Nontrivial solutions and least energy nodal solutions for a class of fourth-order elliptic equations

This paper is concerned with the following fourth-order elliptic equation

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \Delta ^{2}u-\Delta u+V(x)u=|u|^{p-1}u,\,\mathrm{in}\,\mathbb {R}^{N},\\ u\in H^{2}\left( \mathbb {R}^{N}\right) , \end{array} \right. \end{aligned}$$

where \(p\in (2,\,2_{*}-1),\,u{\text {:}}\,\mathbb {R}^{N}\rightarrow \mathbb R.\) Under some appropriate assumptions on potential V(x),  the existence of nontrivial solutions and the least energy nodal solution are obtained by using variational methods.

     

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Consider the nonlinear parabolic equation in the form

$$\begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,(|F|^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned}$$

where \(T>0\) and \(\Omega \) is a Reifenberg domain. We suppose that the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) and to more general setting of Lorentz spaces.

     

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$

where \(a>0,~b,~\mu \ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2,3+2\alpha )\), the potential V(x) may be unbounded from below and \(\phi |u|^{p-2}u\) is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.

     

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.

     

Impulsive fractional boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplace operator

In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem

$$\begin{aligned} {_{t}}D_{T}^{\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) + a(t)|u(t)|^{p-2}u(t)= & {} f(t,u(t)),\;\;t\ne t_j,\;\;\hbox {a.e.}\;\;t\in [0,T],\\ \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} I_j(u(t_j))\;\;j=1,2,\ldots ,n,\\ u(0)= & {} u(T) = 0. \end{aligned}$$

where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and

$$\begin{aligned} \Delta \left( {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j)\right) \right)= & {} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right) \\&- {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^-\right) \right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right| ^{p-2}{_{0}}D_{t}^{\alpha }u\left( t_j^+\right) \right)= & {} \lim _{t \rightarrow t_j^+} {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) ,\\ {_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t_j^-)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t_j^-)\right)= & {} \lim _{t\rightarrow t_j^-}{_{t}}I_{T}^{1-\alpha }\left( \left| {_{0}}D_{t}^{\alpha }u(t)\right| ^{p-2}{_{0}}D_{t}^{\alpha }u(t)\right) . \end{aligned}$$

By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.

     

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

     

Existence of Positive Ground-State Solution for Choquard-Type Equations

In this paper, we are concerned with the following nonlocal problem

$$\begin{aligned} -\Delta u+u=q(x)\left( \int _{\mathbb {R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha }}\mathrm{d}y\right) |u|^{p-2}u,\quad x\in \mathbb {R}^N, \end{aligned}$$

where \(N\ge 3, \alpha \in ((N-4)_+,N), 2\le p<\frac{N+\alpha }{N-2}\) and q(x) is a given potential. Using comparison arguments and variational approach, we obtain the existence of positive ground-state solution for the Choquard-type equations with some restrictions on the potential q.

     

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.

     

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|~(p-2)?u) =-|u|~(β-1) u + α|u|~(q-2 )u,where p 1, β 0, q≥1 and α 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.     

On Existence Results for Nonlinear Fractional Differential Equations Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian at Resonance

In this paper, we study the existence result for the nonlinear fractional differential equations with p-Laplacian operator

$$\left\{\begin{array}{ll}D_{0^+}^{\beta} \phi_p( D_{0^+}^{\alpha} u(t))=f(t,u(t),D_{0^+}^{\alpha}u(t)), \quad t\in(0,1),\\ D_{0^+}^{\alpha}u(0)=D_{0^+}^{\alpha}u(1)=0,\end{array}\right.$$

where the p-Laplacian operator is defined as \({\phi_p(s) = |s|^{p-2}s,p > 1, \,\,{\rm and}\,\, \phi_q(s) = \phi_p^{-1}(s), \frac{1}{p}+\frac{1}{q} = 1;\, 0 < \alpha, \beta < 1, 1 < \alpha + \beta < 2 \,\,{\rm and}\,\, D_{0^+}^{\alpha}, D_{0^+}^{\beta}}\) denote the Caputo fractional derivatives, and \({f : [0,1] \times \mathbb{R}^2\rightarrow \mathbb{R}}\) is continuous. Though Chen et al. have studied the same equations in their article, the proof process is not rigorous. We point out the mistakes and give a correct proof of the existence result. The innovation of this article is that we introduce a new definition to weaken the conditions of Arzela–Ascoli theorem and overcome the difficulties of the proof of compactness of the projector K _P (I ? Q)N. As applications, an example is presented to illustrate the main results.

