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.

     

Improved Moser–Trudinger inequality of Tintarev type in dimension <Emphasis Type="Italic">n</Emphasis> and the existence of its extremal functions

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$

for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)

     

Groups with a ternary equivalence relation

We consider in a group \((G,\cdot )\) the ternary relation

$$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ | \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$

and show that \(\kappa \) is a ternary equivalence relation if and only if the set \( \mathfrak Z \) of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure

$$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ | \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$

We study the automorphism group of \((G,\kappa )\), i.e. all permutations \(\varphi \) of the set G such that \( (\alpha , \beta , \gamma ) \in \kappa \) implies \((\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa \). We show \(\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)\), \(\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )\) and if \( \varphi \in \mathrm{Aut}(G,\kappa )\) with \(\varphi (1)=1\) and \(\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}\) for all \(\xi \in G\) then \(\varphi \) is an automorphism of \((G,\cdot )\). This allows us to prove a representation theorem of \(\mathrm{Aut}(G,\kappa )\) (cf. Theorem 6) and that for \(\alpha \in G \) the maps

$$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$

of the corresponding reflection structure \((G, \widetilde{G})\) (with \( \tilde{G} := \{\tilde{\gamma }\ | \ \gamma \in G \}\)) are point reflections. If \((G ,\cdot )\) is uniquely 2-divisible and if for \(\alpha \in G\), \(\alpha ^{1\over 2}\) denotes the unique solution of \(\xi ^2=\alpha \) then with \(\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}\), the pair \((G,\odot )\) is a K-loop (cf. Theorem 5).

     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).

     

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.

     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as

$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.

     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given \(1\le q \le 2\) and \(\alpha \in \mathbb {R}\), we study the properties of the solutions of the minimum problem

$$\begin{aligned} \lambda (\alpha ,q)=\min \left\{ \dfrac{\displaystyle \int _{-1}^{1}|u'|^{2}dx+\alpha \left| \int _{-1}^{1}|u|^{q-1}u\, dx\right| ^{\frac{2}{q}}}{\displaystyle \int _{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not \equiv 0\right\} . \end{aligned}$$

In particular, depending on \(\alpha \) and q, we show that the minimizers have constant sign up to a critical value of \(\alpha =\alpha _{q}\), and when \(\alpha >\alpha _{q}\) the minimizers are odd.

     

Existence of Solution for a Singular Elliptic Equation of Kirchhoff Type

We study both the existence and uniqueness of nonnegative solution to a singular elliptic problem of Kirchhoff type, whose model is:

$$\begin{aligned} {\left\{ \begin{array}{ll} -B\left( \dfrac{1}{2}\displaystyle \int _\Omega |\nabla u|^2\mathrm {d}x\right) \Delta u=\dfrac{h(x)}{u^\gamma }, &{}\quad x\in \Omega ,\\ u>0, &{}\quad x\in \Omega ,\\ u=0, &{}\quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset \mathbb {R}^n(n\ge 1)\) is a smooth bounded domain, \(\gamma >1\), \(h\in L^1(\Omega )\) is positive (i.e., \(h(x)>0\) a.e. in \(\Omega \)), \(B : \mathbb {R}^+\rightarrow \mathbb {R}^+\) is a \(C^1\)-continuous function with positive lower bound. A necessary and sufficient condition will be given for the existence of weak solution of the general nonlocal singular elliptic with strong singularity. In addition, we prove that the solution is unique under some suitable conditions.

     

A characterization of nonlinear Hölder seminorm preserving bijections

For a compact metric space (X, d) and \(\alpha \in (0,1)\), let \(\mathrm{Lip}^\alpha (X)\) be the linear space of all complex-valued functions f on X satisfying

Open image in new window

and \(\mathrm{lip}^\alpha (X)\) be the subspace of \(\mathrm{Lip}^\alpha (X)\) consisting of functions f with \(\lim \frac{f(x)-f(y)}{d^\alpha (x,y)} =0\) as \(d(x,y) \rightarrow 0\). In this paper, we give a characterization of a bijective map \(T:\mathrm{lip}^\alpha (X)\longrightarrow \mathrm{lip}^\alpha (Y)\), not necessarily linear, which is an isometry with respect to the Hölder seminorm \(L(\cdot )\). It is shown that there exist \(K_0>0\), a surjective map \(\Psi : Y \longrightarrow X\) with \(d^\alpha (y,z)= K_0 \, d^\alpha (\Psi (y),\Psi (z))\) for all \(y,z\in Y\), and a function \(\Lambda : \mathrm{lip}^\alpha (X) \longrightarrow {\mathbb {C}}\) (which is linear or real-linear if T is so) such that either

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$

or

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$

where \(\tau =e^{i\theta }\) for some \(\theta \in [0,\pi )\).

