1. Introduction

Nonlinear boundary value problems (NBVPs) for elliptic partial differential equations (PDEs) are widely studied due to the great mathematical interest in themselves and their applications in various areas of science. For example, they arise in the modelling of nonlinear diffusion phenomena and in the theory of nuclear and chemical reactors (see e.g. [Reference Amann2, Reference Cohen and Laetsch15, Reference Keller30]). This class of problems has been addressed through different techniques and approaches, such as variational, penalty, and maximum principle-based methods that have prominent historical roles (see e.g. [Reference Hess27, Reference Rădulescu42, Reference Struwe45]).

In the present work, we are concerned with a class of NBVPs for elliptic equations with singular boundary potentials and nonlinear derivative terms in the half-space $\mathbb {R}_+^{n}$. More precisely, we consider the following nonhomogeneous elliptic problem:

(1.1)\begin{equation} \left\{\begin{array}{@{}ll} -\Delta u=K_{1}(\partial^{\beta}u)^{\rho_{1}}, & \text{in }\mathbb{R}_+^{n} \\ \partial_{\eta}u=V(x^{\prime})u+K_{2}u^{\rho_{2}}+f(x^{\prime}), & \text{in }\mathbb{R}^{n-1} \end{array}\right., \end{equation}

where $n\geq 3$, $\rho _{1},\,\rho _{2}\geq 2$, $u=u(x^{\prime },\,x_{n})$ with $x^{\prime }\in \mathbb {R}^{n-1}$ and $x_{n}>0$, $\partial _{\eta }={\partial }/{\partial \eta }$, $\eta$ is the normal unit outward vector on $\partial \mathbb {R}_+^{n}=\mathbb {R}^{n-1}$, $K_{1},\,K_{2}$ are constants, $0\leq \beta <{2}/{\rho _{1}},$ and the fractional derivative $\partial ^{\beta }$ is defined via the Fourier transform on the $(n-1)$-first variables as

(1.2)\begin{equation} (\partial^{\beta}u)^{{\wedge}}(\xi^{\prime},x_{n})=(2\pi|\xi^{\prime}|)^{\beta }(\widehat{u})(\xi^{\prime},x_{n}). \end{equation}

The case $\beta =0$ corresponds to the power-type nonlinearity $u^{\rho _{1}}$. Moreover, we can treat doubly supercritical variational cases such as $\rho _{1}>2^{\ast }-1$ and $\rho _{2}>2_{\ast }-1$. However, due to technical issues in our approach, the powers $\rho _{1},\,\rho _{2}$ have to be positive integers as well as they and the order $\beta$ of the derivative present a certain relation between them. The boundary potentials $V$ and forcing terms $f$ can be singular such as critical multipolar potentials and Radon measures, respectively.

Our intent is to analyse problem (1.1) via a different approach based on localization-in-frequency arguments and the Littlewood–Paley decomposition. To handle the influences of different frequency bands on each of the terms of (1.1), especially on those coming from singular potentials and forces, we consider a frequency-based setting, namely the Fourier–Besov space $\mathcal {FB}_{p,\infty }^{s}$ (FB-space, for short), whose elements $h$ are such that $\widehat {h}\in L_{loc}^{1}(\mathbb {R}^{n})$ and present the control in frequency:

\[{\parallel}\widehat{\Delta_{j}h}\parallel_{L^{p}(\mathbb{R}^{n})}\leq C2^{{-}sj},\quad \text{for all }j\in\mathbb{Z}, \]

where the Littlewood–Paley operator $\Delta _{j}$ works as a filter in the frequency domain with corresponding passband $\mathcal {A}_{j}=\{\xi \in \mathbb {R}^{n};2^{j-1}\leq |\xi |\leq 2^{j+1}\}$. The parameters $s\in \mathbb {R}$ and $p\in \lbrack 1,\,\infty ]$ stand for the regularity and integrability indexes of the space, respectively. For more details, see (2.8) and (2.9) in § 2.2. This kind of framework, as well as some of its extensions, has been successfully employed in the analysis of the well-posedness of parabolic problems, see e.g. [Reference de Almeida, Ferreira and Lima1, Reference Iwabuchi and Takada40] and references therein.

