On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.

     

A Faber–Krahn Inequality for Solutions of Schrödinger’s Equation on Riemannian Manifolds

We consider a bounded open set with smooth boundary \(\Omega \subset M\) in a Riemannian manifold (M, g), and suppose that there exists a non-trivial function \(u\in C({\overline{\Omega }})\) solving the problem

$$\begin{aligned} -\Delta u=V(x)u, \,\, \text{ in }\,\,\Omega , \end{aligned}$$

in the distributional sense, with \(V\in L^\infty (\Omega )\), where \(u\equiv 0\) on \(\partial \Omega .\) We prove a sharp inequality involving \(||V||_{L^{\infty }(\Omega )}\) and the first eigenvalue of the Laplacian on geodesic balls in simply connected spaces with constant curvature, which slightly generalises the well-known Faber–Krahn isoperimetric inequality. Moreover, in a Riemannian manifold which is not necessarily simply connected, we obtain a lower bound for \(||V||_{L^{\infty }(\Omega )}\) in terms of its isoperimetric or Cheeger constant. As an application, we show that if \(\Omega \) is a domain on a m-dimensional minimal submanifold of \({\mathbb {R}}^n\) which lies in a ball of radius R, then

$$\begin{aligned} ||V||_{L^{\infty }(\Omega )}\ge \left( \frac{m}{2R}\right) ^{2}. \end{aligned}$$

     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: existence

We apply the compactness results obtained in the first part of this work, to prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation

$$-\triangle u + V \left(\left|x\right|\right) u = g \left(\left|x\right|, u\right) \quad {\rm in} \Omega \subseteq \mathbb{R}^{N},\,N \geq 3,$$

where \({\Omega}\) is a radial domain (bounded or unbounded) and u satisfies u =  0 on \({\partial\Omega}\) if \({\Omega \neq\mathbb{R}^{N}}\) and \({u \rightarrow 0}\) as \({\left|x\right| \rightarrow \infty}\) if \({\Omega}\) is unbounded. The potential V may be vanishing or unbounded at zero or at infinity and the nonlinearity g may be superlinear or sublinear. If g is sublinear, the case with a forcing term \({g\left(\left|\cdot\right|, 0\right) \neq 0}\) is also considered. Our results allow to deal with V and g exhibiting behaviours at zero or at infinity which are new in the literature and, when \({\Omega = \mathbb{R}^{N}}\), do not need to be compatible with each other.

     

A note on regularity of solutions for degenerate singular elliptic boundary value problems

We prove the \(C^{1,\beta }\)-boundary regularity and a comparison principle for weak solutions of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^{N},N>1\ \)with smooth boundary \(\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ \)and f is allowed to be unbounded.

     

Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form \(-\Delta _p u = |\nabla u|^p + \sigma \) in a bounded domain \(\Omega \subset \mathbb {R}^n\). Here \(\Delta _p\), \(p>1\), is the standard p-Laplacian operator defined by \(\Delta _p u=\mathrm{div}\, (|\nabla u|^{p-2}\nabla u)\), and the datum \(\sigma \) is a signed distribution in \(\Omega \). The class of solutions that we are interested in consists of functions \(u\in W^{1,p}_0(\Omega )\) such that \(|\nabla u|\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\), a space pointwise Sobolev multipliers consisting of functions \(f\in L^{p}(\Omega )\) such that

$$\begin{aligned} \int _{\Omega } |f|^{p} |\varphi |^p dx \le C \int _{\Omega } (|\nabla \varphi |^p + |\varphi |^p) dx \quad \forall \varphi \in C^\infty (\Omega ), \end{aligned}$$

for some \(C>0\). This is a natural class of solutions at least when the distribution \(\sigma \) is nonnegative and compactly supported in \(\Omega \). We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write \(\sigma =\mathrm{div}\, F\) for a vector field F such that \(|F|^{\frac{1}{p-1}}\in M(W^{1,p}(\Omega )\rightarrow L^p(\Omega ))\). As an important application, via the exponential transformation \(u\mapsto v=e^{\frac{u}{p-1}}\), we obtain an existence result for the quasilinear equation of Schrödinger type \(-\Delta _p v = \sigma \, v^{p-1}\), \(v\ge 0\) in \(\Omega \), and \(v=1\) on \(\partial \Omega \), which is interesting in its own right.

