American Institute of Mathematical Sciences

The aim of this paper is to prove a reverse Hölder inequality for nonnegative adjoint solutions for elliptic operator in non divergence form in \begin{document}$\mathbb R^3$\end{document}. As an application we generalize a Theorem due to Sirazhudinov and Zhikov [24] and, under suitable assumptions, we characterize the \begin{document}$G$\end{document}-limit of a sequence of elliptic operator.The operator \begin{document}$N$\end{document}

\begin{document}$N[v] = \sum\limits_{i,j = 1}^3 {\frac{{{\partial ^2}({a_{ij}}v)}}{{\partial {x_i}\partial {x_j}}}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$ \end{document}

arises naturally as the formal adjoint of the operator in "non divergence form"

\begin{document}$L[u] = \sum\limits_{i,j = 1}^3 {{a_{i,j}}} (x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} = Tr(A{D^2}u).\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 2 \right)$ \end{document}

The reason to study the solutions of the adjoint operator is that they are not only important for the solvability of \begin{document}$Lu = f$\end{document} but for the properties of the Green's function for \begin{document}$L$\end{document}. There is a long literature in this context, see for example Sÿogren [22], Bauman [2], Fabes and Stroock [12], Fabes, Garofalo, Marĺn-Malavé, and Salsa [11], Escauriaza and Kenig [10], and Escauriaza [9].

We are concerned with the existence of nontrivial weak solutions for a class of generalized minimal surface equations with subcritical growth and Dirichlet boundary condition. In relationship with the values of several variable exponents, we establish two sufficient conditions for the existence of solutions. In the first part of this paper, we prove the existence of a non-negative solution. Next, we are concerned with the existence of infinitely many solutions in a symmetric abstract setting.

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian \begin{document}$A_{1/2}$\end{document} in a smooth bounded domain \begin{document}$Ω\subset \mathbb{R}^{n}$\end{document} (\begin{document}$n≥2$\end{document}) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation

\begin{document}$\left\{ \begin{align} &{{A}_{1/2}}u = \lambda f(u) \\ &u = 0 \\ \end{align} \right.\begin{array}{*{35}{l}} {}&\text{in}\ \Omega \\ {}&\text{on }\partial \Omega . \\\end{array}$ \end{document}

The existence of at least two non-trivial \begin{document}$L^{∞}$\end{document}-bounded weak solutions is established for large value of the parameter \begin{document}$λ$\end{document}, requiring that the nonlinear term \begin{document}$f$\end{document} is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.

We consider the following singular semilinear problem

\begin{document}$\left\{ \begin{array}{l} - \Delta u(x) = a(x){u^\sigma }(x),{\rm{ }}x \in \Omega \backslash \{ 0\} ({\rm{in\;the\;distributional\;sense}}),\\\;u > 0,{\rm{ on}}\;\Omega \backslash \{ 0\} ,\\\mathop {\lim }\limits_{\left| x \right| \to 0} \frac{{u(x)}}{{\ln \left| x \right|}} = 0,\\u(x) = 0,\;x \in \partial \Omega ,\end{array} \right.$ \end{document}

where \begin{document}$σ <1,$\end{document} \begin{document}$Ω $\end{document} is a bounded regular domain in \begin{document}$\mathbb{R}^{2}$\end{document} with \begin{document}$0∈ Ω .$\end{document} The weight function \begin{document}$a(x)$\end{document} is requiredto be positive and continuous in \begin{document}$Ω \backslash \{0\}$\end{document} with thepossibility to be singular at \begin{document}$x = 0$\end{document} and/or at the boundary \begin{document}$\partial Ω. $\end{document} When the function \begin{document}$a$\end{document} satisfies sharp estimates related to Karamataclass, we prove the existence and global asymptotic behavior of a positivecontinuous solution on \begin{document}$\overline{Ω }\backslash \{0\}$\end{document} which couldblow-up at \begin{document}$0$\end{document}.

The goal of the paper is to investigate the existence of solutions for semilinear upper diagonal infinite systems of differential equations. We will look for solutions of the mentioned infinite systems in a Banach tempered sequence space. In our considerations we utilize the technique associated with the Hausdorff measure of noncompactness and some existence results from the theory of ordinary differential equations in abstract Banach spaces.

We consider solutions \begin{document}$u$\end{document} to the Navier-Stokes equations in the whole space. We set \begin{document}$\omega = \nabla × u, $\end{document} the vorticity of \begin{document}$u$\end{document}. Our study concerns relations between \begin{document}$\beta -$\end{document}Hölder continuity assumptions on the direction of the vorticity and induced integrability regularity results, a significant research field starting from a pioneering 1993 paper by P. Constantin and Ch. Fefferman. Nowadays it is know that if \begin{document}$\beta = \frac{1}{2}$\end{document} then \begin{document}$\omega ∈ L^{∞}(L^2), $\end{document} a 2002 result by L.C. Berselli and the author. This conclusion implies smoothness of solutions. Assume now that one is able to prove that a strictly decreasing perturbation of \begin{document}$\beta $\end{document} near \begin{document}$\frac{1}{2}$\end{document} induces a strictly decreasing perturbation for \begin{document}$r$\end{document} near \begin{document}$2$\end{document}. Since regularity holds if merely \begin{document}$\omega ∈ L^{∞}(L^r), $\end{document} for some \begin{document}$r≥ \frac32, $\end{document} the above assumption would imply regularity for values \begin{document}$\beta <\frac{1}{2}.$\end{document} The aim of the present note is to go deeper into this study and related open problems. The approach developed below reinforces the conjecture on the particular significance of the value \begin{document}$\beta = \frac{1}{2}.$\end{document}

We consider the boundary value problem associated to the curl operator, with vanishing Dirichlet boundary conditions. We prove, under mild regularity of the data of the problem, existence of classical solutions.