Varying the levels of regularity and integrability, we are able to cover singular classes of boundary potentials $V$ and forcing terms $f$ as well as obtain properties for solutions such as axial symmetry, positivity, and homogeneity. Of particular interest, we have the critical boundary potential $V(x^{\prime })=C\left \vert x^{\prime }\right \vert ^{-1}$ as well as its multipolar versions (even infinitely many poles):

(1.3)\begin{equation} V(x^{\prime})=\sum_{j=1}^{l}\frac{\lambda_{j}}{|x^{\prime}-x^{j}|}\quad \text{and}\quad V(x^{\prime})= \sum_{j=1}^{l}\frac{(x^{\prime}-x^{j})\cdot d^{j}}{|x^{\prime }-x^{j}|^{2}}, \end{equation}

where $x^{j}\in \partial \mathbb {R}_+^{n}$ are the poles, $d^{j}\in \partial \mathbb {R}_+^{n}$ are constant vectors, $\lambda _{j}$ are real constants, $j=1,\,\ldots,\,l$, and $l\in \mathbb {N\cup \{\infty \}}$. Indeed, for $0<\sigma < n-1$ and $1\leq p\leq \infty$, a simple computation yields that the potentials $\left \vert x^{\prime }\right \vert ^{-\sigma }$ and $x^{\prime }\left \vert x^{\prime }\right \vert ^{-(\sigma +1)}$ belong to $\mathcal {FB} _{p,\infty }^{n-1-({(n-1)}/{p})-\sigma }(\mathbb {R}^{n-1})$ as well as their translations and (for $\sigma =1$) those in (1.3). These critical potentials can be regarded as boundary versions of the so-called Hardy-type potentials in the whole space $\mathbb {R}^{n}$. The latter has been the object of study in a number of works mainly by combining variational methods, Hardy-type inequalities, and Sobolev spaces (see e.g. [Reference Felli, Marchini and Terracini17, Reference Felli and Terracini18] and references therein). For a study via a contraction argument and a sum of weighted $L^{\infty }$-spaces, see [Reference Ferreira and Mesquita21].

In what follows, we review some works on NBVPs. Chipot et al. [Reference Chipot, Shafrir and Fila13] described the non-trivial and non-negative solutions of the NBVP $-\Delta u=au^{\rho _{1}}$ in $\mathbb {R} _+^{n}$ and $\partial _{\eta }u=bu^{\rho _{2}}$ on $\mathbb {R}^{n-1}$ with $n\geq 3$, $a,\,b\geq 0$, $\rho _{1}={(n+2)}/{(n-2)}$ and $\rho _{2}={n}/{(n-2)}$. For $a\geq 0$ and $b=1$, the existence part was extended by [Reference Chipot, Chlebík, Fila and Shafrir14] to the case $\rho _{1}\geq {(n+2)}/{(n-2)}$ and $\rho _{2} \geq {n}/{(n-2)}$. Harada [Reference Harada39] analysed the same problem with $a=0$ (Laplace equation), $b=1$, and $\rho _{2}>{n}/{(n-2)}$ obtaining results on $x_{n}$-axial symmetry and asymptotic expansion for positive solutions. For $a=0$, $b=1$, and $1<\rho _{2}<{n}/{(n-2)}$, Hu [Reference Hu28] proved the non-existence of non-negative classical solutions. By means of a variational approach and the method of invariant sets, Liu and Liu [Reference Liu and Liu31] studied the existence of positive solutions and sign-changing solutions for the Laplace equation in $\mathbb {R}_+^{n}$ with the nonlinear boundary condition $\partial _{\eta }u=\lambda V(x^{\prime })u+g(u),$ where the potential $V\in L^{\infty }(\mathbb {R}^{n-1})$, $0\leq V\leq 1$, $\lim _{\left \vert x^{\prime }\right \vert \rightarrow \infty }V(x^{\prime })=1$, $\lambda$ is a negative parameter, and $g$ is superlinear at zero and asymptotically linear at infinity. Linked to the self-similarity problem for the semilinear heat equation in $\mathbb {R}_+^{n}$, the authors of [Reference Ferreira, Furtado and Medeiros20, Reference Furtado and da Silva22] analysed the elliptic PDE with drift $-\Delta u=({1}/{2})x\cdot \nabla u+cu+g_{1}(u)$ in $\mathbb {R}_+^{n}$ with $\partial _{\eta }u=g_{2}(u)$ on $\mathbb {R}^{n-1}$ by employing variational techniques along with weighted Sobolev spaces. See also [Reference Furtado and Ruviaro23, Reference Hsu and Lin47] for further related results on NBVPs in the half-space and/or bounded domains.

