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We study the fractional Schr\"odinger equations with a vanishing parameter: $$ (-\Delta)^s u+u =|u|^{p-2}u+\lambda|u|^{q-2}u \text{ in }\mathbb{R}^N,\quad u \in H^s(\mathbb{R}^N),$$ where $s\in(0,1)$, $N>2s$, $20$ is a vanishing parameter. We investigate the asymptotic behaviour of positive ground state solutions for $\lambda$ small, when $p$ is subcritical, or critical Sobolev exponent $2_s^*$. For $p<2_s^*$, the ground state solution asymptotically coincides with unique positive ground state solution of $(-\Delta)^s u+u=u^p$, whereas for $p=2_s^*$ the asymptotic behaviour of the solutions, after a rescaling, is given by the unique positive solution of the nonlocal critical Emden-Fowler type equation. Additionally, for $\lambda>0$ small, we show the uniqueness and nondegeneracy of the positive ground state solution using these asymptotic profiles of solutions., Comment: 37 pages

On the hyperbolic space, we study a semilinear equation with non-autonomous nonlinearity having a critical Sobolev exponent. The Poincar\'e-Sobolev equation on the hyperbolic space explored by Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] resembles our equation. As seen from the profile decomposition of the energy functional associated with the problem, the concentration happens along two distinct profiles: localised Aubin-Talenti bubbles and hyperbolic bubbles. Standard variational arguments cannot obtain solutions because of nontrivial potential and concentration phenomena. As a result, a deformation-type argument based on the critical points at infinity of the associated variational problem has been carried out to obtain solution for $N>6.$ Conformal change of metric is used for proofs, enabling us to convert the original equation into a singular equation in a ball in $\mathbb{R}^N$ and perform a fine blow-up analysis.

In this paper, we deal with the following $(p,q)$-fractional problem $$ (-\Delta)^{s_{1}}_{p}u +(-\Delta)^{s_{2}}_{q}u=\lambda P(x)|u|^{k-2}u+\theta|u|^{p_{s_{1}}^{*}-2}u \, \mbox{ in }\, \Omega,\qquad u=0\, \mbox{ in }\, \mathbb{R}^{N} \setminus \Omega, $$ where $\Omega\subseteq\mathbb{R}^{N}$ is a general open set, $00$, $P$ is a nontrivial nonnegative weight, while $p_{s_{1}}^{*}=Np/(N-ps_{1})$ is the critical exponent. We prove that there exists a decreasing sequence $\{\theta_j\}_j$ such that for any $j\in\mathbb N$ and with $\theta\in(0,\theta_j)$, there exist $\lambda_*$, $\lambda^*>0$ such that above problem admits at least $j$ distinct weak solutions with negative energy for any $\lambda\in (\lambda_*,\lambda^*)$. On the other hand, we show there exists $\overline{\lambda}>0$ such that for any $\lambda>\overline{\lambda}$, there exists $\theta^*=\theta^*(\lambda)>0$ such that the above problem admits a nonnegative weak solution with negative energy for any $\theta\in(0,\theta^*)$.

This paper deals with the global compactness and multiplicity of positive solutions to problems of the type $$ -\Delta_{\mathbb B^N} u -\lambda u=a(x) |u|^{2^*-2}u+f(x) \quad\text{in } \mathbb B^N, \quad u\in H^1(\mathbb B^N),$$ where $\mathbb B^N$ denotes the ball model of the hyperbolic space of dimension $N\geq 4$, $2^*=\frac{2N}{N-2}$, $\frac{N(N-2)}{4}<\lambda<\frac{(N-1)^2}{4}$ and $f\in H^{-1}(\mathbb B^N)$ ($f\not\equiv 0$) is a non-negative functional in the dual space of $H^1(\mathbb B^N)$. The potential $a\in L^\infty(\mathbb B^N)$ is assumed to be strictly positive, such that $\lim_{ d(x,0)\to \infty}a(x)=1$, where $d(x,0)$ denotes the geodesic distance. We establish profile decomposition of the associated functional. We show that concentration takes place along two different profiles, namely along hyperbolic bubbles and localized Aubin-Talenti bubbles. For $f=0$ and $a\equiv 1$, profile decomposition was studied by Bhakta and Sandeep [Calc. Var. PDE, 2012]. However, due to the presence of $a(.)$, an extension of profile decomposition to the present set-up is highly nontrivial and requires several delicate estimates and geometric arguments concerning the isometry group (M\"obius group) of the hyperbolic space. Further, using the decomposition result, we derive various energy estimates involving the interacting hyperbolic bubbles and hyperbolic bubbles with localized Aubin-Talenti bubbles. Finally, combining these estimates with topological and variational arguments, we establish a multiplicity of positive solutions in the cases: $a\geq 1$ and $a<1$ separately. The equation studied in this article can be thought of as a variant of a scalar-field equation with a critical exponent in the hyperbolic space, although such a critical exponent problem in the Euclidean space $\mathbb{R}^N$ has only a trivial solution when $f \equiv 0,$ $a(x)\equiv1$ and $\lambda < 0.$

