuous function. The proof of our main result is based upon bifurcation techniques.

In this paper, we are concerned with the Dirichlet problem of quasilinear differential system, involving the mean curvature operator in Minkowski space ℳ(w)=( w'/√(1-|w'|^2)) '. Using global bifurcation technique, we obtain the existence of an unbounded branch of positive solutions, which is unbounded in the positive λ -direction. We also establish a Calabi-Bernstein type asymptotic property for a pair of one-signed solutions on certain bifurcation branch converge to a piecewise linear function. The proof of our main results is based upon the global bifurcation theory and the Sturm separation theorem.

The aim of the present paper is to study the existence of positive solutions for a second-order boundary value problem of Kirchhoff type in R $\mathbb{R}$ . Our approach is based on continuum theory and cut-off function technique.

We prove several existence and multiplicity results for some nonlinear elliptic Kirchhoff equations {-(1 + gamma G '(integral(Omega)|del u|(2)dx)) Delta u = f(x, u, del u), x is an element of Omega, u(x) = 0, x is an element of partial derivative Omega, where gamma > 0 is a parameter, G : & Ropf;(+) -> & Ropf; and f : Omega x & Ropf;(+)x & Ropf;(N) -> & Ropf; are smooth functions satisfying C(1)t(q-1) <= G'(t) <= C(2)t(q-1) for all t is an element of [0, infinity), C-3|u|(p-1) <= f(x, u, v) <= C-4|u|(p-1) for all (x, u, v) is an element of Omega x & Ropf;(+) x & Ropf;(N), C-i(i = 1, 2, 3, 4) are constants and q > 1, 2 < p < 2(& lowast;) with p <= 2q. The proofs of our main results are based upon bifurcation techniques.

In this paper, we investigate the Calabi-Bernstein type asymptotic property of one-sign solutions for one-dimensional Minkowski-curvature equations. In particular, we show that solutions on two certain bifurcation branches converge to two piecewise linear functions. The proof of our main results is based upon the global bifurcation theory and the Sturm separation theorem.

We are concerned with the global structure of positive solutions of the (k, n - k) conjugate boundary value problem { (-1)(n-k )u((n))(t) = lambda a(t) f (u(t)), 0 < t < 1, u((i) )(0) = 0, 0 <= i <= k - 1, u((j) )(1) = 0, 0 <= j <= n - k - 1, (P) where n >= 2, 1 <= k <= n-1, lambda > 0 is a parameter, a is an element of C([0, 1], (0, infinity)) and f is an element of C( [0, infinity), [0, infinity)). We obtain existence and multiplicity results for positive solutions of problem (P) under suitable growth conditions on f. The proof of main result is based upon bifurcation techniques.

We show the existence of the principal eigenvalues of fourth-order problem {u^''''(x) = λ g(x)u(x) x ∈ (0,1),u(0) = u(1) = u^'(0) = u^'(1) = 0,. where λ ∈ ℝ is a parameter and g changes its sign in [0, 1], via disconjugacy theory and Elias’s eigenvalue theory. We apply our spectrum result to show the existence of unbounded connected components of one-signed solutions for the corresponding nonlinear problems.

We study the existence of positive solutions for the Berger plate equation with Navier boundary condition P {[ Δ ^2 u-m (x, ∫ _Ω |∇ u|^2dx )Δ u =f(x,u,|∇ u|,Δ u), x∈Ω ,; u=Δ u=0, x∈∂Ω ,; ]. where Ω is a bounded domain in ℝ^N , N∈ℕ , with a smooth boundary ∂Ω , m:Ω×ℝ^+ →ℝ is a continuous function, f:Ω×ℝ×ℝ^+×ℝ→ℝ is a Carathéodory function. The proofs of the main results are based on the topological degree and continuum theory.

We are concerned with the quasilinear Neumann problems in Euclidean space {[ -div (∇ u/√(1+|∇ u|^2) )=λ a(|x|)f(u) in 𝒜,; ∂ u/∂ν=0 on ∂𝒜, ]. (P) where ∂ u/∂ν denotes the outward normal derivative of u, 𝒜={x∈ℝ^N| R_1<|x|R_1>0 , a(|x|) is a radial sign-changing function, the function f is superlinear at 0 and λ >0 is a parameter. We show that the existence of an unbounded connected set of positive radial regular solutions (λ ,u) of (P) bifurcating from u=0 as λ→ +∞ . The proof of our main result is based upon the method of lower and upper solutions.

We study the semipositone problem of the elliptic Kirchhoff type equation {-((b)integral(Omega e)|del u|(2)dx)Delta u=lambda K(|x|)f(u), x is an element of Be, u(x)=0, |x|=r(0), (0.1) u(x)-> 0, |x|->infinity, where b is a positive constant,lambda is a positive parameter, B-e={x is an element of R-N:|x|>r(0)},N>2,K:[r(0),+infinity)->(0,+infinity)is continuous withrN+eta K(r)bounded for some eta>0,f:[0,+infinity)-> R is continuous, f(0)<0and lim(u ->infinity)f(u)/u(q)=beta for some q is an element of(0,1]. We show that there exists lambda(& lowast;)>0, such that (0.1) has at least one positive radial solution if lambda>lambda & lowast;. The proof of the main result is based upon bifurcation theory.