In another branch of research, we have the study of boundary value problems (BVPs) with singular data which have been a subject of great interest to elliptic PDEs community, see e.g. [Reference Amann and Quittner3, Reference Marcus and Véron33] and references therein. As a matter of fact, there exists a rich literature about the analysis of such problems with measure as forcing terms and boundary data. By employing comparison principles, monotonicity arguments, Kato inequality, weak compactness in weighted $L^{1}$-spaces, or suitable capacity-based characterizations, we would like to mention the works [Reference Brézis and Ponce4–Reference Brezis, Marcus and Ponce6, Reference Boukarabila and Véron9, Reference Chen and Véron11, Reference Chen and Véron12, Reference Gmira and Véron25, Reference Gkikas and Véron26, Reference Marcus and Veron34, Reference Marcus and Véron35, Reference Marcus and Nguyen37, Reference Véron and Yarur48], where the reader can find results on solvability and qualitative properties for BVPs of coercive type in smooth bounded domains $\Omega$ of $\mathbb {R}^{n}$ (see also the book [Reference Marcus and Véron33] for a nice review). Gmira and Véron [Reference Gmira and Véron25] considered the problem $-\Delta u+g(u)=0$ in $\Omega$ with $u=f$ on $\mathbb {\partial }\Omega$, where the boundary data $f$ is a measure and $g:\mathbb {R\rightarrow }\mathbb {R}$ is a continuous nondecreasing function such that $\int _{1}^{\infty }(\left \vert g(s)\right \vert +\left \vert g(-s)\right \vert )s^{-({2n}/{(n-1)})}{\rm d}s<\infty$. They proved existence of a unique solution $u\in L^{1}(\Omega )$ such that $\rho (x)g(u)\in L^{1}(\Omega )$ where $\rho (x)=d(x,\,\partial \Omega )$. For related results involving the nonlinearity $g(s)=s\left \vert s\right \vert ^{\rho -1}$ and positive measures $f$, see [Reference Marcus and Veron34, Reference Marcus and Véron35]. Brézis and Ponce [Reference Brézis and Ponce4] studied the same problem for a bounded measure $f$ and $g:\mathbb {R\rightarrow }\mathbb {R}$ being a continuous nondecreasing function satisfying $g(s)=0$ for $s\leq 0$. They developed a programme in the spirit of [Reference Brezis, Marcus and Ponce5, Reference Brezis, Marcus and Ponce6] by introducing a concept of reduced measure $f^{\ast }$ and showing that $f^{\ast }$ is the largest measure such that $f^{\ast }\leq f$ and the problem has $L^{1}(\Omega )$-solution with boundary data $f^{\ast }$ (good measure), among other properties. In the case of boundary nonlinearities, Boukarabila and Véron [Reference Boukarabila and Véron9] showed the solvability of the NBVP $-\Delta u=0$ in $\Omega$ with $\partial _{\eta }u+g(u)=f$ on $\mathbb {\partial }\Omega,$ for Radon measures $f$ and $g:\mathbb {R\rightarrow }\mathbb {R}$ a continuous nondecreasing function satisfying $g(0)=0$ and an integral subcritical condition. In the case of problems involving potentials $V$ and Radon measures $f$, we highlight [Reference Véron and Yarur48] where the authors studied nonnegative $L^{1}(\Omega )$-solutions and reduced measure for the BVP $-\Delta u+Vu=0$ in $\Omega$ with $\partial _{\eta }u=f$ on $\mathbb {\partial }\Omega$ by means of an approach with capacity depending on the locally bounded potential $V\geq 0$ (then, it can be singular near $\partial \Omega$), the Poisson kernel and the first positive eigenfunction of $-\Delta$ in $W_{0}^{1,2}(\Omega )$. For results on semilinear problems considering an interplay between measure data and Hardy-type potentials, see [Reference Chen and Véron11, Reference Chen and Véron12, Reference Gkikas and Véron26, Reference Marcus and Nguyen37].