A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincar\'e-Sobolev equation on the hyperbolic space $$ -\Delta_{\mathbb{B}^n} u-\lambda u = |u|^{p-1}u, \quad u\in H^1(\mathbb{B}^n), $$ are unique up to hyperbolic isometries where $n \geq 3,$ $1 < p \leq \frac{n+2}{n-2} $ and $\lambda \leq \frac{(n-1)^2}{4}.$ We prove under certain bounds on $\|\nabla u \|_{L^2(\mathbb{B}^n)}$ the inequality $$ \delta(u) \lesssim \|\Delta_{\mathbb{B}^n} u+ \lambda u + u^{p}\|_{H^{-1}}, $$ holds whenever $p >2$ and hence forcing the dimensional restriction $3 \leq n \leq 5,$ where $\delta(u)$ denotes the $H^1$ distance of $u$ from the manifold of sums of hyperbolic bubbles. Moreover, it fails for any $n \geq 3$ and $p \in (1,2].$ This strengthens the phenomenon observed in the Euclidean case that the (linear) quantitative stability estimate depends only on whether the exponent $p$ is $>2$ or $\leq 2$. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [Arch. Ration. Mech. Anal, 237 (2020)] but we notice a striking dependence on the exponent $p$ in the subcritical regime as well which is not present in the flat case. Our technique is an amalgamation of Figalli and Glaudo's method and builds upon a series of new and novel estimates on the interaction of hyperbolic bubbles and their derivatives and improved eigenfunction integrability estimates. Since the conformal group coincides with the isometry group of the hyperbolic space, we perceive a remarkable distinction in arguments and techniques to achieve our main results compared to that of the Euclidean case., Comment: 70 pages, 4 figures. This is the updated version of our previous submission arXiv:2211.14618. New results have been added e.g., Section 9 and Section 10. The main new result is contained in Theorem~9.1

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-\Delta_p)^s u =|u|^{p^*_s-2}u+ \frac{\gamma\alpha}{p_s^*}|u|^{\alpha-2}u|v|^{\beta}\;\;\text{in}\;\Omega, (-\Delta_p)^s v =|v|^{p^*_s-2}v+ \frac{\gamma\beta}{p_s^*}|v|^{\beta-2}v|u|^{\alpha}\;\;\text{in}\;\Omega, % % u,\;v\in\wsp, \end{cases} \end{equation*} where $s\in(0,1)$, $p\in(1,\infty)$ with $N>sp$, $\alpha,\,\beta>1$ such that $\alpha+\beta = p^*_s:=\frac{Np}{N-sp}$ and $\Omega=\mathbb{R}^N$ or smooth bounded domains in $\mathbb{R}^N$. For $\Omega=\mathbb{R}^N$ and $\gamma=1$, we show that any ground state solution of the above system has the form $(\lambda U, \tau\lambda V)$ for certain $\tau>0$ and $U,\;V$ are two positive ground state solutions of $(-\Delta_p)^s u =|u|^{p^*_s-2}u$ in $\mathbb{R}^N$. For all $\gamma>0$, we establish existence of a positive radial solution to the above system in balls. For $\Omega=\mathbb{R}^N$, we also establish existence of positive radial solutions to the above system in various ranges of $\gamma$., Comment: 27 pages, 10 figures

