First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities
Abstract: We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t … Show more
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Cited by 3 publications
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In this paper, we construct the solutions to the following nonlinear Schrödinger system $$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u+P(x)u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v+Q(x)v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ - ϵ 2 Δ u + P ( x ) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q ( x ) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where $$3< p<+\infty $$ 3 < p < + ∞ , $$N\in \{1,2\}$$ N ∈ { 1 , 2 } , $$\epsilon >0$$ ϵ > 0 is a small parameter, the potentials P, Q satisfy $$0<P_{0} \le P(x)\le P_{1}$$ 0 < P 0 ≤ P ( x ) ≤ P 1 and Q(x) satisfies $$0<Q_{0} \le Q(x)\le Q_{1}$$ 0 < Q 0 ≤ Q ( x ) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When $$x_{0}$$ x 0 is a local maximum point of the potentials P and Q and $$P(x_{0})=Q(x_{0})$$ P ( x 0 ) = Q ( x 0 ) , we construct k spikes concentrating near the local maximum point $$x_{0}$$ x 0 . When $$x_{0}$$ x 0 is a local maximum point of P and $$\overline{x}_{0}$$ x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point $$x_{0}$$ x 0 and m spikes of v concentrating at the local maximum point $$\overline{x}_{0}$$ x ¯ 0 when $$x_{0}\ne \overline{x}_{0}.$$ x 0 ≠ x ¯ 0 . This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case $$N=3$$ N = 3 , $$p=3$$ p = 3 .
In this paper, we construct the solutions to the following nonlinear Schrödinger system $$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u+P(x)u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v+Q(x)v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ - ϵ 2 Δ u + P ( x ) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q ( x ) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where $$3< p<+\infty $$ 3 < p < + ∞ , $$N\in \{1,2\}$$ N ∈ { 1 , 2 } , $$\epsilon >0$$ ϵ > 0 is a small parameter, the potentials P, Q satisfy $$0<P_{0} \le P(x)\le P_{1}$$ 0 < P 0 ≤ P ( x ) ≤ P 1 and Q(x) satisfies $$0<Q_{0} \le Q(x)\le Q_{1}$$ 0 < Q 0 ≤ Q ( x ) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When $$x_{0}$$ x 0 is a local maximum point of the potentials P and Q and $$P(x_{0})=Q(x_{0})$$ P ( x 0 ) = Q ( x 0 ) , we construct k spikes concentrating near the local maximum point $$x_{0}$$ x 0 . When $$x_{0}$$ x 0 is a local maximum point of P and $$\overline{x}_{0}$$ x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point $$x_{0}$$ x 0 and m spikes of v concentrating at the local maximum point $$\overline{x}_{0}$$ x ¯ 0 when $$x_{0}\ne \overline{x}_{0}.$$ x 0 ≠ x ¯ 0 . This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case $$N=3$$ N = 3 , $$p=3$$ p = 3 .
<abstract><p>This paper is devoted to considering the attainability of minimizers of the $ L^2 $-constraint variational problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ m_{\gamma, a} = \inf \, \{J_{\gamma}(u):u\in H^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}} \vert u\vert^2 dx = a^2 \} {, } $\end{document} </tex-math></disp-formula></p> <p>where</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ J_{\gamma}(u) = \frac{\gamma}{2}\int_{\mathbb{R}^{N}} \vert\Delta u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} \vert\nabla u\vert^2 dx+\frac{1}{2}\int_{\mathbb{R}^{N}} V(x)\vert u\vert^2 dx-\frac{1}{2\sigma+2}\int_{\mathbb{R}^{N}} \vert u\vert^{2\sigma+2} dx, $\end{document} </tex-math></disp-formula></p> <p>$ \gamma > 0 $, $ a > 0 $, $ \sigma\in(0, \frac{2}{N}) $ with $ N\ge 2 $. Moreover, the function $ V:\mathbb{R}^{N}\rightarrow [0, +\infty) $ is continuous and bounded. By using the variational methods, we can prove that, when $ V $ satisfies four different assumptions, $ m_{\gamma, a} $ are all achieved.</p></abstract>
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