Combined effects in nonlinear problems arising in the study of anisotropic continuous media

The paper deals with the study of a Lane-Emden-Fowler equation with Dirichlet boundary condition and variable potential functions. The analysis developed in this paper combines monotonicity methods with variational arguments. Remark (April 21, 2020): Our results were later studied also by D.-P. Covei, Quasilinear problems with the competition between convex and concave nonlinearities and variable potentials, Internat. J. Math. 24:1 (2013), art. 1350005), arXiv:1104.4626v1.Comment: arXiv admin note: substantial text overlap with arXiv:1104.4626 by other author

Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

\noi In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity \quad \left\{ \begin{array}{lr} \quad -M\left(\displaystyle\int_\Omega |\nabla^m u|^{\frac{n}{m}}dx\right)\Delta_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^\alpha} \; \text{in}\; \Om{,} \quad \quad u = \nabla u=\cdot\cdot\cdot= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Om{,} \end{array} \right. where \Om\subset \mb R^n is a bounded domain with smooth boundary, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$ as |u|\ra\infty. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution. %{OR}\\ In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using {the} Nehari manifold technique, we show the existence and multiplicity of non-negative solutions. \medskipComment: Communications in pure and applied analysis (2016

Boundary characteristic point regularity for semilinear reaction-diffusion equations: Towards an ODE criterion

Boundary characteristic point regularity is studied for a class of semilinear heat equations and an ODE criterion of regularity is obtained. Extensions to higher-order semilinear parabolic problems are discussed.Comment: 31 pages, 1 figur

A Reduction Method for Semilinear Elliptic Equations and Solutions Concentrating on Spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an annulus D in R^(m+1), invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m-1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity

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

A monotonicity result under symmetry and Morse index constraints in the plane

This paper deals with solutions of semilinear elliptic equations of the type $$\left\{\begin{array}{ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right.$$ where $\Omega$ is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions $u$ that are invariant by rotations of a certain angle $\theta$ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or $u$ is radial, or, else, there exists a direction $e\in \mathcal S$ such that $u$ is symmetric with respect to $e$ and it is strictly monotone in the angular variable in a sector of angle $\frac{\theta}2$. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities $\Phi$ in power form, precisely $\Phi'(u)=c\,u^{-(n+1)}$ for some $0< n<1$. We also include the logarithmic diffusion equation $\partial_t u + (-\Delta)^s \log(u)= 0$, which appears as the case $n=0$. We consider the Cauchy problem with nonnegative and integrable data $u_0(x)$ in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The {\sl limit solutions} we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, that are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as $t\to\infty$) of the solutions with general integrable data. A new comparison principle is introduced.Comment: 35 page

Nonzero positive solutions of nonlocal elliptic systems with functional BCs

We discuss the existence and non-existence of non-negative weak solutions for second order nonlocal elliptic systems subject to functional boundary conditions. Our approach is based on classical fixed point index theory combined with some recent results by the author.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1707.0283

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

Regularity of stable solutions to semilinear elliptic equations on Riemannian models

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian manifolds includes the classical \textit{space forms}, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish $L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend on the nonlinearity $f$. As an application, under standard assumptions on the nonlinearity $\lambda f(u)$, we prove that the corresponding extremal solution $u^*$ is bounded whenever $n\leq9$. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.Comment: 21 page