scholarly journals Elliptic problem driven by different types of nonlinearities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Debajyoti Choudhuri ◽  
Dušan D. Repovš
Keyword(s):  
Nontrivial Solution ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Different Types ◽  
Exponential Critical Growth

AbstractIn this paper we establish the existence and multiplicity of nontrivial solutions to the following problem: $$\begin{aligned} \begin{aligned} (-\Delta )^{\frac{1}{2}}u+u+\bigl(\ln \vert \cdot \vert * \vert u \vert ^{2}\bigr)&=f(u)+\mu \vert u \vert ^{- \gamma -1}u,\quad \text{in }\mathbb{R}, \end{aligned} \end{aligned}$$ ( − Δ ) 1 2 u + u + ( ln | ⋅ | ∗ | u | 2 ) = f ( u ) + μ | u | − γ − 1 u , in  R , where $\mu >0$ μ > 0 , $(*)$ ( ∗ ) is the convolution operation between two functions, $0<\gamma <1$ 0 < γ < 1 , f is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity f is of exponential critical growth.

scholarly journals Multiple solutions for a quasilinear Schrödinger–Poisson system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang
Keyword(s):  
Nontrivial Solution ◽  
Ground State ◽  
Multiple Solutions ◽  
Continuous Functions ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Poisson System ◽  
Subcritical Growth ◽  
Monotonicity Properties

AbstractIn this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

scholarly journals Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Die Hu ◽  
Xianhua Tang ◽  
Qi Zhang
Keyword(s):  
Schrödinger Equation ◽  
Nontrivial Solution ◽  
Ground State ◽  
Schrodinger Equation ◽  
Existence Of Solutions ◽  
Ground State Solution ◽  
State Solution ◽  
Quasilinear Schrödinger Equation ◽  
Kirchhoff Type

<p style='text-indent:20px;'>In this paper, we discuss the generalized quasilinear Schrödinger equation with Kirchhoff-type:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1a"> \begin{document}$\left (1\!+\!b\int_{\mathbb{R}^{3}}g^{2}(u)|\nabla u|^{2} dx \right) \left[-\mathrm{div} \left(g^{2}(u)\nabla u\right)\!+\!g(u)g'(u)|\nabla u|^{2}\right] \!+\!V(x)u\! = \!f( u),(\rm P)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ b&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $\end{document}</tex-math></inline-formula>. Under some "Berestycki-Lions type assumptions" on the nonlinearity <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> which are almost necessary, we prove that problem <inline-formula><tex-math id="M6">\begin{document}$ (\rm P) $\end{document}</tex-math></inline-formula> has a nontrivial solution <inline-formula><tex-math id="M7">\begin{document}$ \bar{u}\in H^{1}(\mathbb{R}^{3}) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M8">\begin{document}$ \bar{v} = G(\bar{u}) $\end{document}</tex-math></inline-formula> is a ground state solution of the following problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1b"> \begin{document}$-\left(1+b\int_{\mathbb{R}^{3}} |\nabla v|^{2} dx \right) \triangle v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \frac{f(G^{-1}(v))}{g(G^{-1}(v))},(\rm \bar{P})$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M9">\begin{document}$ G(t): = \int_{0}^{t} g(s) ds $\end{document}</tex-math></inline-formula>. We also give a minimax characterization for the ground state solution <inline-formula><tex-math id="M10">\begin{document}$ \bar{v} $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

2018 ◽  
Vol 9 (1) ◽  
pp. 108-123 ◽  
Author(s):  
Claudianor O. Alves ◽  
Grey Ercole ◽  
M. Daniel Huamán Bolaños
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Ground State Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

Abstract We prove the existence of at least one ground state solution for the semilinear elliptic problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either {\mathbb{R}^{N}} or a bounded domain, and {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.

scholarly journals Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-21
Author(s):  
Tsung-Fang Wu
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Ground State ◽  
Unbounded Domains ◽  
Ground State Solution ◽  
Multiple Positive Solutions ◽  
Dirichlet Problems ◽  
Existence And Multiplicity ◽  
Sobolev Constant ◽  
Multiplicity Of Positive Solutions

We consider the elliptic problem−Δu+u=b(x)|u|p−2u+h(x)inΩ,u∈H01(Ω), where2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ωis a smooth unbounded domain inℝN, b(x)∈C(Ω), andh(x)∈H−1(Ω). We use the shape of domainΩto prove that the above elliptic problem has a ground-state solution if the coefficientb(x)satisfiesb(x)→b∞>0as|x|→∞andb(x)≥cfor some suitable constantsc∈(0,b∞), andh(x)≡0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficientb(x)also satisfies the above conditions,h(x)≥0and0<‖h‖H−1<(p−2)(1/(p−1))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), whereS(Ω)is the best Sobolev constant of subcritical operator inH01(Ω)andbsup=supx∈Ωb(x).

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