Consider the Poincar\'e-Sobolev inequality on the hyperbolic space: for every $n \geq 3$ and $1 < p \leq \frac{n+2}{n-2},$ there exists a best constant $S_{n,p, \lambda}(\mathbb{B}^{n})>0$ such that $$S_{n, p, \lambda}(\mathbb{B}^{n})\left(~\int \limits_{\mathbb{B}^{n}}|u|^{{p+1}} \, {\rm d}v_{\mathbb{B}^n} \right)^{\frac{2}{p+1}} \leq\int \limits_{\mathbb{B}^{n}}\left(|\nabla_{\mathbb{B}^{n}}u|^{2}-\lambda u^{2}\right) \, {\rm d}v_{\mathbb{B}^n},$$ holds for all $u\in C_c^{\infty}(\mathbb{B}^n),$ and $\lambda \leq \frac{(n-1)^2}{4},$ where $\frac{(n-1)^2}{4}$ is the bottom of the $L^2$-spectrum of $-\Delta_{\mathbb{B}^n}.$ It is known from the results of Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] that under appropriate assumptions on $n,p$ and $\lambda$ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant $S_{n,p,\lambda}(\mathbb{B}^n).$ In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in $\mathbb{R}^n$ of Bianchi-Egnell [J. Funct. Anal. 100 (1991)] and Ciraolo-Figalli-Maggi [Int. Math. Res. Not. IMRN 2018] to the Poincar\'{e}-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a refined smoothing estimates, we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-\Delta)^s u -\theta\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\, \mathbb{R}^N, \quad u \in \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), \end{align*} where $s\in (0,1)$, $N>2s$, $q>p\geq{(N+2s)}/{(N-2s)}$, $\theta\in(0, \Lambda_{N,s})$ and $\Lambda_{N,s}$ is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions we mean, both the radial symmetry that is obtained by using the moving plane method in a nonlocal setting on the whole $\mathbb{R}^N$, and upper bound behavior of the solutions. To this last end we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space., Comment: 25 pages

We study boundary value problems with measure data in smooth bounded domains $\Omega$, for semilinear equations involving Hardy type potentials. Specifically we consider problems of the form $-L_V u + f(u) = \tau$ in $\Omega$ and $\mathrm{tr}^*u=\nu$ on $\partial \Omega$, where $L_V= \Delta+V$, $f\in C(\mathbb{R})$ is monotone increasing with $f(0)=0$ and $\mathrm{tr}^*u$ denotes the normalized boundary trace of $u$ associated with $L_V$. The potential $V$ is typically a H\"older continuous function in $\Omega$ that explodes as $\mathrm{dist}(x,F)^{-2}$ for some $F \subset \partial \Omega$. In general the above boundary value problem may not have a solution. We are interested in questions related to the concept of 'reduced measures', introduced by Brezis, Marcus and Ponce for $V=0$. For positive measures, the reduced measures $\tau^*, \nu^*$ are the largest measures dominated by $\tau$ and $\nu$ respectively such that the boundary value problem with data $(\tau^*,\nu^*)$ has a solution. Our results extend results for the case $V=0$, including a relaxation of the conditions on $f$. In the case of signed measures, some of the present results are new even for $V=0$., Comment: In the second version, we used the notion of L_V boundary trace instead of normalized boundary trace, made several major modifications, and changed the structure of the paper

In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics., Comment: 23 pages

In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N}, &(-\Delta)^s v = \frac{\beta}{2_s^*}|v|^{\beta-2}v|u|^{\alpha}+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, v >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*} where $N>2s$, $\alpha,\,\beta>1$, $\alpha+\beta=2N/(N-2s)$, and $f,\, g$ are nonnegative functionals in the dual space of $\dot{H}^s(\mathbb{R}^{N})$. When $f=0=g$, we show that the ground state solution of the above system is {\it unique}. On the other hand, when $f$ and $g$ are nontrivial nonnegative functionals with ker$(f)$=ker$(g)$, then we establish the existence of at least two different positive solutions of the above system provided that $\|f\|_{(\dot{H}^s)'}$ and $\|g\|_{(\dot{H}^s)'}$ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the above system., Comment: 27 pages

The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation., Comment: 32 pages

