Consider the following elliptic system: \begin{equation*} \left\{\aligned&-\ve^2\Delta u_1+\lambda_1u_1=\mu_1u_1^3+\alpha_1u_1^{p-1}+\beta u_2^2u_1\quad&\text{in }\Omega,\\ &-\ve^2\Delta u_2+\lambda_2u_2=\mu_2u_2^3+\alpha_2u_2^{p-1}+\beta u_1^2u_2\quad&\text{in }\Omega,\\ &u_1,u_2>0\quad\text{in }\Omega,\quad u_1=u_2=0\quad\text{on }\partial\Omega,\endaligned\right. \end{equation*} where $\Omega\subset\bbr^4$ is a bounded domain, $\lambda_i,\mu_i,\alpha_i>0$ $(i=1,2)$ and $\beta\not=0$ are constants, $\ve>0$ is a small parameter and $2<p<2^*=4$. By using variational methods, we study the existence of ground state solutions to this system for sufficiently small $\ve>0$. The concentration behaviors of least-energy solutions as $\ve\to0^+$ are also studied. Furthermore, by combining elliptic estimates and local energy estimates, we obtain the locations of these spikes as $\ve\to0^+$.

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where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion. Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13, 14]. The original model was considered in 2D,

We study the minimization problem of a non-convex sparsity promoting penalty func- tion, the transformed l1 (TL1), and its application in compressed sensing (CS). The TL1 penalty inter- polates l0 and l1 norms through a nonnegative parameter a ∈ (0, +∞), similar to lp with p ∈ (0, 1], and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of l0 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l0 norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized con- strained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an l1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value a = 1, and compare DCATL1 with other CS algorithms on two classes of sensing ma- trices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on l1/2 norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on l1 minus l2 penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also com- petitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.