Exact Multiplicity Result Research Articles

Results of Ambrosetti–Prodi type for non-selfadjoint elliptic operators

The well-known Ambrosetti–Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non-self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.

Exact multiplicity of stationary limiting problems of a cell polarization model

We show existence, nonexistence, and exact multiplicity for stationary limiting problems of a cell polarization model proposed by Y. Mori, A. Jilkine and L. Edelstein-Keshet. It is a nonlinear boundary value problem with total mass constraint. We obtain exact multiplicity results by investigating a global bifurcation sheet which we constructed by using complete elliptic integrals in a previous paper.

Exact Multiplicity of Sign-Changing Solutions for a Class of Second-Order Dirichlet Boundary Value Problem with Weight Function

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problemsu″+a(t)f(u)=0,t∈(0,1),u(0)=0, andu(1)=0, wheref∈C(ℝ,ℝ)satisfiesf(0)=0and the limitsf∞=lim|s|→∞(f(s)/s),f0=lim|s|→0(f(s)/s)∈{0,∞}. Weight functiona(t)∈C1[0,1]satisfiesa(t)>0on[0,1].

Exact multiplicity results for a singularly perturbed Neumann problem

In this paper we study the number of the boundary single peak solutions of the problem $$\left\{\begin{array}{ll} -\varepsilon^{2} \Delta u + u = u^{p}, \quad {\rm in}\, \Omega u > 0, \quad\quad\quad\quad\quad\quad {\rm in}\, \Omega \frac{\partial u}{\partial {\nu}} = 0, \quad\quad\quad\quad\quad\,\,\, {\rm on}\, \partial {\Omega}\end{array}\right.$$ for \({\varepsilon}\) small and p subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.

Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations

We investigate various properties of time maps for one-dimensional prescribed mean curvature equations. Using these properties, we obtain some exact multiplicity results of positive solutions and sign-changing solutions. As it turned out, these quasilinear problems show many different phenomena from semilinear problems. Our methods are based on a detailed analysis of time maps.

Global solution curves for a class of elliptic systems

We study curves of positive solutions for a system of elliptic equations of Hamiltonian type on a unit ball. We give conditions for all positive solutions to lie on global solution curves, allowing us to use the analysis similar to the case of one equation, as developed in P. Korman, Y. Li and T. Ouyang [An exact multiplicity result for a class of semilinear equations, Commun. PDE 22 (1997), pp. 661–684.] (see also T. Ouyang and J. Shi [Exact multiplicity of positive solutions for a class of semilinear problems, II, J. Diff. Eqns. 158(1) (1999), pp. 94–151].). As an application, we obtain some non-degeneracy and uniqueness results. For the one-dimensional case we also prove the positivity for the linearized problem, resulting in more detailed results.

Exact multiplicity of solutions to a diffusive logistic equation with harvesting

An Ambrosetti–Prodi type exact multiplicity result is proved for a diffusive logistic equation with harvesting. We show that a modified diffusive logistic mapping has exactly either zero, or one, or two pre-images depending on the harvesting rate. It implies that the original diffusive logistic equation with harvesting has at most two positive steady state solutions.

Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function

An exact multiplicity result of positive solutions for the boundary value problems , , , is achieved, where is a positive parameter. Here the function is and satisfies , for for some . Moreover, is asymptotically linear and can change sign only once. The weight function is and satisfies , for . Using bifurcation techniques, we obtain the exact number of positive solutions of the problem under consideration for lying in various intervals in . Moreover, we indicate how to extend the result to the general case.

An exact multiplicity result for a class of symmetric problems

We consider positive solutions of a class of semilinear problems u'' + λa(x)f(u) = 0, \qquad −1 < x < 1,\qquad u(−1) = u(1) = 0, with even and positive a(x) , depending on a positive parameter λ . In case f(u) is convex, an exact multiplicity result was given in P. Korman, Y. Li and T. Ouyang [6]; see also P. Korman [4] for the details. It was observed by P. Korman and J. Shi [7] that convexity requirement can be relaxed for large u (see also [5]). We show that convexity requirement can also be relaxed for small u .

Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations

In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275–283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method.

Exact multiplicity results and bifurcation for nonhomogeneous elliptic problems

In this paper we consider the following problem: (⋆) { − Δ u ( x ) + u ( x ) = λ ( f ( x , u ) + h ( x ) ) in R N , u ∈ H 1 ( R N ) , u > 0 in R N , where λ > 0 is a parameter. We assume lim | x | → ∞ f ( x , u ) = f ̄ ( u ) uniformly on any compact subset of [ 0 , ∞ ) , but we do not require f ( x , u ) ≥ f ̄ ( u ) for all x ∈ R N . We prove that there exists + ∞ > λ ∗ > 0 such that (⋆) has exactly two positive solutions for λ ∈ ( 0 , λ ∗ ) , no solution for λ > λ ∗ , a unique positive solution u ∗ for λ = λ ∗ , and ( λ ∗ , u ∗ ) is a bifurcation point in C 2 , α ( R N ) ∩ W 2 , 2 ( R N ) .