     

$${\sigma_2}$$-Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

Let \({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional

$${{\mathbb F}_{\sigma_2}}[u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$

over the space of admissible maps

$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$

where the integrand \({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when \({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of \({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to \({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group \({\mathbf {SO}(n)}\). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.

     

A threshold phenomenon for embeddings of $${H^m_0}$$ into Orlicz spaces

Given an open bounded domain \({\Omega\subset\mathbb {R}^{2m}}\) with smooth boundary, we consider a sequence \({(u_k)_{k\in\mathbb{N}}}\) of positive smooth solutions to

$\left\{\begin{array}{ll} (-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2} \quad\quad\quad\quad\quad {\rm in}\,\Omega\\ u_k=\partial_\nu u_k=\cdots =\partial_\nu^{m-1} u_k=0 \quad {\rm on }\, \partial \Omega, \end{array}\right.$

where λ _k → 0⁺. Assuming that the sequence is bounded in \({H^m_0(\Omega)}\) , we study its blow-up behavior. We show that if the sequence is not precompact, then

$\liminf_{k\to\infty}\|u_k\|^2_{H^m_0}:=\liminf_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx\geq \Lambda_1,$

where Λ₁ = (2m ? 1)!vol(S ^2m ) is the total Q-curvature of S ^2m .

     

Existence results for a singular quasilinear elliptic equation

Let \(\Omega \subset \mathbb R^N\) be a bounded domain with smooth boundary. Existence of a positive solution to the quasilinear equation

$$\begin{aligned} -\text {div}\left[ \left( a(x)+|u|^\theta \right) \nabla u\right] +\frac{\theta }{2}|u|^{\theta -2}u|\nabla u|^2=|u|^{p-2}u \quad \text {in}\ \Omega \end{aligned}$$

with zero Dirichlet boundary condition is proved. Here \(\theta >0\) and a(x) is a measurable function satisfying \(0<\alpha \le a(x)\le \beta \). The equation involves singularity when \(0<\theta \le 1\). As a main novelty with respect to corresponding results in the literature, we only assume \(\theta +2<p<\frac{2^*}{2}(\theta +2)\). The proof relies on a perturbation method and a critical point theory for E-differentiable functionals.

     

Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

In this paper, we shall be concerned with the existence result of the following problem,

$$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\left( a(x,u,\nabla u)\right) -\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(0.1)

with the second term f belongs to \(L^1(\Omega )\). The growth and the coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M. We assume any restriction on M, therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term \(\Phi \) is a Carathéodory function satisfying only a growth condition.

     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

Existence of Multi-bump Solutions for a Class of Quasilinear Schrödinger Equations in $${\mathbb{R}^{N}}$$ Involving Critical Growth

In this paper we prove the existence of multi-bump solutions for a class of quasilinear Schrödinger equations of the form \({-\Delta{u} + (\lambda{V} (x) + Z(x))u - \Delta(u^{2})u = \beta{h}(u) + u^{22*-1}}\) in the whole space, where h is a continuous function, \({V, Z : \mathbb{R}^{N} \rightarrow \mathbb{R}}\) are continuous functions. We assume that V(x) is nonnegative and has a potential well \({\Omega : = {\rm int} V^{-1}(0)}\) consisting of k components \({\Omega_{1}, \ldots , \Omega{k}}\) such that the interior of Ω _i is not empty and \({\partial\Omega_{i}}\) is smooth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that for any given non-empty subset. \({\Gamma \subset \{1, \ldots ,k\}}\), a bump solution is trapped in a neighborhood of \({\cup_{{j}\in\Gamma}\Omega_{j}}\) for\({\lambda > 0}\) large enough.     

Local Maximum Modulus Property for Polyanalytic Functions

Let \(\Omega \subset {\mathbb {C}}\) be an open subset and let \({\mathcal {F}}\) be a space of functions defined on \(\Omega \). \({\mathcal {F}}\) is said to have the local maximum modulus property if: for every \(f\in {\mathcal {F}},p_0\in \Omega ,\) and for every sufficiently small domain \(D\subset \Omega ,\) with \(p_0\in D,\) it holds true that \(\max _{z\in \overline{D}}\left| f(z)\right| = \max _{z\in \Sigma \cup \partial D}\left| f(z)\right| ,\) where \(\Sigma \subset \Omega \) denotes the set of points at which \(\left| f\right| \) attains strict local maximum. This property fails for \({\mathcal {F}}=C^{\infty }.\) We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever \({\mathcal {F}}\) is the set of n-analytic functions on \(\Omega \), \(n\ge 2,\) in the sense that locality cannot be removed as a condition and independently \(\Sigma \) cannot be removed from the conclusion.     

