Existence of multiple solutions of pp-fractional Laplace operator with sign-changing weight function

In this article, we study the following $p$-fractional Laplacian equation \begin{equation*} (P_{\la}) \left\{ \begin{array}{lr} - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dy = \la |u(x)|^{p-2}u(x) + b(x)|u(x)|^{\ba-2}u(x)\; \text{in}\; \Om \quad \quad\quad\quad \quad\quad\quad\quad\quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om,\quad u\in W^{\al,p}(\mb R^n).\\ \end{array} \quad \right. \end{equation*} where \Om is a bounded domain in \mb R^n with smooth boundary, n> p\al, $p\geq 2$, \al\in(0,1), \la>0 and b:\Om\subset\mb R^n \ra \mb R is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (P_{\la}) with respect to the parameter \la, which changes according to whether 1<\ba or p< \ba< p^{*}=\frac{np}{n-p\al} respectively. We discuss both the cases separately. Non-existence results are also obtained

Critical growth elliptic problems with Choquard type nonlinearity:A survey

This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involve the nonlinearity of convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.Comment: 28 page

On the Second Eigenvalue of Combination Between Local and Nonlocal pp-Laplacian

In this paper, we study Mountain Pass Characterization of the second eigenvalue of the operator -\De_p u -\De_{J,p}u and study shape optimization problems related to these eigenvalues.Comment: 15 page

On doubly nonlocal pp-fractional coupled elliptic system

\noi We study the following nonlinear system with perturbations involving p-fractional Laplacian \begin{equation*} (P)\left\{ \begin{split} (-\De)^s_p u+ a_1(x)u|u|^{p-2} &= \alpha(|x|^{-\mu}*|u|^q)|u|^{q-2}u+ \beta (|x|^{-\mu}*|v|^q)|u|^{q-2}u+ f_1(x)\; \text{in}\; \mb R^n,\\ (-\De)^s_p v+ a_2(x)v|v|^{p-2} &= \gamma(|x|^{-\mu}*|v|^q)|v|^{q-2}v+ \beta (|x|^{-\mu}*|u|^q)|v|^{q-2}v+ f_2(x)\; \text{in}\; \mb R^n, \end{split} \right. \end{equation*} where $n>sp$, $0<s<1$, $p\geq2$, $\mu \in (0,n)$, $\frac{p}{2}\left( 2-\frac{\mu}{n}\right) < q <\frac{p^*_s}{2}\left( 2-\frac{\mu}{n}\right)$, $\alpha,\beta,\gamma >0$, 0< a_i \in C^1(\mb R^n, \mb R), $i=1,2$ and f_1,f_2: \mb R^n \to \mb R are perturbations. We show existence of atleast two nontrivial solutions for $(P)$ using Nehari manifold and minimax methods.Comment: 26 page

On Dirichlet problem for fractional pp-Laplacian with singular nonlinearity

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om. \end{equation*} where \Om is a bounded domain in \mb{R}^n with smooth boundary \partial \Om, n > sp, s \in (0,1), \la >0, 0 < q \leq 1 and $\alpha\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter \la.Comment: 26 page

A Nehari manifold for non-local elliptic operator with concave-convex non-linearities and sign-changing weight function

In this article, we study the existence and multiplicity of non-negative solutions of following $p$-fractional equation: \quad \left\{\begin{array}{lr}\ds \quad - 2\int_{\mb R^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\al}} dxdy = \la h(x)|u|^{q-1}u+ b(x)|u|^{r-1} u \; \text{in}\; \Om \quad \quad \quad \quad u \geq 0 \; \mbox{in}\; \Om,\quad u\in W^{\al,p}(\mb R^n), \quad \quad\quad \quad\quad u =0\quad\quad \text{on} \quad \mb R^n\setminus \Om \end{array} \right. where \Om is a bounded domain in \mb R^n, $p\geq 2$, n> p\al, \al\in(0,1), $00$ and $h$, $b$ are sign changing smooth functions. We show the existence of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists \la_0 such that for \la\in (0,\la_0), it has at least two solutions.Comment: 14 page

On The Fu\v{c}ik Spectrum Of Non-Local Elliptic Operators

In this article, we study the Fu\v{c}ik spectrum of fractional Laplace operator which is defined as the set of all (\al,\ba)\in \mb R^2 such that \begin{equation*} \quad \left. \begin{array}{lr} \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\ \end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$, where \Om is a bounded domain in \mb R^n with Lipschitz boundary, $n>2s$, $s\in(0,1)$. The existence of a first nontrivial curve \mc C of this spectrum, some properties of this curve \mc C, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to Fu\v{c}ik spectrum.Comment: 22 pages in NoDEA: Nonlinear differential equations and applications, 201

Fractional Choquard Equation with Critical Nonlinearities

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{\mu,s}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu,s}-2}u +\la u \; \text{in } \Om, where \Om is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, \la is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{\mu,s}= (2n-\mu)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.Comment: 32 pages. arXiv admin note: text overlap with arXiv:1604.00826 by other author

Existence and multiplicity results for fractional pp-Kirchhoff equation with sign changing nonlinearities

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-\Delta)^{s}_p u &=\lambda f(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} \Omega,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus \Omega, \end{array} \end{equation*} where $(-\Delta)^{s}_p$ is the fractional $p$-Laplace operator, $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(\Omega)$ and $g\in L^\infty(\Omega)$ are sign changing, $M$ is continuous function, $ps<n<2ps$ and $1<q<p<r\leq p_s^*=\frac{np}{n-ps}$.Comment: Advances in pure and applied Mathematics 201

Critical growth fractional elliptic problem with singular nonlinearities

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity \quad (-\Delta)^s u = \lambda a(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; \Omega,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminus\Omega, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s,\; s \in (0,1),\; \lambda >0,\; 0 < q \leq 1$, $\theta \leq a(x) \in L^\infty(\Omega)$, for some $\theta>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $\lambda$.Comment: 25 pages. arXiv admin note: substantial text overlap with arXiv:1602.0087