     

Blow-up phenomena for a nonlocal <Emphasis Type="Italic">p</Emphasis>-Laplace equation with Neumann boundary conditions

This paper is concerned with the blow-up of solutions to the following nonlocal p-Laplace equation:

$$u_t-\mathrm{div}(|\nabla{u}|^{p-2}\nabla{u})=|u|^{q-1}u-\frac{1}{|\Omega|} \int\limits_\Omega{|u|^{q-1}u}dx,\quad x\in\Omega,\quad 0 < t < T,$$

under homogeneous Neumann boundary conditions in a bounded smooth domain \({\Omega\subset\mathrm{R}^N}\). For all \({p > 2, q > p-1}\), a blow-up result for the solutions to the above equation with positive initial energy is established. This result improves a recent result by Qu and Liang (Abstr Appl Anal 3:551–552, 2013) which asserts the blow-up of solutions for \({p-1 < q\leq\frac{Np}{(N-p)_+}-1}\).

     

Mean Square Estimates for Coefficients of Symmetric Power <Emphasis Type="Italic">L</Emphasis>-Functions

Let L(sym ^j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$

and

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$

     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional (p, q)-Laplacian operators:

$$\left\{ {\begin{array}{*{20}{c}} {\left( { - \Delta } \right)_p^su = \lambda a\left( x \right){{\left| u \right|}^{p - 2}}u + \lambda b\left( x \right){{\left| u \right|}^{\alpha - 2}}{{\left| v \right|}^\beta }u + \frac{{\mu \left( x \right)}}{{\alpha \delta }}{{\left| u \right|}^{\gamma - 2}}{{\left| v \right|}^\delta }uin\Omega ,} \\ {\left( { - \Delta } \right)_q^sv = \lambda c\left( x \right){{\left| v \right|}^{q - 2}}v + \lambda b\left( x \right){{\left| u \right|}^\alpha }{{\left| v \right|}^{\beta - 2}}v + \frac{{\mu \left( x \right)}}{{\beta \gamma }}{{\left| u \right|}^\gamma }{{\left| v \right|}^{\delta - 2}}vin\Omega ,} \\ {u = v = 0on{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$

where λ > 0 is a real parameter, Ω is a bounded domain in R ^N , with boundary ?Ω Lipschitz continuous, s ∈ (0, 1), 1 < p ≤ q < ∞, sq < N, while (?Δ) _p ^s u is the fractional p-Laplacian operator of u and, similarly, (?Δ) _q ^s v is the fractional q-Laplacian operator of v. Since possibly p ≠ q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalue λ₁ for a related system, they prove that there exists a positive solution for the problem when λ < λ₁ by the modified definitions. Moreover, the authors obtain the bifurcation property when λ → λ₁^-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ ≥ λ₁.

     

A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system

$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$

Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k _i -Hessian operator, a ₁, p ₁, f ₁, a ₂, p ₂ and f ₂ are continuous functions.

     

A note on skew characters of symmetric groups

In this paper we establish the following estimate:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$

where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log⁺(t)). This inequality relies upon the following sharp L ^p estimate:

$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$

where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$

We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.

     

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.

     

The identification problem for complex-valued Ornstein–Uhlenbeck operators in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$

In this paper we study perturbed Ornstein–Uhlenbeck operators

$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$

for simultaneously diagonalizable matrices \(A,B\in \mathbb {C}^{N,N}\). The unbounded drift term is defined by a skew-symmetric matrix \(S\in \mathbb {R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by

$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$

One key assumption is a new \(L^p\)-dissipativity condition

$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$

for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). Then we prove that the Schwartz space is a core of \(A_p\) and apply an \(L^p\)-solvability result of the resolvent equation for \(A_p\). In addition, we derive \(W^{1,p}\)-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

     

On the <Emphasis Type="Italic">p</Emphasis>-Laplacian with Robin boundary conditions and boundary trace theorems

Let \(\Omega \subset \mathbb {R}^\nu \), \(\nu \ge 2\), be a \(C^{1,1}\) domain whose boundary \(\partial \Omega \) is either compact or behaves suitably at infinity. For \(p\in (1,\infty )\) and \(\alpha >0\), define

$$\begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega |\nabla u|^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } |u|^p\mathrm {d}\sigma }{\displaystyle \int _\Omega |u|^p\mathrm {d} x}, \end{aligned}$$

where \(\mathrm {d}\sigma \) is the surface measure on \(\partial \Omega \). We show the asymptotics

$$\begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned}$$

where \(H_\mathrm {max}\) is the maximum mean curvature of \(\partial \Omega \). The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.

     

Positive solutions with single and multi-peak for semilinear elliptic equations with nonlinear boundary condition in the half-space

We consider the existence of single and multi-peak solutions of the following nonlinear elliptic Neumann problem

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u+\lambda ^{2} u&=Q(x)|u|^{p-2}u \qquad&\text {in} ~~~~\mathbb {R}^{N}_{+}, \\ \frac{\partial u }{\partial n}&=f(x,u) \qquad&\text {on}~~\partial \mathbb {R}^{N}_{+}, \end{aligned}\right. \end{aligned}$$

where \(\lambda \) is a large number, \(p\in (2,\frac{2N}{N-2})\) for \(N\ge 3\), f(x, u) is subcritical about u and Q is positive and has some non-degenerate critical points in \(\mathbb {R}^{N}_{+}\). For \(\lambda \) large, we can get solutions which have peaks near the non-degenerate critical points of Q.

     

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 \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).