Helicoidal surfaces satisfying {\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}

In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.     

Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

We study the existence and multiplicity of sign-changing solutions of the following equation

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$

where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).

     

The exact asymptotic behavior of boundary blow-up solutions to infinity Laplacian equations

In this paper, we study the asymptotic behavior of viscosity solutions to boundary blow-up elliptic problem \({\Delta_{\infty}u=b(x)f(u),\, x\in\Omega,\,u|_{\partial\Omega}=+\infty,}\) where \({\Omega}\) is a bounded domain with C²-boundary in \({\mathbb{R}^{N}}\), \({b\in \rm C(\bar{\Omega})}\) is positive in \({\Omega}\), which may be vanishing on the boundary, \({f\in C^{1}([0, \infty))}\) is regularly varying or is rapidly varying at infinity.     

Interpolation of sum and intersection spaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-type and applications to the Stokes problem in general unbounded domains

In a general unbounded uniform C ²-domain \({\Omega \subset \mathbb{R}^n, n \geq 3}\) , and \({1\leq q\leq \infty}\) consider the spaces \({\tilde{L}^q(\Omega)}\) defined by \({\tilde{L^q}(\Omega) := \left\{\begin{array}{ll}L^q(\Omega)+L^2(\Omega),\quad q < 2, \\ L^q(\Omega)\cap L^2(\Omega),\quad q\geq 2, \end{array}\right.}\) and corresponding subspaces of solenoidal vector fields, \({\tilde{L}^q_\sigma(\Omega)}\) . By studying the complex and real interpolation spaces of these we derive embedding properties for fractional order spaces related to the Stokes problem and L ^p ? L ^q -type estimates for the corresponding semigroup.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

Gelfand type problem for singular quadratic quasilinear equations

In this paper, we study the existence of positive solutions for the quasilinear elliptic singular problem

$$\left\{\begin{array}{ll}-\Delta u + c\,\frac{|\nabla u|^2}{u^\gamma} = \lambda\,f(u), \quad \quad \mbox{in $\Omega$},\\ u=0, \quad \qquad \qquad \qquad \quad \, \, \, \, \, \mbox{on $\partial$$\Omega$},\end{array}\right.$$

where \({c,\lambda >0}\), \({\gamma \in (0,1)}\), f is strictly increasing and derivable in \({[0,\infty)}\) with \({f(0)>0}\). We show that there exists \({\lambda^*>0}\) such that \({(0,\lambda^*]}\) is the maximal set of values such there exists solution. In addition, we prove that for \({\lambda<\lambda^*}\) there exists minimal and bounded solutions. Moreover, we give sufficient conditions for existence and regularity of solutions for \({\lambda=\lambda^*}\).

     

Global $$C^{1,\al pha }$$ regularity and e xistence of multi ple solutions for singular p( x)-La placian equations

In this paper we study the following singular p(x)-Laplacian problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \text{ div } \left( |\nabla u|^{p(x)-2} \nabla u\right) =\frac{ \lambda }{u^{\beta (x)}}+u^{q(x)}, &{} \text{ in }\quad \Omega , \\ u>0, &{} \text{ in }\quad \Omega , \\ u=0, &{} \text{ on }\quad \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 2\), with smooth boundary \(\partial \Omega \), \(\beta \in C^1(\bar{\Omega })\) with \( 0< \beta (x) <1\), \(p\in C^1(\bar{\Omega })\), \(q \in C(\bar{\Omega })\) with \(p(x)>1\), \(p(x)< q(x) +1 <p^*(x)\) for \(x \in \bar{\Omega }\), where \( p^*(x)= \frac{Np(x)}{N-p(x)} \) for \(p(x) <N\) and \( p^*(x)= \infty \) for \( p(x) \ge N\). We establish \(C^{1,\alpha }\) regularity of weak solutions of the problem and strong comparison principle. Based on these two results, we prove the existence of multiple (at least two) positive solutions for a certain range of \(\lambda \).