This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm., Comment: 11 pages

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-\Delta)^s u = a(x) |u|^{2^*_s-2}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, \quad u \in \dot{H}^s(\mathbb{R}^{N}), \end{equation} where $s \in (0,1)$, $N>2s$, $2_s^*:=\frac{2N}{N-2s}$, $0< a\in L^\infty(\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a continuous function, $a(x)\geq 1$ in $\mathbb{R}^{N}$, $a(x)\to 1$ as $|x|\to\infty$ and $\|f\|_{\dot{H}^s(\mathbb{R}^{N})'}$ is small enough (but $f\not\equiv 0$), we establish existence of at least two positive solutions to ($\mathcal E$)., Comment: 21 pages. arXiv admin note: text overlap with arXiv:1910.07919

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: \begin{equation} \tag{$\mathcal{P}$} \left\{\begin{aligned} (-\Delta)^s u + u &= a(x) |u|^{p-1}u+f(x)\;\;\text{in}\;\mathbb{R}^{N}, u &\in H^{s}{(\mathbb{R}^{N})} \end{aligned} \right. \end{equation} where $s \in (0,1)$, $N>2s$, $1

Our main aim is to study Green function for the fractional Hardy operator $P:=(-\Delta)^s -\frac{\theta}{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<\theta<\Lambda_{N,s}$ and $\Lambda_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds., Comment: 16 pages

We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-\Delta)^s u &= v^p+\mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q+\nu \quad &&\text{in } \Omega\\ u = v &= 0 \quad &&\text{in } \Omega^c={\mathbb R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a $C^2$ bounded domains in ${\mathbb R}^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $\mu,\, \nu$ are positive measures in $\Omega$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $\mu$ and $\nu$. Furthermore, if $p,q \in (1,\frac{N+s}{N-s})$ and $0 \leq \mu,\, \nu\in L^r(\Omega)$ for some $r>\frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions., Comment: 24 pages

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 pages

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 pages

We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$ in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha \in(0,1)$. We establish a universal a priori estimate for positive solutions of (E), as well as for their gradients. Then for $C^2$ bounded domain $\Omega$, we prove the existence of positive solutions of (E) with prescribed boundary value $\rho \nu$, where $\rho>0$ and $\nu$ is a positive Radon measure on $\partial \Omega$ with total mass $1$, and discuss regularity property of the solutions. When $f(u)=u^p$, we demonstrate that there exists a critical exponent $p_s:=\frac{N+s}{N-s}$ in the following sense. If $p\geq p_s$, the problem does not admit any positive solution with $\nu$ being a Dirac mass. If $p\in(1,p_s)$ there exits a threshold value $\rho^*>0$ such that for $\rho\in (0, \rho^*]$, the problem admits a positive solution and for $\rho>\rho^*$, no positive solution exists. We also show that, for $\rho>0$ small enough, the problem admits at least two positive solutions., Comment: We withdraw the paper because there was a flaw in the proof of Theorem 3.1 in Section 3. As a consequence, Theorem 1.2, Theorem 3.4 and Theorem A.1 are not valid

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, $0ps$ ., Comment: 28 pages. arXiv admin note: text overlap with arXiv:1603.05554

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega, where $s\in(0,1)$, $q>p\geq \frac{N+2s}{N-2s}$ and $\epsilon>0$ is a parameter. Here $\Omega\subseteq\mathbb{R}^N$ is a bounded star-shaped domain with smooth boundary and $N> 2 s$. We establish existence of a variational positive solution $u_{\epsilon}$ and characterize the asymptotic behaviour of $u_{\epsilon}$ as $\epsilon\to 0$. When $p=\frac{N+2s}{N-2s}$, we describe how the solution $u_{\epsilon}$ concentrates and blows up at a interior point of the domain. Furthermore, we prove the local uniqueness of solution of the above problem when $\Omega$ is a convex symmetric domain of $\mathbb{R}^N$ with $N>4s$ and $p=\frac{N+2s}{N-2s}$., Comment: 30 pages

