We consider the following singularly perturbed Kirchhoff type equations $$-\varepsilon^2 M\left(\varepsilon^{2-N}\int_{\R^N}|\nabla u|^2 dx\right)\Delta u +V(x)u=|u|^{p-2}u~\hbox{in}~\R^N, u\in H^1(\R^N),N\geq 1,$$ where $M\in C([0,\infty))$ and $V\in C(\R^N)$ are given functions. Under very mild assumptions on $M$, we prove the existence of single-peak or multi-peak solution $u_\varepsilon$ for above problem, concentrating around topologically stable critical points of $V$, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. in 2014 [ARMA,213].
Abstract We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: −Δu=∣u∣2∗−2u+λu+μulogu2x∈Ω,u=0x∈∂Ω, \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}-\Delta u={| u| }^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2}\hspace{1.0em}& x\in \Omega ,\\ u=0\hspace{1.0em}& x\in \partial \Omega ,\end{array}\right. where Ω⊂RN \Omega \subset {{\mathbb{R}}}^{N} is a bounded open domain, λ,μ∈R \lambda ,\mu \in {\mathbb{R}} , N≥3 N\ge 3 and 2∗≔2NN−2 {2}^{\ast }:= \frac{2N}{N-2} is the critical Sobolev exponent for the embedding H01(Ω)↪L2∗(Ω) {H}_{0}^{1}\left(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{{2}^{\ast }}\left(\Omega ) . The uncertainty of the sign of slogs2 s\log {s}^{2} in (0,+∞) \left(0,+\infty ) has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided λ∈R,μ>0 \lambda \in {\mathbb{R}},\mu \gt 0 and N≥4 N\ge 4 . While the case of μ<0 \mu \lt 0 is thornier. However, for N=3,4 N=3,4 , λ∈(−∞,λ1(Ω)) \lambda \in \left(-\infty ,{\lambda }_{1}\left(\Omega )) , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for μ<0 \mu \lt 0 and −(N−2)μ2+(N−2)μ2log−(N−2)μ2+λ−λ1(Ω)≥0 -\frac{\left(N-2)\mu }{2}+\frac{\left(N-2)\mu }{2}\log \left(-\frac{\left(N-2)\mu }{2}\right)+\lambda -{\lambda }_{1}\left(\Omega )\ge 0 if N≥3 N\ge 3 . Comparing with the results in the study by Brézis and Nirenberg ( Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents , Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter μ \mu on logarithmic perturbation is not zero.
We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schr\"odinger equation of the form \begin{equation*} -\Delta u+\lambda u=g(u), \quad u \in H^1(\mathbb{R}^N), \, N \geq 1. \end{equation*} Our approach permits to handle in a unified way nonlinearities $g(s)$ which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as $\lambda\rightarrow 0^+$ or $\lambda\rightarrow +\infty$ and the existence of an unbounded continuum of solutions in $(0, + \infty) \times H^1(\mathbb{R}^N)$.
Abstract We study the following nonlinear Dirichlet boundary value problem: where Ω is a bounded domain in ℝ N ( N ≥ 2) with a smooth boundary ∂Ω and g ∈ C (Ω × ℝ) is a function satisfying $\displaystyle \underset{|t|\rightarrow 0}{\lim}\frac{g(x, t)}{t}= \infty$ for all x ∈ Ω. Under appropriate assumptions, we prove the existence of infinitely many solutions when g ( x , t ) is not odd in t .
Ultrasonication assisted extraction (UAE) was employed to extract total flavonoids (TF) from Kaempferia galanga L. (KGL) roots. Single-factor experiments were performed to evaluate the effects of the process parameters (solvent-to-material ratio, ethanol concentration, extraction time, ultrasonic power, and extraction temperature) on the TF yield and the response surface methodology (RSM) was used to optimize the extraction parameters. The optimum conditions to obtain the maximum TF yield were solvent-to-material ratio, 25; ethanol concentration, 62%; extraction time, 27.0 min; extraction temperature, 50℃ and ultrasonic power, 225 W. The yield of TF under these conditions reached 3.22 ± 0.12 mg/g of dry root. The analysis of variance and validation test indicated that the proposed method is useful and reliable. The antioxidant capacities of TF of KGL was determined by DPPH· and ABTS+· methods, which revealed that it has good antioxidant activity in both systems.
In present paper, we study the following Schr\"odinger systems $$\begin{cases} -\Delta u_1+V_1(x)u_1+\lambda_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\R^N,\\ -\Delta u_2+V_2(x)u_2+\lambda_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\R^N,\\ 00, ~2+\frac{4}{N}>r_{1,k}+r_{2,k}, p_i, q_j>2, ~r_{1,k}, r_{2,k}>1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{cases} $$ Under some assumptions on $V_\iota$ $(\iota =1,2)$ and the parameters, we can show the existence of ground state normalized solution $(\lambda_1,\lambda_2; u_1, u_2)\in \R^2 \times H^1(\R^N, \R^2) $ to the above mass sub-critical problem for any given $a_1>0,a_2>0$. Here our nonlinearities are general. The potentials $V_1(x)$ and $V_2(x)$ are also very general such that $\inf ess~\sigma(-\Delta+V_\iota)>-\infty$, which are allowed to be singular at some points.
In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad &\hbox{in}\;\RN, \\ -\Delta v + \lambda_2 v= \mu_2|v|^{q-2}v+\beta u \quad &\hbox{in}\;\RN, \end{cases}$$ satisfying the normalization constraints $\displaystyle\int_{\RN}u^2 =a, ~ \int_{\RN}v^2 =b$. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed and the masses $a,b>0$. Here $2+\frac{4}{N}0$ and $\beta>0$, provided $2\leqslant N\leqslant 4$. For the Sobolev critical case with $N=3,4$, it can be viewed as a counterpart of the Brezis-Nirenberg critical semilinear elliptic problem for the system case in the context of normalized solutions. Under some suitable assumptions, we obtain the existence or non-existence of positive normalized ground state solution.
We consider the existence of solutions $(\lambda_1,\lambda_2, u, v)\in \mathbb{R}^2\times (H^1(\mathbb{R}^N))^2$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1 u^{p-1}+\beta r_1 u^{r_1-1}v^{r_2}\quad &\hbox{in}~\mathbb{R}^N,\\ -\Delta v+\lambda_2 v=\mu_2 v^{q-1}+\beta r_2 u^{r_1}v^{r_2-1}\quad &\hbox{in}~\mathbb{R}^N,\\ 00$ and the prescribed masses $a,b>0$. We focus on the coupled purely mass super-critical case, i.e., $$2+\frac{4}{N}0$ and $\beta>0$.
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