Positivity for the linearized problem for semilinear equations

Using recent results of M. Tang [ Uniqueness of positive radial solutions for $\Delta u − u + u^p = 0$ on an annulus , J. Differential Equations 189 (2003), no. 1, 148–160], we provide a simple approach to proving positivity for the linearized problem of semilinear equations, which is crucial for establishment of exact multiplicity results, and for symmetry breaking.

Exact multiplicity result for the perturbed scalar curvature problem in $\mathbb{R}^N $ $(N \geq 3)$

Let D 1,2 (R N ) denote the closure of C ∞ 0 (R N ) in the norm ∥u∥ 2 D 1,2 (R N ) = |∇u| 2 . Let N > 3 and define the constants α N = N(N - 2) and CN = (N(N - 2)) N-2 4 . Let K ∈ C2(RN). We consider the following problem for e > 0: Find u ∈ D 1,2 (R N ) solving: (P e ) -Δu = α N (1+eK(x))u N+2 N-2 , in R N . We show an exact multiplicity result for (Pe) for all small e > 0.

On the exact structure of positive solutions of an Ambrosetti-Brezis-Cerami problem and its generalization in one space variable

We study the exact structure of positive solutions of an Ambrosetti-Brezis-Cerami problem and its generalization in one space variable and in the classical Laplacian case. We prove the exact multiplicity result when $\lambda $ ranges over the whole interval $\left( 0,\infty \right) $ and get more detailed results of the solution curve. The proof of our exact multiplicity result uses the modified time-map techniques which can be adapted, and the exact multiplicity result can be extended to a more general $k$-Laplacian problem with $k>1.$

Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives

This paper concerns a class of nonlinear variational problems involving pointwise constraints on the second derivatives. Our aim is to describe the set of data for which these problems have solutions and to analyse the structure of the set of solutions under suitable assumptions on the asymptotic behaviour of the nonlinear term. In particular, if this term is assumed to be convex, then we can specify the number of solutions and obtain exact multiplicity results. The existence, nonexistence and multiplicity results we obtain show that the presence of constraints of this kind produces some phenomena which are typical of nonlinear elliptic equations with “jumping” nonlinearities.

On the number of single-peak solutions of the nonlinear Schrödinger equation

In this paper we consider uniqueness and multiplicity results for single-peak solutions for the nonlinear Schrödinger equation. For a suitable class of potentials V and critical points P we are able to establish exact multiplicity results. In particular we show that for any nondegenerate critical point of V there are only one solution concentrating at P. Moreover some examples of single-peak solutions concentrating at the same point P are provided.

Vortex structures for an SO(5) model of high-TC superconductivity and antiferromagnetism

We study the structure of symmetric vortices in a Ginzburg–Landau model based on Zhang's SO(5) theory of high-temperature superconductivity and antiferromagnetism. We consider both a full Ginzburg–Landau theory (with Ginzburg–Landau scaling parameter κ < ∞) and a κ → ∞ limiting model. In all cases we find that the usual superconducting vortices (with normal phase in the central core region) become unstable (not energy minimizing) when the chemical potential crosses a threshold level, giving rise to a new vortex profile with antiferromagnetic ordering in the core region. We show that this phase transition in the cores is due to a bifurcation from a simple eigenvalue of the linearized equations. In the limiting large-κ model, we prove that the antiferromagnetic core solutions are always non-degenerate local energy minimizers and prove an exact multiplicity result for physically relevant solutions.

Equilibrium solutions of viscous scalar balance laws with a convex flux

These boundary conditions turn out to make the problem analytically rather difficult but most of our methods apply to the study of other boundary conditions, too. Our goal is a complete description of all equilibrium (i.e. time-independent) solutions, including exact multiplicity results and statements about the stability of the solutions. Without providing any details, we just mention that the information we collect during the proofs is sufficient to give a fairly good picture of the global attractor of (1) using the work of Fiedler and Rocha [2, 3]. Throughout the paper we make the following assumptions on f and g:

Exact Multiplicity of Positive Solutions for a Class of Semilinear Problems

We establish an exact multiplicity result of positive solutions ofΔu+λf(u)=0inBn,u=0on∂Bn,whereBnis the unit ball,fsatisfiesf″ changes sign only once and asymptotically sublinear or linear. The nonlinearities which we are concerned with here includef(u)=u(u−b)(c−u) for 0<2b<candf(u)=up−uqfor 1<p<q. Precise global bifurcation diagrams are obtained.

An Exact Multiplicity Result for a class of Semilinear Equations

For a class of Dirichlet problems in two dimensions, generalizing the model case we show existence of a critical so that there are exactly 0, 1 or 2 nontrivial solutions (in fact, positive), depending on whether We show that all solutions lie on a single smooth solution curve, and study some properties of this curve. We use bifurcation approach. The Curial thing is to show that any nontrivial solution of the corresponding linearized problem is of one sign.