We study the problem \begin{eqnarray*} (-\Delta)^s u &=& u^p - u^q \quad\text{in }\quad \mathbb{R}^N, u &\in& \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), u&>0& \quad\text{in}\quad\mathbb{R}^N, \end{eqnarray*} where $s\in(0,1)$ is a fixed parameter, $(-\Delta)^s$ is the fractional laplacian in $\mathbb{R}^N$, $q>p\geq \frac{N+2s}{N-2s}$ and $N>2s$. For every $s\in(0,1)$, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all $s\in(0,1)$. Using those decay estimates, we prove Pohozaev type identity in $\mathbb{R}^N$ and show that the above problem does not have any solution when $p=\frac{N+2s}{N-2s}$. We also discuss radial symmetry and decreasing property of the solution and prove that when $p>\frac{N+2s}{N-2s}$, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every $p\geq \frac{N+2s}{N-2s}$ and every solution is a classical solution., Comment: 27 pages (In this new version, some of the proofs of earlier version have been generalized.)

We study the problem \begin{equation*} (I_{\epsilon}) \left\{\begin{aligned} -\Delta u- \frac{\mu u}{|x|^2}&=u^p -\epsilon u^q \quad\text{in }\quad \Omega, \\ u&>0 \quad\text{in }\quad \Omega, \\ u &\in H^1_0(\Omega)\cap L^{q+1}(\Omega), \end{aligned} \right. \end{equation*} where $q>p\geq 2^*-1$, $\epsilon>0$ is a parameter, $\Omega\subseteq\mathbb{R}^N$ is a bounded domain with smooth boundary, $0\in \Omega$, $N\geq 3$ and $0<\mu<\bar\mu:=\big(\frac{N-2}{2}\big)^2$. We prove at $0$, any solution of $(I_{\epsilon})$ has the singularity of order $|x|^{-\nu}$ when $q<\frac{2+\nu}{\nu}$ and of the order $|x|^{-\frac{2}{q-1}}$, when $q>\frac{2+\nu}{\nu}$, where $\nu=\sqrt{\bar\mu}-\sqrt{\bar\mu-\mu}$. Moreover, we show that when $q=\frac{2+\nu}{\nu}$ and $u$ is radial, $u\sim |x|^{-\nu}|\log|x||^{-\frac{\nu}{2}}$. This gives the complete classification of singularity at $0$ in the supercritical case. We also obtain gradient estimate. Using the transformation $v=|x|^{\nu}u$, we reduce the problem $(I_{\epsilon})$ to $(J_{\epsilon})$ \begin{equation*} (J_{\epsilon}) \left\{\begin{aligned} -div(|x|^{-2\nu} \nabla v)&=|x|^{-(p+1)\nu} v^p -\epsilon |x|^{-(q+1)\nu} v^q \quad\text{in }\quad \Omega, \\ v&>0 \quad\text{in }\quad \Omega, \\ v& \in H^1_0(\Omega, |x|^{-2\nu} )\cap L^{q+1}(\Omega, |x|^{-(q+1)\nu} ), \end{aligned} \right. \end{equation*} and then formulating a variational problem for $(J_{\epsilon})$, we establish the existence of a variational solution $v_{\epsilon}$. Furthermore, we characterize the asymptotic behavior of $v_{\epsilon}$ as $\epsilon\to 0$ by variational arguments and when $p=2^*-1$, we show how the solution $v_{\epsilon}$ blows-up at $0$. This is the first paper where the results have been established with super critical exponents for $\mu>0$., Comment: 52 pages

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 $\mathbb{R}^N$, $N>2s$, $s\in(0, 1)$, $06s$, $\lambda=1$, 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 version

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\ u=0=\Delta u \quad\textrm{on $\partial\Omega$,} \end{cases} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 5$, $a, b, f$ are nonnegaive functions satisfying certain hypothesis which we will specify later. $\mu,\lambda$ are positive constants. Under some suitable conditions on functions $a, b, f$ and the constant $\mu$, we show that there exists $\lambda^*>0$ such that when $0<\lambda<\lambda^*$, ($P_{\lambda}$) admits a solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ and for $\lambda>\lambda^*$, it does not have any solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$. Moreover as $\lambda\uparrow\lambda^*$, minimal positive solution of ($P_{\lambda}$) converges in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ to a solution of ($P_{\lambda^*}$). We also prove that there exists $\tilde{\lambda}^*<\infty$ such that $\lambda^*\leq\tilde{\lambda}^*$ and for $\lambda>\tilde{\lambda}^*$, the above problem ($P_{\lambda}$) does not have any solution even in the distributional sense/very weak sense and there is complete {\it blow-up}. Under an additional integrability condition on $b$, we establish the uniqueness of positive solution of ($P_{\lambda^*}$) in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$., Comment: 21 pages