In [Reference Amann and Quittner3], Amann and Quittner considered the doubly nonlinear problem $-\Delta u=g_{1} (x,\,u)+\varepsilon f_{1}$ in $\Omega$ with $\partial _{\eta }u=g_{2}(x,\,u)+\varepsilon f_{2}$ on $\mathbb {\partial }\Omega$ in the noncoercive case (i.e. $g_{i}(x,\,z)$ nondecreasing in $z$), where $f_{1} ,\,f_{2}$ are finite Radon measures and $\varepsilon >0$. Among others, they obtained existence and multiplicity results by assuming suitable smallness conditions on $\varepsilon$ and employing a mix of sub-super solution method, Sobolev–Slobodeckij spaces, and techniques of fixed points in ordered Banach spaces. In [Reference Bidaut-Véron, Hoang, Nguyen and Véron7], Bidaut-Véron et al. treated the problem $-\Delta u=g(u,\,\nabla u)$ in $\mathbb {R}_+^{n}$ with the Dirichlet condition $u=\varepsilon f$ on $\mathbb {R}^{n-1}$, where $n\geq 3$ and $f$ is a finite Radon measure. For $g(u,\,\nabla u)=u^{\rho }$ with $\rho >1,$ they proved existence of positive solution for small $\varepsilon >0$, as well as some sharp pointwise estimates of the solutions, by assuming suitable conditions involving the Riesz capacity on $\mathbb {R}^{n-1}$ and employing some ideas by Kalton and Verbitsky [Reference Kalton and Verbitsky29] who developed an extensive study about a class of integral equations with measure data. The authors of [Reference Bidaut-Véron, Hoang, Nguyen and Véron7] also analysed the case of smooth bounded domains $\Omega$ (see [Reference Bidaut-Véron and Vivier8] for related results) as well as the mixed gradient-power case $g(u,\,\nabla u)=u^{\rho _{1}}\left \vert \nabla u\right \vert ^{\rho _{2}},$ where $\rho _{1},\,\rho _{2}\geq 0$, $\rho _{1}+\rho _{2}>1$, and $\rho _{2}<2$, both considering small boundary data $\varepsilon f$.

In [Reference Castañeda-Centurión and Ferreira10], the authors considered a class of weighted $L^{\infty }$-spaces in Fourier variables, namely the pseudomeasure spaces $\mathcal {PM} ^{a}$ in the half-space $\mathbb {R}_+^{n}$, and obtained results on solvability and regularity for (1.1) with $\beta =0$ (nonlinearity independent of derivatives) and Robin boundary conditions by means of suitable weighted-type estimates and convolution properties of homogeneous functions. Their approach in $\mathcal {PM}^{a}$ is also based on Fourier analysis and employs an integral formulation similar to ours, nevertheless without using localization arguments and the Littlewood–Paley decomposition as in the case of $\mathcal {FB}_{p,\infty }^{s}$-spaces. Moreover, we have that $\mathcal {PM} ^{a}$ $\subset \mathcal {FB}_{p,\infty }^{s}$ for $s=a-{n}/{p}$ and $1\leq p<\infty,$ and then our results allow more singular potentials and forcing terms. For a Fourier analysis approach and an application of the $\mathcal {PM}^{a}$-framework in the study of elliptic problems in the whole space $\mathbb {R}^{n}$ with nonlinear derivative terms, see [Reference Ferreira and Castañeda-Centurión19]. In this context, difficulties related to the trace and boundary terms are not present in the integral formulation of the problem, and handling the Fourier transform is relatively simpler as the transform can be applied to the whole $\mathbb {R}^{n}$ and not just to some components of $x=(x_{1},\,x_{2},\,\ldots,\,x_{n})$.