The Leray-Lions operators are versatile enough to be particularized to various elliptic operators, so they receive a lot of attention. This paper introduces to the mathematical literature Leray-Lions type operators that are appropriate for the study of the variable exponent problems of higher order. We establish some properties concerning these general operators and then we apply them to a fourth order problem with variable exponents.

In this note we exploit nonlinear capacity estimates in the spirit of Mitidieri-Pohozaev [15] in the context of Lorentz spaces. This from one side yields a simple proof, though non-optimal, of non-attainability of Hardy's inequality in \begin{document}$\mathbb{R}^N$\end{document}, on the other side gives a partial positive answer to a conjecture raised in [15].

We are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let \begin{document}$Ω $\end{document} be an open subset of \begin{document}$\mathbb{R}^{n}$\end{document}. Let \begin{document}$f≤\left( {x, \xi } \right) $\end{document} be a real function defined in \begin{document}$Ω × \mathbb{R}^{n}$\end{document} satisfying the growth condition \begin{document}$|{f_{\xi x}}\left( {x, \xi } \right)| \le h\left( x \right)|\xi {{\rm{|}}^{p - 1}}$\end{document}, for \begin{document}$x∈ Ω $\end{document} and \begin{document}$\xi ∈ \mathbb{R}^{n}$\end{document} with \begin{document}$|\xi {\rm{|}} \ge {M_0}$\end{document} for some \begin{document}$M_{0}≥ 0$\end{document}, with \begin{document}$h \in L_{{\rm{loc}}}^r\left( \Omega \right) $\end{document} for some \begin{document}$r>n$\end{document}. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to \begin{document}$f\left( {x, \xi } \right) $\end{document} the so-called natural \begin{document}$p-$\end{document}growth conditions on the second derivatives \begin{document}${f_{\xi \xi }}\left( {x, \xi } \right)$\end{document}; i.e., \begin{document}$\left( {p - 2} \right) - $\end{document}growth for \begin{document}$|{f_{\xi \xi }}\left( {x, \xi } \right)| $\end{document} from above and \begin{document}$\left( {p - 2} \right) - $\end{document}growth from below for the quadratic form \begin{document}$({f_{\xi \xi }}\left( {x, \xi } \right)\lambda , \lambda {\rm{ }})$\end{document}; for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer \begin{document}$u \in W_{{\rm{loc}}}^{1, p}\left( \Omega \right) $\end{document} of the energy integral \begin{document}$\int_\Omega {f(x, Du\left( x \right)){\mkern 1mu} dx} $\end{document}.

In this paper we prove an existence theorem for positive solutions of a nonlinear Dirichlet problem involving the p-Laplacian operator on a smooth bounded domain when a nonlinearity depending on the gradient is considered. Our main theorem extends a previous result by Ruiz in [19], in which a slight modification of the celebrated blowup technique due to Gidas and Spruck, [11] and [12] is introduced.

We consider a nonlinear elliptic equation with Robin boundary condition driven by the p-Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray-Schauder alternative principle, we show the existence of a smooth solution.

We study the system

\begin{document}$\left\{\begin{split}-\Delta u+u&=(I_\alpha*|u|^p)|u|^{p-2}u+K(x) \phi |u|^{q-2}u & \qquad \mbox{ in }\mathbb{R}^N,\\-\Delta \phi&=K(x)|u|^q& \qquad \mbox{ in }\mathbb{R}^N,\end{split}\right.$ \end{document}

where \begin{document}$N≥ 3$\end{document}, \begin{document}$α∈ (0,N)$\end{document}, \begin{document}$p,q>1$\end{document} and \begin{document}$K≥ 0$\end{document}. Using a Pohozaev type identity we first derive conditions in terms of \begin{document}$p,q,N,α$\end{document} and \begin{document}$K$\end{document} for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity

\begin{document}$\begin{equation*} \quad (P_{t}^s) \left\{\begin{split} \quad u_t + (-\Delta)^s u & = u^{-q} + f(x,u), \;u >0\; \text{in}\;(0,T) \times \Omega, \\ u & = 0 \; \mbox{in}\; (0,T) \times (\mathbb{R}^n \setminus \Omega ),\\ \quad \quad \quad \quad u(0,x)& = u_0(x) \; \mbox{in} \; {\mathbb{R}^n},\end{split}\quad \right.\end{equation*}$ \end{document}

