On a singular Riemann–Liouville fractional boundary value problem with parameters

We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

A multiparameter fractional Laplace problem with semipositone nonlinearity

<p style='text-indent:20px;'>In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u&amp; = &amp; \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&amp;&gt;&amp;0 \text{ in } \Omega\\ u&amp;\equiv &amp;0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>when the positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> belong to certain range. Here <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is assumed to be a bounded open set with smooth boundary, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0, 1), N&gt; 2s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;q&lt;1&lt;r\leq \frac{N+2s}{N- 2s}. $\end{document}</tex-math></inline-formula> First we consider <inline-formula><tex-math id="M6">\begin{document}$ (P_ \lambda^\mu) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula> and prove that there exists <inline-formula><tex-math id="M8">\begin{document}$ \lambda_0\in(0, \infty) $\end{document}</tex-math></inline-formula> such that for all <inline-formula><tex-math id="M9">\begin{document}$ \lambda&gt; \lambda_0 $\end{document}</tex-math></inline-formula> the problem <inline-formula><tex-math id="M10">\begin{document}$ (P_ \lambda^0) $\end{document}</tex-math></inline-formula> admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of <inline-formula><tex-math id="M11">\begin{document}$ (P_\lambda^0) $\end{document}</tex-math></inline-formula> emanating from infinity. Next for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda&gt;\lambda_0 $\end{document}</tex-math></inline-formula> and for all <inline-formula><tex-math id="M13">\begin{document}$ 0&lt;\mu&lt;\mu_{\lambda} $\end{document}</tex-math></inline-formula> we establish the existence of at least one positive solution of <inline-formula><tex-math id="M14">\begin{document}$ (P_\lambda^\mu) $\end{document}</tex-math></inline-formula> using variational method. Also in the sub critical case, i.e., for <inline-formula><tex-math id="M15">\begin{document}$ 1&lt;r&lt;\frac{N+2s}{N-2s} $\end{document}</tex-math></inline-formula>, we show the existence of second positive solution via mountain pass argument.</p>

Existence of positive solutions for a class of semipositone problem in whole ℝN

In this paper we show the existence of solution for the following class of semipositone problem P$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$ where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.

On Semipositone Non-Local Boundary-Value Problems with Nonlinear or Affine Boundary Conditions

We consider the boundary-value problemwhereH: [0,+∞) → ℝ andf: [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that ifHsatisfies a boundedness condition on a specified compact set, then this, together with an assumption thatHis either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions onf. Finally, since it can hold thatf(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.