We study the existence and multiplicity of sign changing solutions of the following equation $ \begin{cases} -\Delta u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad \Omega, u=0 \quad\text{on}\quad\partial\Omega, \end{cases} $where $\Omega$ is a bounded domain in $R^N$, $0\in\partial\Omega$, all the principal curvatures of $\partial\Omega$ at $0$ are negative and $\mu\geq 0, \ \ a>0, \ \ N\geq 7, \ \ 0

We study the existence and non-existence of nontrivial weak solution of $$ {\Delta^2u-\mu\frac{u}{|x|^{4}} = \frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$, $q_{\beta}=\frac{2(N-\beta)}{N-4}$, $0<\beta<4$, $1

We study the existence/nonexistence of positive solution of $$ {\Delta^2u-\mu\frac{u}{|x|^4}=\frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}\quad\textrm{in $\Omega$,}} $$ when $\Omega$ is a bounded domain and $N\geq 5$, $q_{\beta}=\frac{2(N-\beta)}{N-4}$, $0\leq \beta<4$ and $0\leq\mu<\big(\frac{N(N-4)}{4}\big)^2$. We prove the nonexistence result when $\Omega$ is an open subset of $\mathbf R^N$ which is star shaped with respect to the origin. We also study the existence of positive solution in $\Omega$ when $\Omega$ is a bounded domain with non trivial topology and $\beta=0$, $\mu\in(0,\mu_0)$, for certain $\mu_0<\big(\frac{N(N-4)}{4}\big)^2$ and $N\geq 8$. Different behavior of PS sequences have been obtained depending on $\beta=0$ or $\beta>0$., Comment: 21 pages

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations $$ \Delta_{p}^2 u = \lambda|x|^{-2p}|u|^{p-2}u + |x|^{-\beta}|u|^{q-2} u\quad\text{in} \quad \mathbb R^N, $$ where $N> 2p$, $p>1$, $q>p$, $\beta = N - \frac{q}{p}(N-2p)$ and $\lambda\in\mathbb R$ is smaller than the Rellich constant., Comment: 12 pages

In this paper we study the equation $-\Delta u+\rho^{-(\alpha+2)}h(\rho^{\alpha}u)=0$ in a smooth bounded domain $\Omega$ where $\rho(x)=\textrm{dist}\,(x,\partial \Omega)$, $\alpha>0$ and $h$ is a non-decreasing function which satisfies Keller-Osserman condition. We introduce a condition on $h$ which implies that the equation is subcritical, i.e. the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on $h$ we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of \cite{MV1} treating equations of the form $-\Delta u+\rho^\beta u^q=0$ with $q>1$, $\beta>-2$., Comment: 30 pages

We study boundary value problems for semilinear elliptic equations of the form $-\Delta u+g\circ u=\mu$ in a smooth bounded domain $\Omega\subset R^N$. Let $\{\mu_n\}$ and $\{\tau_n\}$ be sequences of measure in $\Omega$ and $\partial \Omega$ respectively. Assume that there exists a solution $u_n$ of the equation with $\mu=\mu_n$ subject to boundary data $\tau_n$. Further assume that the sequences of measures converge in a weak sense to $\mu$ and $\tau$ respectively and $\{u_n\}$ converges to $u$ in $L^1(\Omega)$. In general $u$ is not a solution of the boundary value problem with data $(\mu,\tau)$. However there exist measures $(\mu^*,\tau^*)$ such that $u$ satisfies the equation with $\mu$ replaced by $\mu^*$ and with $u=\tau^*$ on the boundary. The pair $(\mu^*,\tau^*)$ is called the reduced limit of the sequence $\{(\mu_n,\tau_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence., Comment: 17 pages

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights., Comment: 19 pages

We study the a priori estimates,existence/nonexistence of radial sign changing solution, and the Palais-Smale characterisation of the problem $-\De_{\Bn}u - \la u = |u|^{p-1}u, u\in H^1(\Bn)$ in the hyperbolic space $\Bn$ where $1