where \begin{document}$\Omega $\end{document} is a bounded domain in \begin{document}$\mathbb{R}^n$\end{document} with smooth boundary \begin{document}$\partial \Omega $\end{document}, \begin{document}$n> 2s, \;s ∈ (0,1)$\end{document}, \begin{document}$q>0$\end{document}, \begin{document}${q(2s-1)<(2s+1)}$\end{document}, \begin{document}$u_0 ∈ L^∞(\Omega )\cap X_0(\Omega )$\end{document} and \begin{document}$T>0$\end{document}. We suppose that the map \begin{document}$(x,y)∈ \Omega × \mathbb{R}^+ \mapsto f(x,y)$\end{document} is a bounded from below Carathéodary function, locally Lipschitz with respect to the second variable and uniformly for \begin{document}$x ∈ \Omega $\end{document} and it satisfies

0.1 \begin{document}$ \begin{equation}\label{cond_on_f}{ \limsup\limits_{y \to +\infty} \frac{f(x,y)}{y}<\lambda_1^s(\Omega)}, \end{equation}$ \end{document}

where \begin{document}$\lambda_1^s(\Omega )$\end{document} is the first eigenvalue of \begin{document}$(-\Delta )^s$\end{document} in \begin{document}$\Omega $\end{document} with homogeneous Dirichlet boundary condition in \begin{document}$\mathbb{R}^n \setminus \Omega $\end{document}. We prove the existence and uniqueness of a weak solution to \begin{document}$(P_t^s)$\end{document} on assuming \begin{document}$u_0$\end{document} satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results.We also show additional regularity on the solution of \begin{document}$(P_t^s)$\end{document} when we regularize our initial function \begin{document}$u_0$\end{document}.

Let \begin{document}$Ω\subset\mathbb{R}^n$\end{document} be a bounded smooth open set. We prove that the extremal solution of the system

\begin{document}$\begin{equation*}-Δ u = μ e^{θ u +(1-θ)v} , ~~~- Δ v = λ e^{θ v + (1-θ)u}~~~\mbox{ in }Ω,\end{equation*}$ \end{document}

with \begin{document}$u = v = 0$\end{document} on \begin{document}$\partial Ω$\end{document}, \begin{document}$θ$\end{document} in \begin{document}$[0,1]$\end{document} and \begin{document}$μ,λ≥0$\end{document} are smooth if \begin{document}$n≤ 9$\end{document}.

We consider a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document}-Laplacian and a Laplacian and a reaction term which is \begin{document}$(p-1)$\end{document}-linear near \begin{document}$\pm \infty$\end{document} and resonant with respect to any nonprincipal variational eigenvalue of \begin{document}$(-\Delta_p,W^{1,p}_0(\Omega))$\end{document}. Using variational tools together with truncation and comparison techniques and Morse Theory (critical groups), we establish the existence of six nontrivial smooth solutions. For five of them we provide sign information and order them.

The paper is focused on existence of nontrivial solutions of a Schrödinger-Hardy system in the Heisenberg group, involving critical nonlinearities. Existence is obtained by an application of the mountain pass theorem and the Ekeland variational principle, but there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the Hardy terms as well as critical nonlinearities.

We are concerned with the existence of infinitely many radial symmetric solutions for a nonlinear stationary problem driven by a new class of nonhomogeneous differential operators. The proof relies on the symmetric version of the mountain pass theorem.

The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional \begin{document}$p$\end{document}-Kirchhoff type problem involving the critical Sobolev exponent and discontinuous nonlinearity:

\begin{document}$\begin{align*}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)(-\Delta)_p^su = \lambda|u|^{p_s^*-2}u+f(x,u)~~\mbox{in }\,\,\mathbb{R}^N,\end{align*}$ \end{document}

where \begin{document}$M(t) = a+bt^{\theta-1}$\end{document} for \begin{document}$t\geq 0$\end{document}, \begin{document}$a\geq 0, b>0,\theta>1$\end{document}, \begin{document}$(-\Delta)_p^s$\end{document} is the fractional \begin{document}$p$\end{document}--Laplacian with \begin{document}$0<s<1$\end{document} and \begin{document}$1<p<N/s$\end{document}, \begin{document}$p_s^* = Np/(N-ps)$\end{document} is the critical Sobolev exponent, \begin{document}$\lambda>0$\end{document} is a parameter, and \begin{document}$f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$\end{document} is a function. Under suitable assumptions on \begin{document}$f$\end{document}, we show that there exists \begin{document}$\lambda_0>0$\end{document} such that the above equation admits at least one nontrivial nonnegative solution provided \begin{document}$\lambda<\lambda_0$\end{document} by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any \begin{document}$k\in\mathbb{N}$\end{document}, there exists \begin{document}$\Lambda_k>0$\end{document} such that the above equation has \begin{document}$k$\end{document} pairs of nontrivial solutions if \begin{document}$\lambda<\Lambda_k$\end{document}. The main feature is that our paper covers the degenerate case, that is the coefficient of \begin{document}$(-\Delta)_p^s$\end{document} may be zero at zero. Moreover, the existence results are obtained when \begin{document}$f$\end{document} is discontinuous. Thus, our results are new even in the semilinear case.