In [Reference Quittner and Reichel41], Quittner and Reichel addressed the problem $-\Delta u=0$ in $\Omega$ with $\partial _{\eta }u+u=g(x,\,u)$ on $\mathbb {\partial }\Omega$, where $n\geq 3$ and $\Omega \subset \mathbb {R}^{n}$ is a bounded domain. Considering the growth condition $\left \vert g(x,\,s)\right \vert \lesssim (1+\left \vert s\right \vert ^{p})$ for some $p\in (1,\,{(n-1)}/{(n-2)}),$ and developing suitable a priori estimates, they proved that all positive very weak solution belongs to $L^{\infty }(\Omega )$ (see also [Reference Takahashi46] for related results). In addition, they provided examples showing that $\bar {p}={(n-1)}/{(n-2)}$ is a sharp critical exponent. In fact, for $n=3,\,4$, some exponents $p> \bar {p}$ and $g(x,\,u)=u^{p}+f$ with some $f\in L^{\infty }(\partial \Omega )$, they constructed two unbounded very weak solutions blowing-up at a prescribed point on $\partial \Omega$, where $\Omega$ is taken within a half-space and with a flat boundary piece. In turn, Merker and Rakotoson [Reference Merker and Rakotoson36] analysed very weak solutions of the Poisson equation $-\Delta u=h$ in a bounded domain $\Omega$ for singular forcing terms $h$ and singular Neumann boundary conditions, by means of a framework based on Lorentz-spaces $L^{p,q}(\Omega ),$ with $p\in (1,\,\infty )$ and $q\in \lbrack 1,\,\infty )$, and an approach relying on a suitable duality formulation for the BVP. They proved an existence and uniqueness result covering the following classes of forces: (i) $h\in L^{1}(\Omega )$ with $\partial _{n}u=(-\int _{\Omega }h{\rm d}x)\delta _{x_{0}}$ and (ii) non-integrable $h$ with $h\cdot \left \vert x-x_{0}\right \vert \in L^{1}(\Omega )$ and $x_{0}\in \partial \Omega$. Moreover, a generalization for finite Radon measures $\mu$ in place of $\delta _{x_{0}}$ was also discussed by them.

We analyse the unique solvability of (1.1) in a setting based on Fourier–Besov spaces $\mathcal {FB}_{p,\infty }^{s}$ in which localization-in-frequency arguments play a key role (see theorem 3.2). Moreover, the regularity of solutions is investigated with the help of Fourier–Sobolev spaces which naturally provide further decay in Fourier variables for them (see theorem 3.5). Due to the scaling analysis, the power $\rho _{2}$ is connected to $\rho _{1}$ and $\beta$ via the relation (see (3.13))

(1.4)\begin{equation} (\rho_{2}-\rho_{1})\gamma=\rho_{1}\beta-1\text{ with }\gamma=\frac{2-\rho _{1}\beta}{\rho_{1}-1}. \end{equation}

So, we can think that $\rho _{1}$ and $\beta$ are free and determine $\rho _{2}$. Or, alternatively, that $\rho _{1}$ and $\rho _{2}$ are free and determine $\beta$. However, in the case $K_{2}=0$, BVP (1.1) does not depend on $\rho _{2}$ and then we no longer have condition (1.4), and thus $\rho _{1}$ and $\beta$ are free from each other (except for natural conditions involving the parameter ranges). Assuming that $V$ and $f$ are radially symmetric in $\mathbb {R}^{n-1},$ we show that the obtained solutions are $x_{n}$-axial symmetric (see theorem 3.7). Our solvability result can also be adapted to the case of (1.1) with Robin boundary conditions in place of the Neumann one (see remark 3.1). With a slight modification in statements and proofs, our results work well for (1.1) with an additional forcing term $h$ acting within the domain $\mathbb {R} _+^{n},$ namely considering $-\Delta u=K_{1}(\partial ^{\beta }u)^{\rho _{1} }+h$ as the first equation in (1.1) (see remark 3.3(iii)).

