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In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in \begin{document}$ \mathbb{R}^N $\end{document} involving \begin{document}$ p $\end{document} -Laplacian \begin{document}$ \begin{equation*} \label{eq:1} -\Delta_pu = \frac{|u|^{p^\ast_{\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{p^\ast_{\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document} where \begin{document}$ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1 and \begin{document}$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $\end{document} is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian \begin{document}$ \begin{equation*} (-\Delta)^{s}u = \frac{|u|^{2^\ast_{s,\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{2^\ast_{s,\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document} where \begin{document}$ 0 and \begin{document}$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $\end{document} .

Let Ω e be a bounded domain in R N with small holes. In this paper, we study the existence of solutions for the following problem { ( − Δ ) s u = u 2 s ⁎ − 1 , u > 0 , in Ω e , u = 0 , in R N ∖ Ω e , where 0 s 1 and 2 s ⁎ = 2 N N − 2 s . We construct solutions which blow up like a volcano near the center of each hole in Ω e

In this paper, we establish Liouville type theorems for nonlinear elliptic systems related to Hartree and Hartree–Fock equations in quantum mechanics. The main ingredient we use is the moving plane method.

In this paper, we consider the following elliptic problem: (0.1) − Δ u + V ( x ) u + λ ( κ α ⁎ | u | p ) | u | p − 2 u = ( κ β ⁎ | u | q ) | u | q − 2 u , x ∈ R 3 , where λ > 0 , 0 α 3 , 0 β 3 , 1 + α 3 p 3 + α , q > 1 and κ α ( x ) = | x | α − 3 , κ β ( x ) = | x | β − 3 . If V = 1 , we show the existence of a ground state solution and multiple solutions, as well as the nonexistence result for (0.1) . Furthermore, we investigate the limit behavior of ground state solutions of (0.1) as λ → 0 + . Ground state solutions are also obtained if the potential V is not a constant.

In this paper, we consider the existence of non-trivial solutions for the following equation \begin{document}$\mathcal{H}_{0J}u = |u|^{p-2}u+λ u\;\;\;\;{\rm in}\,\,\mathbb{R}^3,\;\;\;\;\;\;\;\;(1)$ \end{document} where \begin{document} $\mathcal{H}_{0J}$ \end{document} is the higher-order Schrodinger operator with \begin{document} $J∈\mathbb{N}$ \end{document} , \begin{document} $2 , and \begin{document} $λ∈\mathbb{R}$ \end{document} is a parameter. Let \begin{document} $E(u)$ \end{document} be the corresponding variational functional of problem (1). We look for solutions of equation (1) by finding minimizers of the minimization problem \begin{document}$E_ρ = \inf\{E(u)|u∈ H^{J}(\mathbb{R}^3):\,\,\|u\|_{L^2(\mathbb{R}^3)} = ρ\}.$ \end{document} We show that problem (1) admits at least a solution provided that in the case \begin{document} $J$ \end{document} being odd, \begin{document} $2 and \begin{document} $ρ>0$ \end{document} small or \begin{document} $2+J and \begin{document} $ρ>0$ \end{document} large; and for the case \begin{document} $J$ \end{document} being even, \begin{document} $3 and \begin{document} $ρ>0$ \end{document} small.

In this paper, we show that the following nonlocal equation (0.1) { − Δ u + λ u = ( | x | − μ ⁎ | u | q ) | u | q − 2 u in Ω , u ∈ H 0 1 ( Ω ) has at least c a t Ω ( Ω ) positive solutions under certain condition on q if λ > λ 0 for some λ 0 > 0 , where Ω ⊂ R N , N ≥ 3 , is an unbounded domain with the boundary ∂ Ω ≠ ∅ , the set R N ∖ Ω is bounded.

In this paper, we investigate the existence of nodal solutions for the following problem where 1 2 α 1 , 0 μ ≤ 2 and 2 N − μ N − 1 p 2 N − μ N − 2 α . We show that for any positive integer k, problem ( P 0 ) possesses a radially symmetrical solution changing sign exactly k-times.