We study the existence of positive solutions for the semipositone biharmonic equation with Navier boundary conditions P {[ Δ ^2 u=λ f(t,u), t∈Ω ,; u=Δ u=0, t∈∂Ω , ]. where Ω⊂ℝ^n (n≥ 1) is a smooth bounded domain, λ >0 and f:Ω×ℝ^+→ℝ is a continuous function with f(t,0)<0 in Ω , ℝ^+=[0,∞ ) . We obtain existence results for positive solutions of problem (P) under different growth conditions. The proofs of the main results are based on bifurcation theory and degree theory combined with a rescaling argument.

In this paper, we study the spectrum of the one-dimensional fractional Laplace operator with a definite weight { (-d2 dx2)alpha u(x) =lambda a(x)u(x), x E (-1, 1), u(x)= 0, x in R \ (-1, 1), where alpha E (0, 1), a E C([-1, 1],[0,+infinity)) and (- d2 dx2)alpha is the one-dimensional fractional Laplace nonlocal operator. By virtue of Gamma-convergence arguments, we investigate, in the singular limit, that the eigenvalue and eigenfunction of the nonlocal operator converge to those of the corresponding classical second-order two-point boundary value problem in the first place, and then, building upon the continuity of the eigenvalues and eigenfunctions as a function of fractional index, we derive the simplicity of the eigenvalues of the fractional Laplace nonlocal operator for all fractional index alpha E (0, 1) by adopting a stet-by-step iterative approach. Furthermore, using the alpha-harmonic extension, we receive that the corresponding eigenfunction phi k alpha has at most 2k-2 zeros in (-1, 1). At last, from an experiment point of view, we give the numerical eigenvalues and eigenfunctions of the weighted fractional Laplace problem by means of the finite element method in some special cases, which enriches the theoretical results. The spectral results presented here, especially the simplicity of eigenvalues lambda alpha kresolve a conjecture of Ba & ntilde;uelos and Kulczycki.

In this paper, the propagation dynamics of a reaction-diffusion system with saturated reaction is studied. For multi-species models, it is possible that all species possess the same spreading speeds (short for asymptotic spreading speed), or that different species spread at different speeds. We focus on the speed selection mechanism and the traveling wave connection in the two cases mentioned above, which are different. Based on the comparison principle, we give the range of spreading speed of each species, and obtain the conditions for the existence of multiple speeds and single speed. By means of the upper and lower solution method, the conditions for linear and nonlinear selection are derived.

We are concerned with the principal eigenvalue of (P) { − Δ p u = λ θ 1 φ p ( v ) , x ∈ Ω , − Δ p v = λ θ 2 φ p ( u ) , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω and the global structure of positive solutions for the system (Q) { − Δ p u = λ f ( v ) , x ∈ Ω , − Δ p v = λ g ( u ) , x ∈ Ω , u = 0 = v , x ∈ ∂ Ω , where φ p ( s ) = | s | p − 2 s , Δ p s = div ( | ∇ s | p − 2 ∇ s ) , λ > 0 is a parameter, Ω ⊂ R N , N > 2 , is a bounded domain with smooth boundary ∂ Ω , f , g : R → ( 0 , ∞ ) are continuous functions with p -superlinear growth at infinity. We obtain the principal eigenvalue of ( P ) by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for ( Q ) via bifurcation technology.

The aim of this paper is to establish an Ambrosetti–Prodi type result involving Dirac weights {[ u””(x)+q(x)u(x)=(c (x)+∑ _i=1^pc_iδ (x-x_i))(g(u(x))+f(x)), x∈ (0,1),; u(0)=u(1)=u”(0)=u”(1)=0,; ]. where δ (x-x_i) is the canonical Dirac delta function at the point x_i , i=1,2,… ,p , p∈ℕ , 0=x_0

We are concerned with the existence of homoclinic solutions for the nonlinear problems P {[ u”+ω u'-ku=f(t,u,u'), t∈ℝ,; lim _|t|→ +∞u(t)=0, ]. where ω∈ℝ, k>0 are real constants, and f: ℝ^3→ℝ is an L^1- Carathéodory function. Under some suitable conditions, the existence of homoclinic solutions for problem (P) and the corresponding coupled systems are provided. The proofs of the main results are based on the method of upper and lower solutions.

We investigate the local stability for fixed-depth rotational equatorial flows. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation point. In particular, their signs can be assessed when vorticity is small enough and the mean depth is small or large enough. Furthermore, we study the stability property for periodic traveling waves of the thermocline near the equator with a fixed mean depth. For this aim, we obtain a new stability exchange formula which improves the famous Crandall-Rabinowitz stability exchange theorem.