American Institute of Mathematical Sciences

In this paper we prove a priori estimates for positive solutions of elliptic equations of the \begin{document}$ p $\end{document}-Laplacian type on arbitrary domains of \begin{document}$ \mathbb {R}^N $\end{document}, when a nonlinearity depending on the gradient is considered. Also the case of systems with very general nonlinearities is considered. Our main theorems extend previous results by Polacik, Quitter and Souplet in [26] in which either the case \begin{document}$ p = 2 $\end{document} with a nonlinearity depending on the gradient or the \begin{document}$ p $\end{document}-Laplacian case with a nonlinearity not depending on the gradient is treated. The technique is based on the use of a method developed in [26] whose main tools are rescaling arguments combined with a key "doubling" property, which is different from the celebrated blow up technique due to Gidas and Spruck in [16]. A discussion on the sharpness of the main result in the scalar case is presented.

We obtain existence results for mild solutions of a fractional differential inclusion subjected to impulses and nonlocal initial conditions. By means of a technique based on the weak topology in connection with the Glicksberg-Ky Fan Fixed Point Theorem we are able to avoid any hypotheses of compactness on the semigroup and on the nonlinear term and at the same time we do not need to assume hypotheses of monotonicity or Lipschitz regularity neither on the nonlinear term, nor on the impulse functions, nor on the nonlocal condition. An application to a fractional diffusion process complete the discussion of the studied problem. 200 words.

We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of \begin{document}$ \mathbb R^N $\end{document}, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs.

In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem

\begin{document}$ \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = f(x,u) &\hbox{in $\Omega$,}\\ u \ge 0 &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} \right. $\end{document}

where \begin{document}$ \Omega \subset \mathbb {R}^N $\end{document} is an open bounded domain, \begin{document}$ N\ge 3 $\end{document}, and \begin{document}$ A(x, t, \xi) $\end{document}, \begin{document}$ f(x, t) $\end{document} are given functions, with \begin{document}$ A_t = \frac{\partial A}{\partial t} $\end{document}, \begin{document}$ a = \nabla_\xi A $\end{document}.

To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami–Palais–Smale condition.

Furthermore, if \begin{document}$ A(x, t, \xi) $\end{document} grows fast enough with respect to \begin{document}$ t $\end{document}, then the nonlinear term related to \begin{document}$ f(x, t) $\end{document} may have also a supercritical growth.

The existence of \begin{document}$ L^{2} $\end{document}-nonnegative solutions for nonlinear quadratic integral equations on a bounded closed interval is investigated. Two existence results for different classes of functions are shown. As a consequence an existence theorem for the Chandrasekhar integral quadratic equation, well-known in theory of radiative transfer, is obtained. The aim is achieved by means of a new fixed point theorem for multimaps in locally convex linear topological spaces.

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

This paper is devoted to the study of the following Schrödinger–Kirchhoff–Hardy equation in \begin{document}$ \mathbb R^n $\end{document}

\begin{document}$ M\left(\iint_{\mathbb R^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\right)(-\Delta)^{s}_pu+V(x)|u|^{p-2}u-\mu\frac{|u|^{p-2}u}{|x|^{ps}} = f(x, u), $\end{document}

where \begin{document}$ (-\Delta)^s_p $\end{document} is the fractional \begin{document}$ p $\end{document}–Laplacian, with \begin{document}$ s\in(0, 1) $\end{document} and \begin{document}$ p>1 $\end{document}, dimension \begin{document}$ n>ps $\end{document}, \begin{document}$ M $\end{document} models a Kirchhoff coefficient, \begin{document}$ V $\end{document} is a positive potential, \begin{document}$ f $\end{document} is a continuous nonlinearity and \begin{document}$ \mu $\end{document} is a real parameter. The main feature of the paper is the combination of a Kirchhoff coefficient and a Hardy term with a suitable function \begin{document}$ f $\end{document} which does not necessarily satisfy the Ambrosetti–Rabinowitz condition. Under different assumptions for \begin{document}$ f $\end{document} and restrictions for \begin{document}$ \mu $\end{document}, we provide existence and multiplicity results by variational methods.

This paper deals with existence and regularity in variational problems related to partial differential equations and systems - both in the elliptic and in the parabolic contexts - and to calculus of variations, under general and \begin{document}$ p,q- $\end{document} growth conditions. The manuscript is dedicated to my friend and colleague Patrizia Pucci, with great esteem and sympathy.