In this paper, we consider the existence of static solutions to the nonlinear Chern-Simons-Schrodinger system 1 \begin{document}$ \begin{equation} \left\{\begin{array}{ll} -iD_0\Psi-(D_1D_1+D_2D_2)\Psi+V\Psi = |\Psi|^{p-2}\Psi,\\ \partial_0A_1-\partial_1A_0 = -\frac 12i\lambda[\overline{\Psi}D_2\Psi-\Psi\overline{D_2\Psi}],\\ \partial_0A_2-\partial_2A_0 = \frac 12i\lambda[\overline{\Psi}D_1\Psi-\Psi\overline{D_1\Psi}],\\ \partial_1A_2-\partial_2A_1 = -\frac12\lambda|\Psi|^2.\\ \end{array} \right. \end{equation} $\end{document} with an external potential \begin{document}$ V(x) $\end{document} , where \begin{document}$ D_{0} = \partial_{t}+i\lambda A_{0} $\end{document} and \begin{document}$ D_{k} = \partial_{x_k}-i\lambda A_{k}, \, k = 1,2, $\end{document} for \begin{document}$ (x_1,x_2,t)\in \mathbb{R}^{2,1} $\end{document} are covariant derivatives, \begin{document}$ \lambda $\end{document} is the coupling number. Suppose that \begin{document}$ V(x) $\end{document} satisfies \begin{document}$ \lim_{|x|\to\infty}V(x) = +\infty $\end{document} , we show for \begin{document}$ 2 that there exists \begin{document}$ \lambda^*>0 $\end{document} such that if \begin{document}$ 0 , problem (1) has two nontrivial static solutions \begin{document}$ (\Psi_\lambda, A_0^\lambda, A_1^\lambda,A_2^\lambda) $\end{document} . Moreover, there also exists \begin{document}$ \tilde\lambda>0 $\end{document} such that if \begin{document}$ \lambda>\tilde\lambda $\end{document} , problem (1) has no nontrivial solutions. While for \begin{document}$ p>4 $\end{document} we assume in addition that \begin{document}$ x\cdot \nabla V(x)\geq 0 $\end{document} , then problem (1) admits a mountain pass solution for all \begin{document}$ \lambda>0 $\end{document} .

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad \Omega, \\ \\ u=0 &{\rm on }\quad \partial\Omega, \end{array} \right. \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$ with $0\in \Omega$, $\lambda,\mu>0,0, Comment: 21 pages

In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_\alpha-2)$, $06\alpha$ and $\lambda>0$. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions., Comment: 34 pages

In this paper, we show that the minimizing problem (1) Λ s , α = inf u ∈ H s ( R N ) , u ≢ 0 ∫ R N | ( − Δ ) s 2 u ( x ) | 2 d x ( ∫ R N | u ( x ) | 2 s , α ∗ | x | α d x ) 2 2 s , α ∗ is achieved by a positive, radially symmetric and strictly decreasing function provided 0 s N 2 , 0 α 2 s .

In this paper, we are concerned with properties of positive solutions of the following fractional elliptic system \begin{eqnarray} {(-\Delta+I)}^{\frac{\alpha}{2}}u=\frac{u^pv^q}{|x|^\beta}, \quad {(-\Delta+I)}^{\frac{\alpha}{2}}v=\frac{v^pu^q}{|x|^\beta}\quad in\quad R^n, \end{eqnarray} where $n \geq 3$, $0 \le \beta 1$ and $p+q

We consider the following problem $$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$ where \({ \mu \ge 0, 0 0}\) and all the principle curvatures of ∂Ω at 0 are negative, then the above problem has infinitely many solutions.

In this paper, we investigate the existence of multiple solutions for critical inhomogeneous elliptic problem in non-contractible domains. We show that for f epsilon H(-1) satisfying a suitable condition and 0

In this paper, we consider semilinear elliptic systems with both singular and critical growth terms in bounded domains. The existence of a nontrivial solution is obtained by variational methods. (C) 2007 Elsevier Ltd. All rights reserved.

