Radial solutions for a Neumann elliptic system with quadratic growth in the gradient

The existence of multiple positive periodic solutions for functional differential equations

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

Radial positive solutions of nonlinear elliptic systems with Neumann boundary conditions

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

In this paper, we deal with the following nonlinear fractional differential problem in the half-line \({\mathbb{R}^{+}=(0,+ \infty)}\) $$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$ where \({m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}\), the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in \({\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}\) satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schauder fixed point theorem. Some examples illustrating our main result are also given.

In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phase-separation regime. As an example, we prove that the system $ \begin{cases} -\Delta u = u-u^3-\Lambda uv^2 \\ -\Delta v = v-v^3-\Lambda u^2v \\ u,v \gt 0 \end{cases} \qquad \text{in }\mathbb{R}^N, \text{with }\Lambda \gt 1, $ has infinitely many saddle-shape solutions in dimension $2$ or higher. This is in sharp contrast with the case $\Lambda \in (0, 1]$, for which, on the contrary, only constant solutions exist.

On the Existence of Multiple Positive Solutions for a Semilinear Problem in Exterior Domains

Existence of multiple positive solutions for one-dimensional p-Laplacian

Existence of multiple positive solutions for Sturm–Liouville-like four-point boundary value problem with [formula omitted]-Laplacian

Existence of multiple positive solutions for inhomogeneous Neumann problem

The existence of multiple positive solutions is presented for the singular second-order boundary value problems $$\left\{\begin{array}{ll} x^{\prime\prime}+\Phi(t)f(t,x,x') =0, 0 < t < 1\\ x(0) = 0, x'(1) = 0 \end{array}\right.$$ using the fixed point index, where f may be singular at x = 0 and x′ = 0.

The existence of multiple positive solutions is presented for the singular second-order boundary value problem $$\left\{\begin{array}{rrr} x^{\prime\prime} + \Phi(t)f(t,x(t), x^\prime(t)) &=& 0, \quad 0< t< 1\\ \alpha x(0) - \beta x^\prime(0) &=& 0, \quad x^{\prime}(1) = 0 \end{array}\right. $$ using the fixed point index; here f may be singular at x = 0 and x′ = 0.

The multiple positive solutions for p-Laplacian multipoint BVP with sign changing nonlinearity on time scales

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian: $$ \left\{ \begin{gathered} (\varphi (u'))' + a(t)f(u(t)) = 0,0 < t < 1, \hfill \\ \alpha \varphi (u(0)) - \beta \varphi (u'(\xi )) = 0,\gamma \varphi (u(1)) + \delta \varphi (u'(\eta )) = 0, \hfill \\ \end{gathered} \right. $$ where φ(x) = |x|p−2x, p > 1, a(t) may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple (at least three) positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.

We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are presented to illustrate the theoretical results.