Branch Of Positive Solutions Research Articles

A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system

The paper is concerned with the local and global bifurcation structure of positive solutions \({u,v\in H^1_0(\Omega)}\) of the system $$\left\{\begin{array}{lll}-\Delta u + u = \mu_1u^3+\beta v^2u &\quad\text{in \(\Omega\)}\\-\Delta v + v = \mu_2v^3+\beta u^2v &\quad\text{in \(\Omega\)}\nonumber\end{array}\right.$$ of nonlinear Schrodinger (or Gross-Pitaevskii) type equations in \({\Omega\subset\mathbb {R}^N}\), N ≤ 3. The system arises in nonlinear optics and in the Hartree–Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches.

Spectral asymptotics for inverse nonlinear Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem −u ′′ (t) + f(u(t),u ′ (t)) = �u(t), u(t) > 0, t ∈ I := (−1/2,1/2), u(±1/2) = 0, where f(x,y) = |x| p−1 x − |y| m , p > 1,1 ≤ m 0 is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in R+ ×L q (I) (1 ≤ q < ∞) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue � = �q(�) as � :=

Asymptotic Behavior of Bifurcation Branch of Positive Solutions for Semilinear Sturm–Liouville Problems

We consider the nonlinear eigenvalue problem $$-u"(t) + f(u(t)) = \lambda u(t),\quad u(t) > 0,\quad t \in I := (0, 1),\quad u(0) = u(1) = 0$$ , where f(u) = up + h(u) (p > 1) and λ > 0 is a parameter. Typical example of h(u) is \(h(u) = \pm u^{q}\) with 1 0 is the Lm-norm of the positive solution associated with \(\lambda \gg 1\).

Global behavior of the branch of positive solutions to a logistic equation of population dynamics

We consider the nonlinear problem arising in population dynamics: - + u(t) p = λu(t), u(t) > 0, t ∈ I:= (0,1), u(0) = u(1) = 0, where p > 1 is a constant and λ > 0 is a positive parameter. We establish the crucial asymptotic formula for the branch of positive solutions λ q (α) in L q -framework as α → oo, where a:= ∥u λ ∥ q (1 < q < ∞). Especially, for the original logistic equation, namely the case where p = 2 and q = 1, we obtain not only the asymptotic expansion formula for λ 1 (α) but also the remainder estimate. Such a formula for the bifurcation branch seems to be new.

Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity

We consider the following problem <p align="center"> $-\Delta u=\frac{\lambda}{(1-u)^2}$ in $\Omega$, $u=0$ on $\partial \Omega$, $0 < u < 1$ in $\Omega$ <p align="left" class="times"> where $\Omega$ is a rather symmetric domain in $\mathbb R^2$. We prove that there exists a $\lambda_\star>0$ such that for $\lambda \in (0, \lambda_\star)$ the minimal solution is unique. Then we analyze the asymptotic behavior of touch-down solutions, i.e., solutions with max$_\Omega u_i (0) \to 1$. We show that after a rescaling, the solution will be asymptotically symmetric. As a consequence, we show that the branch of positive solutions must undergo infinitely many bifurcations as the maximums of the solutions on the branch go to 1 (possibly only changes of direction). This gives a positive answer to some open problems in [12]. Our result is new even in the radially symmetric case. Central to our analysis is the monotonicity formula, one-dimensional Sobloev inequality, and classification of solutions to a supercritical problem <p align="center"> $ \Delta U=\frac{1}{U^2}\quad$ in $\mathbb R^2, U(0)=1, U(z) \geq 1.$ <p align="left" class="times">

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem $$ -u''(t) + f(u(t)) = \lambda u(t),\quad u(t) > 0, \quad t \in I := (0, 1),\quad u(0) = u(1) = 0, $$ (1) where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R+ × L2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction uα with ‖ uα ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/up satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ αp−1h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by limu → ∞uh′(u).

$L^q$ spectral asymptotics for nonlinear Sturm-Liouville problems

We consider the nonlinear Sturm-Liouville problem $$ -u''(t) + f(u(t)) = \lambda u(t), \ \ u(t) > 0, \quad t \in I := (0, 1), \ \ u(0) = u(1) = 0, $$ where $\lambda > 0$ is an eigenvalue parameter. For better understanding of the global behavior of the branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q \le \infty$), we establish precise asymptotic formulas for the eigenvalue $\lambda$ with respect to $\Vert u_\lambda\Vert_q$, where $u_\lambda$ is the unique solution associated with given $\lambda > \pi^2$.

Branches of positive solutions of quasilinear elliptic equations on non-smooth domains

We prove existence of an unbounded global branch (i.e. connected set) of weak solutions of a second order quasilinear equation depending on a real parameter λ on an arbitrary (possibly non-smooth) bounded domain in R N , with a Leray–Lions operator as the leading part. Here, we can allow lower order nonlinearities which depend on first derivatives, satisfying appropriate growth conditions including the critical case. Furthermore, we give sufficient conditions for the existence of a branch consisting entirely of nonnegative solutions for positive λ . Our approach also yields a new existence result in the case of critical growth in derivatives of lower order.

Some elliptic semilinear indefinite problems on RN

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem The function f is allowed to change sign and has an asymptotically linear or a superlinear behaviour.