In comparison with previous works, we are treating an NBVP in the half-space $\mathbb {R}_+^{n}$ with singular boundary potentials as (1.3) (see remark 3.4(i)) and a nonlinearity involving a fractional derivative. The solvability theory is developed via a contraction argument in a new setting for the context of elliptic PDEs providing new classes of solutions, potentials, and forcing terms. Moreover, it covers cases of variational supercritical powers on the boundary and (when $\beta =0$) within the domain. Note that the results are new even for other relevant subcases of the model problem (1.1) such as the simpler one $\beta =0$, $K_{1}=1$, $V=0$, and $K_{2}=0$, that is, $-\Delta u=u^{\rho _{1}}$ in $\mathbb {R}_+^{n}$ with $\partial _{\eta }u=f$ on $\mathbb {R}^{n-1}$. Another feature is that $\mathcal {FB}_{p,\infty }^{s}$-spaces lack of good compactness properties and they are non-reflexive (consequently, neither uniformly convex nor $q$-convex spaces), making it very difficult to employ capacity approaches, variational techniques, Leray–Schauder theory, among others, and thus motivating an analysis based on a non-topological fixed point argument.

In view of the strict continuous inclusions (see property (2.11) in § 2.2)

(1.5)\begin{equation} \mathcal{FB}_{\infty,\infty}^{s+({n}/{p_{1}})} \subset\mathcal{FB} _{p_{2},\infty}^{s+(({n}/{p_{1}})-({n}/{p_{2}}))}\subset\mathcal{FB} _{p_{1},\infty}^{s}\subset\mathcal{FB}_{1,\infty}^{s-(n-({n}/{p_{1}}))}, \end{equation}

where $s\in \mathbb {R}$ and $1\leq p_{1}\leq p_{2}\leq \infty,$ we can feel the breadth of the family of spaces $\mathcal {FB}_{p,\infty }^{s}$, especially for negative regularity indexes $s$. By Hausdorff–Young inequality and (1.5), we can see that $\dot {H}^{s}\subset$ $\mathcal {FB} _{2,\infty }^{s}\subset \mathcal {FB}_{p,\infty }^{s}$ for $1\leq p\leq 2$ and $s\in \mathbb {R},$ where $\dot {H}^{s}$ stands for the homogeneous Sobolev spaces. Also, denoting the space of finite Radon measures in $\mathbb {R}^{n}$ by $\mathcal {M}=\mathcal {M}(\mathbb {R}^{n})$ and taking $s=-({n}/{p_{1}})$ in (1.5), we arrive at

(1.6)\begin{equation} \mathcal{M}\subset\mathcal{FB}_{\infty,\infty}^{0}\subset\mathcal{FB} _{p_{1},\infty}^{-{n}/{p_{1}}}\subset\mathcal{FB}_{1,\infty}^{{-}n}, \end{equation}

which allows us to cover measure data by considering the space $\mathcal {FB} _{p,\infty }^{s}$ with $s=-({n}/{p})$ and $1\leq p\leq \infty$ (see remark 3.4(ii)). Moreover, if $f\in \mathcal {M}$ with $\textrm {supp}(f)$ contained in a set of Hausdorff dimension $s\in \lbrack 0,\,n)$, it follows that $|\hat {f}(\xi )|\lesssim \left \vert \xi \right \vert ^{-s/2}$ and $f\in \mathcal {FB}_{\infty,\infty }^{s/2}$ (see [Reference Mattila38, p. 40]). By considering suitable indexes, we point out that smallness conditions involving $\mathcal {FB}_{p,\infty }^{s}$-norms allow us to consider some functions with large $L^{p}$ and $H^{s}$-norms, as well as large Radon measures.

This paper is organized as follows. Section 2 is devoted to some preliminaries by recalling basic notations of Fourier analysis as well as reviewing basic definitions and properties on Littlewood–Paley decomposition, Fourier–Sobolev spaces, and Fourier–Besov spaces. In § 3, we state our results on solvability, regularity, and symmetry for (1.1). The purpose of § 4 is to develop key estimates for the terms of the integral formulation associated with (1.1). In § 5, with the estimates in hand, we show the proofs of our results.