Positive solutions of a boundary value problem with integral boundary conditions

We consider boundary-value problems studied in a recent paper. We show that some existing theory developed by Webb and Infante applies to this problem and we use the known theory to show how to find improved estimates on parameters μ*, λ so that some nonlinear differential equations, with nonlocal boundary conditions of integral type, have two positive solutions for all λ with μ*< λ < λ

Spectral properties of stationary solutions of the nonlinear heat equation

In this paper, we prove that if ψ is a radially symmetric, signchanging stationary solution of the nonlinear heat equation (NLH) u - ∆u = │u │ α u, in the unit ball of RN, N=3, with Dirichlet boundary conditions, then the solution of (NLH) with initial value λψ blows up infinite time if │λ - 1│ > 0 is sufficiently small and if α > 0 is sufficiently small. The proof depends on showing that the inner product of ψ with the first eigenfunction of the linearized operator L= - ∆ - (α + 1) │ψ│α is nonzero

A Minmax Principle, Index of the Critical Point, and Existence of Sign Changing Solutions to Elliptic Boundary Value Problems

In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at least five solutions (two of which change sign), and iii) the existence of precisely two sign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is sufficiently thin, and ii) the existence of arbitrarily many sign-changing non-radial solutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial sign-changing solutions is established when the underlying region is a ball

Sign changing solutions of p-fractional equations with concave-convex nonlinearities

In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-\Delta)^s_pu &=& \mu |u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad \Omega, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0<q<p-1$, $\mu\in\mathbb{R}^+$ and $p_s^*=\frac{Np}{N-ps}$. $\Omega$ is an open, bounded domain in $\mathbb{R}^N$ with smooth boundary with $N>ps$ .Comment: 28 pages. arXiv admin note: text overlap with arXiv:1603.0555

Study of a logistic equation with local and non-local reaction terms

In this work we examine a logistic equation with local and nonlocal reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results depend strongly on the size and sign of the parameters on the local and non-local terms.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConselho Nacional de Desenvolvimento Científico e TecnológicoFundação de Amparo à Pesquisa do Estado de São Paul