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For the solution of the Poisson problem with an L∞ right hand side
we derive an optimal estimate of the form
\begin{cases} -\Delta u(x) = f (x) & {\rm in}\ D, \\ u=0 & {\rm on}\ \partial D \end{cases}
where σD is a modulus of continuity defined in the interval [0, |D|] and depends only on the domain D. The inequality is optimal for any domain D and for any values of 
\|u\|_\infty\leq \|f\|_\infty \sigma_D(\|f\|_1/\|f\|_\infty),
$\|f\|_1$ and 
$\|f\|_\infty .$ We also show that
where B is a ball and |B| = |D|. Using this optimality property of σD, we derive Brezis–Galloute–Wainger type inequalities on the L∞ norm of u in terms of the L1 and L∞ norms of f. As an application we derive L∞ − L1 estimates on the k-th Laplace eigenfunction of the domain D.
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
Let
$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form
where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, 
$$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$
$\mu \in L^{\infty}$ and 
$\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by
Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.
$$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$
In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation: where 
$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$
$\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in 
$\mathbb{R}^{N}$, 
$N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.
This paper is concerned with the following nonlinear Schrödinger system in 
${\mathbb R}^3$
where 
$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\in {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x\in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$
$\beta \in {\mathbb R}$ is a coupling constant, 
$\mu ,\nu $ are positive constants, P,Q are weight functions decaying exponentially to zero at infinity, α can be regarded as a parameter. This type of system arises, in particular, in models in Bose–Einstein condensates theory and Kerr-like photo refractive media.
We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as 
$\alpha \to +\infty $ (or 
$0^+$). Moreover, we also construct prescribed number of sign-changing solutions.
In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝN under a nonlinear Steklov type boundary condition, namelyFor positive weight functions a and b satisfying appropriate integrability and boundedness assumptions, we show that, for all p>1, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigenvalues located away from zero.
\[\left\{ \begin{aligned} -\Delta_pu-\Delta u & =\lambda a(x)u \quad {\rm in}\ \Omega,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partial\nu} & =\lambda b(x)u \quad {\rm on}\ \partial\Omega . \end{aligned} \right.\]
In this paper, we study both the existence and uniqueness of nonnegative solutions for the nonlocal 
$p$-Laplace equation with singular term where 
$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}-B\Bigl(\frac{1}{p}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{p}\text{d}x\Bigr)\unicode[STIX]{x1D6E5}_{p}u=\frac{h(x)}{u^{\unicode[STIX]{x1D6FE}}}+k(x)u^{q},\quad & x\in \unicode[STIX]{x1D6FA},\\ u>0,\quad & x\in \unicode[STIX]{x1D6FA},\\ u=0,\quad & x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$
$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}(N\geqslant 1)$ is a bounded domain with smooth boundary 
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$, 
$h\in L^{1}(\unicode[STIX]{x1D6FA})$, 
$h>0$ almost everywhere in 
$\unicode[STIX]{x1D6FA}$, 
$k\in L^{\infty }(\unicode[STIX]{x1D6FA})$ is a non-negative function, 
$B:[0,+\infty )\rightarrow [m,+\infty )$ is continuous for some positive constant 
$m$, 
$p>1$, 
$0\leqslant q\leqslant p-1$, and 
$\unicode[STIX]{x1D6FE}>1$. A “compatibility condition” on the couple 
$(h(x),\unicode[STIX]{x1D6FE})$ will be given for the problem to admit at least one solution. To be a little more precise, it is shown that the problem admits at least one solution if and only if there exists a 
$u_{0}\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ such that 
$\int _{\unicode[STIX]{x1D6FA}}h(x)u_{0}^{1-\unicode[STIX]{x1D6FE}}\text{d}x<\infty$. When 
$k(x)\equiv 0$, the weak solution is unique.
In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation
where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.
\[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]
$k$-HESSIAN IN FRACTIONAL SOBOLEV SPACES
We introduce the notion of a distributional 
$k$-Hessian (
$k=2,\ldots ,n$) associated with fractional Sobolev functions on 
$\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in 
$\mathbb{R}^{n}$. We show that the distributional 
$k$-Hessian is weakly continuous on the fractional Sobolev space 
$W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional 
$k$-Hessian is well defined in 
$W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if 
$W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.
In this paper, we study the existence of weak solutions of the quasilinear equation
where a : ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f : Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments.
\begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}
This paper is concerned with the fractional system
where n ⩾ 2, 0 < α, β < 2, a > −α, b > −β and p, q ⩾ 1. By exploiting a direct method of scaling spheres for fractional systems, we prove that if 
\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}
$p \leqslant \frac {n+\alpha +2a}{n-\beta }$, 
$q \leqslant \frac {n+\beta +2b}{n-\alpha }$, 
$p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$ and (u, v) is a nonnegative strong solution of the system, then u ≡ v ≡ 0.
$(p,q)$-LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS
We study the existence of positive ground state solutions for the following class of 
$(p,q)$-Laplacian coupled systems where 
$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$
$1<p\leq q<N$. Here the coefficient 
$\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition 
$|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where 
$\unicode[STIX]{x1D6FF}\in (0,1)$ and 
$\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.
We prove the global logarithmic stability of the Cauchy problem for 
$H^{2}$-solutions of an anisotropic elliptic equation in a Lipschitz domain. The result is based on existing techniques used to establish stability estimates for the Cauchy problem combined with related tools used to study an inverse medium problem.
In this paper we show the existence of solution for the following class of semipositone problem
P
where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.
$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$
In this paper we characterize the boundedness on the product of Sobolev spaces Hs(𝕋) × Hs(𝕋) on the unit circle 𝕋, of the bilinear form Λb with symbol b ∈ Hs(𝕋) given by
$${{\Lambda}_b} (\varphi, \psi): = \int_{\open T} {\left( {{{( - {\Delta })}^s} + I} \right)(\varphi \psi )(\eta )b(\eta ) {\rm d}\sigma (\eta ).}$$
