Multiplicity results and sign changing solutions of non-local equations with concave-convex nonlinearities

In this paper we prove the existence of infinitely many nontrivial solutions of the following equations driven by a nonlocal integro-differential operator $L_K$ with concave-convex nonlinearities and homogeneous Dirichlet boundary conditions \begin{eqnarray*} \mathcal{L}_{K} u + \mu\, |u|^{q-1}u + \lambda\,|u|^{p-1}u &=& 0 \quad\text{in}\quad \Omega, \\[2mm] u&=&0 \quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $0<q<1<p\leq \frac{N+2s}{N-2s}$. Moreover, when $L_K$ reduces to the fractional laplacian operator $-(-\Delta)^s$, $p=\frac{N+2s}{N-2s}$, $\frac{1}{2}(\frac{N+2s}{N-2s})6s$, $\lambda=1$, we find $\mu^*>0$ such that for any $\mu\in(0,\mu^*)$, there exists at least one sign changing solution.Comment: 32 pages. Proof of Claim 4 in Theorem 4.1 has been modified in this versio

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

Multiplicity results for (p, q)(p,\, q) fractional elliptic equations involving critical nonlinearities

In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_{\Omega}(\Omega)$ nonnegative solutions.Comment: 36 page

On the existence of multiple solutions for fractional Brezis Nirenberg type equations

The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, \begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-\Delta\right)^s u-\lambda u &= \alpha |u|^{p-2}u + \beta|u|^{2^*-2}u \quad\mbox{in}\quad \Omega,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{cases} \end{align*} where $(-\Delta)^s$ is the fractional Laplace operator, $s \in (0,1)$, with $N \geq 3s$, $20, \lambda, \alpha \in \mathbb{R}$ and establish the existence of nontrivial solutions and sign-changing solutions for the problem $(\mathcal{P})$

Multiplicity results for non-local elliptic problems with jumping nonlinearity

The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable pseudo-gradient vector field of the corresponding energy functional

On fractional multi-singular Schr\"odinger operators: positivity and localization of binding

In this work we investigate positivity properties of nonlocal Schr\"odinger type operators, driven by the fractional Laplacian, with multipolar, critical, and locally homogeneous potentials. On one hand, we develop a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, while, on the other hand, we study the localization of binding for this kind of potentials. Combining these two tools and performing an inductive procedure on the number of poles, we establish necessary and sufficient conditions for the existence of a configuration of poles that ensures the positivity of the corresponding Schr\"odinger operator

On the existence of three non-negative solutions for a (p,q)(p,q)-Laplacian system

The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-\Delta)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-\Delta)^{s_2}_{p_2} v &=\la f_2\,(x,u,v) +g_2(x,v) \,\mbox{ in }\, \Om, \\ u=v &= 0 \,\mbox{in }\, \Rn \setminus \Om, \\ \end{aligned} \right. \end{equation*} where \Om \subset \Rn is a smooth bounded domain with smooth boundary, $s_1,s_2 \in (0,1)$, $1<p_i<\frac{N}{s_i}$, $i=1,2$, $f_i$ and $g_i$ has certain growth assumptions for $i=1,2$. We prove existence of at least three non negative solutions of $(\mathcal P)$ under restrictive range of $\lambda$ using variational methods. As a consequence, we also conclude that a similar result can be obtained when we consider a more general non local operator $\mathcal L_{\phi_i}$ instead of $(-\Delta)^{s_i}_{p_i}$ in $(\mathcal P)$

Nonlocal scalar field equations: qualitative properties, asymptotic profiles and local uniqueness of solutions

We study the nonlocal scalar field equation with a vanishing parameter $$\left\{\begin{array}{lll} (-\Delta)^s u+\epsilon u &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right.$$ where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $\epsilon>0$ is a vanishing parameter. For $\epsilon>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic behavior of ground state solutions when $p$ is subcritical, supercritical or critical Sobolev exponent $2^*=\frac{2N}{N-2s}$. For $p<2^*$, the solution asymptotically coincides with unique positive ground state solution of $(-\Delta)^s u+u=u^p$. On the other hand, for $p=2^*$ the asymptotic behaviour of the solutions is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. For $p>2^*$, the solution asymptotically coincides with a ground-state solution of $(-\Delta)^s u=u^p-u^q$. Furthermore, using these asymptotic profile of solutions, we prove the \textit{local uniqueness} of solution in the case $p\leq 2^*$.Comment: 49 page

A Carleman estimate for the fractional heat equation and its application in final state observability

In the paper, we show a global Carleman estimate for the non-local heat equation. To be more precise, let \Omega\subset\RR^d be a bounded domain and \CO\subset\Omega an open subdomain, $s\in(0,1)$. We show that there exist constants $C_1,C_2,r_0, T_0>0$ and a weight function $\alpha:\Omega\to(0,\infty)$ such that any solution $u$ of %consider the following system % \begin{eqnarray}\label{oben1} \left\{ \begin{array}{rcl} \timed u(x,t)+(-\De)^s u (x,t) &=&f(x,t) \quad\mbox{for}\quad (x,t)\in \Om \times (0,\infty), \\ u(x,t) &=& 0 \quad\mbox{for}\quad(x,t)\in \partial \Om \times (0,\infty), \end{array}\right. \end{eqnarray} satisfies for all $r\ge r_0$ and $T>0$ \begin{eqnarray}\label{Carle} % \lefteqn{ \int_0^T\Big[ \int_\Om e^{-2r\frac {\alpha(x)}{t(T-t)}} |f(x,t)|^2\,dx+C_1\int_\CO e^{-2r\frac {\alpha(x)}{t(T-t)}} \frac {r^2}{t^4(T-t)^4}|u(x,t)|^2dx\,\Big] dt \vspace{2cm} }&& \\ \nonumber &\ge & C_2 \Bigg[\int_0^T \int_\Om e^{-2r\frac {\alpha(x)}{t(T-t)}}\Big\{ \big|(-\Delta)^s u(x,t)\big|^2 + \frac 12 \Big|\timed u(x,t)\Big|^2+ \frac r{t^4(T-t)^4}\,|u(t,x)|^2\Big\} dx\, dt. %\del{\\ %&&\vspace{2cm}}+ r^3\int_0^T \int_\mathcal{O} \frac {r^3}{t^3(T-t)^3} \Phi^2(x,t) |u(x,t)|^2 \, dx\, dt\Bigg]. \end{eqnarray} % In order to prove this result, we use the Caffarelli-Silvestre extension procedure. To illustrate the applicability of the result, we prove as a second main result the final state observability of the non-local heat equation.Comment: a gap was pointed out. We hope to repair i

Uniqueness of the critical point for semi-stable solution in R2\mathbb{R}^2

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -\Delta u=f(u)&\text{in }\Omega\\ u>0&\text{in }\Omega\\ u=0&\text{on } \partial\Omega,\end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabr\'e-Chanillo to the case where the curvature of $\partial\Omega$ vanishes