We consider in this paper the problem (0.1) { − Δ u = | x | α v p , − Δ v = | x | β u q e , x ∈ Ω , u > 0 , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , where Ω is the unit ball in R N centered at the origin, 0 ⩽ α p N , β > 0 , N ⩾ 8 , p > 1 , q e > 1 . Suppose q e → q > 1 as e → 0 + and q e , q satisfy respectively N p + 1 + N q e + 1 > N − 2 , N p + 1 + N q + 1 = N − 2 , we investigate the asymptotic behavior of the ground state solutions ( u e , v e ) of (0.1) as e → 0 + . We show that the ground state solutions concentrate at a point, which is located at the boundary. In addition, the ground state solution is non-radial provided that e > 0 is small.

In this paper, we consider the existence of multiple positive solutions for an inhomogeneous critical semilinear elliptic problem. We show that the problem possesses at least four positive solutions. (C) 2007 Elsevier Ltd. All rights reserved.

In this paper, we consider the existence of solutions for a critical elliptic system. We show that the problem possesses at least a high energy positive solution in non-contractible domains.

We, prove the existence of nontrivial solutions for the Schrodinger equation -Delta u + V(x)u = a gamma (x) f (u) in R-N. where f is superlinear and suberitical at zero and infinity respectively, V is periodic and a(x) chances sign. (c) 2007 Elsevier Inc. All rights reserved.

In this paper, we show that the semilinear elliptic systems of the form (0.1) { − Δ u − μ Δ v = g ( v ) , − Δ v − λ Δ u = f ( u ) , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , possess at least one nonnegative nontrivial solution pair ( u , v ) ∈ H 0 1 ( Ω ) × H 0 1 ( Ω ) , where Ω is a smooth bounded domain in R N .

In this paper, we study the following elliptic problem with Dirac mass \begin{equation}\label{eq 0.1} -\Delta u=Vu^p+k \delta_0\quad {\rm in}\quad \mathbb{R}^N, \qquad \lim_{|x|\to+\infty}u(x)=0, \end{equation} where $N>2$, $p>0$, $k>0$, $\delta_0$ is Dirac mass at the origin, the function $V$ is a locally Lipchitz continuous in $\mathbb{R}^N\setminus\{0\}$ satisfying $$ V(x)\le \frac{c_1}{|x|^{a_0}(1+|x|^{a_\infty-a_0})} $$ with $a_0a_0 $ and $c_1>0$. We obtain two positive solutions of (\ref{eq 0.1}) with additional conditions for parameters on $a_\infty, a_0$, $p$ and $k$. The first solution is a minimal positive solution and the second solution is constructed by Mountain Pass theorem., Comment: 23 pages

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent. We prove that this problem has solutions blowing up near the boundary of the domain. (c) 2006 Elsevier Inc. All rights reserved.

In this paper we study the existence of positive solutions of an inhomogeneous semilinear elliptic equation. Allowing the inhomogeneous term f (x) to change sign, that is, f is an element of C-1(Omega/{0}), we show under further assumptions that the problem has exact one or two positive solutions according to the classification of f. (C) 2004 Elsevier Ltd. All rights reserved.

In this paper, we prove, under natural assumptions, that the semilinear elliptic system -Deltau + u = g(x, v), x is an element of R-N, -Deltav + v = f(x, u), x is an element of R-N possesses at least one positive solution pair (u, v) is an element of H-1(R-N) x H-1(R-N), where f(x, t) and g(x, t) are continuous functions on R-N x R and asymptotically linear as t --> +infinity.

We establish an existence result for strongly indefinite semilinear elliptic systems with Neumann boundary condition, and we study the limiting behavior of the positive solutions of the singularly perturbed problem. (C) 2003 Elsevier Science (USA). All rights reserved.