In this paper we analyze the porous medium equation

\begin{document}$ \begin{equation} u_t = \Delta u^m + a\int_\Omega u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad {\rm{in}}\quad \Omega \times I,\;\;\;\;\;\;(◇) \end{equation} $\end{document}

where \begin{document}$ \Omega $\end{document} is a bounded and smooth domain of \begin{document}$ \mathbb R^N $\end{document}, with \begin{document}$ N\geq 1 $\end{document}, and \begin{document}$ I = [0,t^*) $\end{document} is the maximal interval of existence for \begin{document}$ u $\end{document}. The constants \begin{document}$ a,b,c $\end{document} are positive, \begin{document}$ m,p,q $\end{document} proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of \begin{document}$ u $\end{document}. Under some hypotheses on the data, including intrinsic relations between \begin{document}$ m,p $\end{document} and \begin{document}$ q $\end{document}, and assuming that for some positive and sufficiently regular function \begin{document}$ u_0({\bf x}) $\end{document} the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution \begin{document}$ u = u({\bf x},t) $\end{document} on \begin{document}$ \Omega \times I $\end{document}:

\begin{document}$ \triangleright $\end{document} when \begin{document}$ p>q $\end{document} and in 2- and 3-dimensional domains, we determine a lower bound of \begin{document}$ t^* $\end{document} for those \begin{document}$ u $\end{document} becoming unbounded in \begin{document}$ L^{m(p-1)}(\Omega) $\end{document} at such \begin{document}$ t^* $\end{document};

\begin{document}$ \triangleright $\end{document} when \begin{document}$ p<q $\end{document} and in \begin{document}$ N $\end{document}-dimensional settings, we establish a global existence criterion for \begin{document}$ u $\end{document}.

In this paper we study a initial-boundary value problem for 4th order hyperbolic equations with weak and strong damping terms and superlinear source term. For blow-up solutions a lower bound of the blow-up time is derived. Then we extend the results to a class of equations where a positive power of gradient term is introduced.

We study the existence of nontrivial weak solutions for a class of generalized \begin{document}$ p(x) $\end{document}-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.

The paper is concerned with existence, multiplicity and asymptotic behavior of (weak) solutions for nonlocal systems involving critical nonlinearities. More precisely, we consider

\begin{document}$\left\{ \begin{array}{*{35}{l}} \begin{align} & M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ & =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ & {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} & \ & \ & {} \\\end{array} \right.$\end{document}

where \begin{document}$ (-\Delta )^s $\end{document} is the fractional Lapalcian, \begin{document}$ [u]_{s} $\end{document} is the Gagliardo seminorm of \begin{document}$ u $\end{document}, \begin{document}$ M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0 $\end{document} is a continuous function satisfying certain assumptions, \begin{document}$ V(x) = {|x|^{-2s}} $\end{document} is the Hardy potential function, \begin{document}$ 2_{s, t}^* = {(3+2t)}/{(3-2s)} $\end{document}, \begin{document}$ s, t\in (0, 1) $\end{document}, \begin{document}$ \lambda, \mu $\end{document} are two positive parameters, \begin{document}$ 1<p<2_s^* = {6}/{(3-2s)} $\end{document} and \begin{document}$ h\in L^{{2_s^*}/{(2_s^*-p)}}(\mathbb{R}^3) $\end{document}. By using topological methods and the Krasnoleskii's genus theory, we obtain the existence, multiplicity and asymptotic behaviour of solutions for above problem under suitable positive parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \mu $\end{document}. Moreover, we also consider the existence of nonnegative radial solutions and non-radial sign-changing solutions. The main novelties are that our results involve the possibly degenerate Kirchhoff function and the upper critical exponent in the sense of Hardy–Littlehood–Sobolev inequality. We emphasize that some of the results contained in the paper are also valid for nonlocal Schrödinger–Maxwell systems on Cartan–Hadamard manifolds.

In this paper, we study blow up and blow up time of solutions for initial boundary value problem of Kirchhoff-type wave equations involving the fractional Laplacian

\begin{document}$\left\{ \begin{align} & {{u}_{tt}}+[u]_{s}^{2(\theta -1)}{{(-\Delta )}^{s}}u=f(u),\ \ \ \ \text{in}\ \Omega \times {{\mathbb{R}}^{+}}, \\ & u(x,0)={{u}_{0}},\ \ {{u}_{t}}(x,0)={{u}_{1}},\ \ \ \ \ \ \text{in}\ \Omega , \\ & u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ ({{\mathbb{R}}^{N}}\backslash \Omega )\times \mathbb{R}_{0}^{+}, \\ \end{align} \right.$\end{document}

where \begin{document}$ [u]_s $\end{document} is the Gagliardo seminorm of \begin{document}$ u $\end{document}, \begin{document}$ s\in(0, 1) $\end{document}, \begin{document}$ \theta\in[1, 2_s^*/2) $\end{document} with \begin{document}$ 2_s^* = \frac{2N}{N-2s} $\end{document}, \begin{document}$ (-\Delta)^s $\end{document} is the fractional Laplacian operator, \begin{document}$ f(u) $\end{document} is a differential function satisfying certain assumptions, \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded domain with Lipschitz boundary \begin{document}$ \partial \Omega $\end{document}. By introducing a new auxiliary function and an adapted concavity method, we establish some sufficient conditions on initial data such that the solutions blow up in finite time for the arbitrary positive initial energy. Moreover, as \begin{document}$ f(u) = |u|^{p-1}u $\end{document}, we estimate the upper and lower bounds for blow up time with arbitrary positive energy.