Stable solutions on $\R^n$ and the primary branch of some non-self-adjoint convex problems

We prove that if $n = 2$ or $3$, the problem $- \Delta u = e^u$ on $R^n$ has no stable negative solution. We then use this to remove self-adjointness conditions in a paper of Crandall and Rabinowitz on the primary branch of positive solutions of a nonlinear boundary-value problem.

Multiple coexistence states for a prey–predator system with cross-diffusion

We study the multiple existence of positive solutions for the following strongly coupled elliptic system: Δ[(1+αv)u]+u(a−u−cv)=0 in Ω, Δ[(1+βu)v]+v(b+du−v)=0 in Ω, u=v=0 on ∂Ω, where α, β, a, b, c, d are positive constants and Ω is a bounded domain in R N . This is the steady-state problem associated with a prey–predator model with cross-diffusion effects and u (resp. v) denotes the population density of preys (resp. predators). In particular, the presence of β represents the tendency of predators to move away from a large group of preys. Assuming that α is small and that β is large, we show that the system admits a branch of positive solutions, which is S or ⊃ shaped with respect to a bifurcation parameter. So that the system has two or three positive solutions for suitable range of parameters. Our method of analysis uses the idea developed by Du-Lou (J. Differential Equations 144 (1998) 390) and is based on the bifurcation theory and the Lyapunov–Schmidt procedure.

Stability of standing waves for some nonlinear Schrödinger equations

The paper concerns the monotonicity with respect to $\lambda$ of the $L^2$-norm of the branch of positive solutions of the nonlinear eigenvalue problem $$ u''(x)+ g(x, u(x)^2) \ u(x) + \lambda u(x) = 0, \ x\in {\bf R}, \ \lim\limits_{|x|\rightarrow \infty} \ u(x) = 0.$$ For the particular case $ g(x) = p(x) + s^{\sigma},$ with $\sigma > 0$ and $p(x)$ an even function, decreasing for $x > 0$ and with $p(\infty) = 0$, the main theorem implies that the $L^2$-norm decreases as we increase $\lambda$ if $\sigma \leq 2$. It is also shown that this is no longer true if $\sigma > 2$. The result has implications for the orbital stability of standing waves of the nonlinear Schrödinger equation.

Non-local investigation of bifurcations of solutions of non-linear elliptic equations

We justify the projective fibration procedure for functionals defined on Banach spaces. Using this procedure and a dynamical approach to the study with respect to parameters, we prove that there are branches of positive solutions of non-linear elliptic equations with indefinite non-linearities. We investigate the asymptotic behaviour of these branches at bifurcation points. In the general case of equations with -Laplacian we prove that there are upper bounds of branches of positive solutions with respect to the parameter.

On positive solutions of indefinite elliptic equations

This paper concerns the elliptic equation of the form Δ p u+ λ| u| p−2 u+ f( x)| u| γ−2 u=0, where Δ p is the p-Laplacian, p∈(1,∞). We discuss the problem of existence and nonexistence of positive solutions depending on λ. The least upper bound of a branch of positive solutions is found. To solve the problem we present a new approach.

Existence of global branches of positive solutions for semilinear elliptic degenerate problems

In this paper, we are dealing with the following semilinear elliptic degenerate problem: (P) −|x| 2Δu=λ u+g(u) inB 1, u≡0∈∂B 1; u≥0 where B 1={ x∈ R N ; ‖ x‖=1} and g is nonlinear. Under appropriate assumptions on g, we prove the existence of global and connected branches of solutions to ( P) in R×L ∞(B 1) or in R×H 1 0(B 1) . Our results cover a large class of sublinear and some nonlinearities.

Global positive solution branches of positone problems with nonlinear boundary conditions

In this paper we consider a class of semilinear elliptic boundary value problems with nonlinear boundary conditions. The continuation method or the implicit function theorem is used to prove the existence of smooth branches of positive solutions. The characterization of the branches, and the uniqueness and asymptotic behavior of positive solutions are also studied by using some comparison principles with semilinear elliptic boundary value problems with linear boundary conditions.

Positive solutions of quasilinear elliptic equations

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and the paper of Ambrosetti and Hess [4] for the case p = 2,

Branches of positive solutions for some semilinear Schrödinger equations

Branches of positive solutions for some semilinear Schrödinger equations

Positive solution curves of semipositone problems with concave nonlinearities

SynopsisWe consider the positive solutions to the semilinear equation:where Ω denotes a smooth bounded region in ℝN(N&gt; 1) and ℷ 0. Heref:[0, ∞)→ℝ is assumed to be monotonically increasing, concave and such thatf(0)&lt;0 (semipositone). Assuming thatf′(∞)≡limt→∞f(t)&gt; 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)′ When Ω is a ball, we determine the exact number of positive solutions for each λ &gt; 0. We also obtain the geometry of the branches of positive solutions completely and establish how they evolve. This work extends and complements that of [3, 7] wheref′(∞)≦0.

A global branch of positive solutions above the continuous spectrum for problems with indefinite nonlinearities

We prove the existence and bifurcation of a global branch of positive solutions for a nonlinear Neumann eigenvalue problem on the half axis [0, ∞). The nonlinearity is assumed to have a superlinear growth multiplied by a weight function changing sign. This leads to the existence of nontrivial solutions above the continuous spectrum of the linearised problem.