An existence result for semilinear elliptic problems whose associated functionals do not satisfy a Palais-Smale condition is proved. The nonlinearity of our problem fits none of the conditions in Ambrosetti and Rabinowitz (J. Funct. Anal. 14 (1973) 349), de Figueiredo et al. (J. Math. Pures Appl. 61 (1982) 41) and Gidas and Spruck (Comm. Part. Diff. Eq. 6 (1981) 883). Some truncation happens to be essential, and in the argument some new results on Liouville-type theorems are established. (C) 2002 Elsevier Science (USA). All rights reserved.

In this paper, we investigate the limiting behavior of solutions $u_p$ of the following subcritical problem \begin{equation}\label{eq:0.1} (-\Delta)^{\frac \alpha 2} u = |x|^\beta u ^ {p-1},\quad u > 0, \quad x\in\Omega ;\quad u = 0,\quad x \in \partial \Omega \end{equation} as $p$ tending to the critical exponent $2^{*}_{\alpha}$, where $\Omega$ is the unit ball in $\mathbb{R}^n$ centered at the origin, and $\beta>0,0

This paper is concerned with the existence of a nontrivial solution for \begin{equation} -\Delta u=\lambda u+|x|^{\alpha}|u|^{2^*-2}u,\,\,\, {\rm in}\ \ \Omega, \quad u=0,\,\,\, {\rm on} \ \ \partial \Omega, \tag*{(0.1)} \end{equation} where $\lambda > 0$ and $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain. Let $\lambda_k,$ $ k =1,2,\ldots$, be eigenvalues of the operator $-\Delta$; we show for $\lambda_k < \lambda < \lambda_{k+1}$ that problem (0.1) possesses at least a solution and each $\lambda_k$ is a bifurcation point.

In this paper, we establish the existence of ground state solutions and bound state solutions for fractional field equations (0.1) ( − Δ + i d ) 1 2 u = f ( u ) in R n , u ∈ H 1 2 ( R n ) . Such solutions decay exponentially at infinity.

In this paper, an elliptic system with sign-changing weights is considered. By variational methods, we show that the problem has at least three nontrivial solutions.

We proved a multiplicity result for a nonlinear elliptic system in R-N. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions and the topology of the set of the global maxima of the coefficients.

We consider the solvability of the Neumann problem for the equation $$ -\Delta u+\lambda u =0, \quad \frac {\partial u}{\partial \nu}=Q(x)|u|^{q-2}u $$ on $\partial \Omega$, where $Q$ is a positive and continuous coefficient on $\partial \Omega$, $\lambda$ is a parameter and $q= {2(N-1)}/{(N-2)}$ is a critical Sobolev exponent for the trace embedding of $H^1(\Omega)$ into $L^q(\partial \Omega)$. We investigate the joint effect of the mean curvature of $\partial \Omega$ and the shape of the graph of $Q$ on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter $\lambda$ interferes with the spectrum of $-\Delta$ with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

We establish in this paper existence results for critical strongly indefinite semilinear elliptic systems defined on both bounded domains and $\mathbb R^N$.

The main purpose of this paper is to establish the existence of a solution of the semilinear Schr¨odinger equation (1) −u + V (x)u = K(x)|u|2−2u + f(x, u) in RN, involving a critical Sobolev exponent 2 = 2N/(N − 2) with N 4 and a subcritical nonlinearity f : RN × R ! R. Throughout this paper it is assumed that (A) The coefficients V and K are continuous and 1-periodic functions in each variable xi, i = 1, . . . ,N. Moreover, we assume that K 0 on RN. In this case it is known that the operator − + V on L2(RN) has a purely continuous spectrum consisting of closed disjoint intervals. In this paper we consider the case: (B) 0 is in the spectral gap of the operator − + V . There are many existence results in the case K 0 on RN and we refer to the papers [4], [5], [9], [10], [13]. 1991 Mathematics Subject Classification. 35J65.

We study an elliptic problem involving critical Sobolev exponent in domains with small holes. We prove the existence of solutions which blow up like a volcano near the centre of each hole. (C) 2003 Elsevier Inc. All rights reserved.

We consider an elliptic problem of Ambrosetti–Prodi type involving critical Sobolev exponent. We prove that this problem has solutions blowing up near the boundary of the domain.