     

Infiltration Equation with Degeneracy on the Boundary

This paper is mainly about the infiltration equation

$$ {u_{t}}= \operatorname{div} \bigl(a(x)|u|^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$

where \(p>1\), \(\alpha >0\), \(a(x)\in C^{1}(\overline{\Omega })\), \(a(x)\geq 0\) with \(a(x)|_{x\in \partial \Omega }=0\). If there is a constant \(\beta \) such that \(\int_{\Omega }a^{-\beta }(x)dx\leq c\), \(p>1+\frac{1}{\beta }\), then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any \(\beta >\frac{1}{p-1}\), \(\int_{\Omega }a^{-\beta }(x)dxdt=\infty \), then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition.

     

Overpartition function modulo powers of 2

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, congruences modulo powers of 2 for \(\overline{p}(n)\) were widely studied. In this paper, we prove several new infinite families of congruences modulo powers of 2 for \(\overline{p}(n)\). For example, for \(\alpha \ge 1\) and \(n\ge 0\),

$$\begin{aligned} \overline{p}(8\cdot 3^{4\alpha +4}n+5\cdot 3^{4\alpha +3})\equiv 0 \quad (\mathrm{mod}\,\,{2^8}). \end{aligned}$$

     

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Let V be a locally bounded measurable function on \({\mathbb {R}}^d\) such that \(\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x\) is a probability measure. Explicit criteria are presented for weighted Poincaré inequalities of the following non-local Dirichlet form

$$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$

Taking \(\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}\) with \(0<\alpha <2\) and \(\delta \geqslant 0\), we get new conclusions for (exponentially) tempered fractional Dirichlet forms, which not only complete our recent work (Chen and Wang in Stoch Process Their Appl 124:123–153, 2014; Wang and Wang in J Theor Probab 28:423–448, 2015), but also improve the main result in Mouhot et al. (J Math Pures Appl 95:72–84, 2011).

     

Jump and Variational Inequalities for Rough Operators

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families \(\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}\) of truncated singular integrals and \(\mathcal M:=\{M_t\}_{t>0}\) of averaging operators with rough kernels, which are defined respectively by

$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$

and

$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$

where the kernel \(\Omega \) belongs to \(L\log ^+\!\!L(\mathbf S^{n-1})\) or \(H^1(\mathbf S^{n-1})\) or \(\mathcal {G}_\alpha (\mathbf S^{n-1})\) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

     

On a Fractional $$ p \& q$$ Laplacian Problem with Critical Sobolev–Hardy Exponents

We consider the following fractional \( p \& q\) Laplacian problem with critical Sobolev–Hardy exponents

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p} u + (-\Delta )^{s}_{q} u = \frac{|u|^{p^{*}_{s}(\alpha )-2}u}{|x|^{\alpha }}+ \lambda f(x, u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^{N}{\setminus } \Omega , \end{array} \right. \end{aligned}$$

where \(0<s<1\), \(1\le q<p<\frac{N}{s}\), \((-\Delta )^{s}_{r}\), with \(r\in \{p,q\}\), is the fractional r-Laplacian operator, \(\lambda \) is a positive parameter, \(\Omega \subset \mathbb {R}^{N}\) is an open bounded domain with smooth boundary, \(0\le \alpha <sp\), and \(p^{*}_{s}(\alpha )=\frac{p(N-\alpha )}{N-sp}\) is the so-called Hardy–Sobolev critical exponent. Using concentration-compactness principle and the mountain pass lemma due to Kajikiya [23], we show the existence of infinitely many solutions which tend to be zero provided that \(\lambda \) belongs to a suitable range.

     

Sign-changing blowing-up solutions for a non-homogeneous elliptic equation at the critical exponent

We consider the equation \(-\Delta u = |u| ^{\frac{4}{n-2}}u + \varepsilon f(x) \) under zero Dirichlet boundary conditions in a bounded domain \(\Omega \) in \(\mathbb {R}^{n}\), \(n \ge 3\), with \(f\ge 0\), \(f\ne 0\). We find sign-changing solutions with large energy. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem

$$\begin{aligned} -\Delta w = |w|^{\frac{4}{n-2}} w, \quad w \in {\mathcal D}^{1,2} (\mathbb {R}^n) \end{aligned}$$

recently constructed in del Pino et al. (J Differ Equ 251(9):2568–2597, 2011).

     

The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1),(-\Delta )^{s}\) is the fractional Laplacian operator, \(\mu \) is a positive real parameter, \(q\in [2, 2^*_s)\) and \(2^*_s=2n/(n-2s)\) is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of \(\Omega \). Precisely, we show that the problem has at least \(cat_{\Omega }(\Omega )\) nontrivial solutions, provided that \(q=2\) and \(n\geqslant 4s\) or \(q\in (2, 2^*_s)\) and \(n>2s(q+2)/q\), extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

     

Multilicity results for some nonlinear ellitic roblems with asymtotically $${{\varvec{}}}$$-linear terms

Taking any \(p > 1\), we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in [14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\).

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.