• Title/Summary/Keyword: mountain pass solutions

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INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH

  • Zhou, Chenxing;Liang, Sihua
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.137-152
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    • 2014
  • In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

SOLUTIONS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEM WITH MIXED NONLINEARITIES

  • Zhang, Ziheng
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1585-1596
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    • 2016
  • In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem (FBVP) $$\{_tD_T^{\alpha}(_0D_t^{\alpha}u(t))={\nabla}W(t,u(t)),\;t{\in}[0,T],\\u(0)=u(T)=0,$$ where ${\alpha}{\in}(1/2,1)$, $u{\in}{\mathbb{R}}^n$, $W{\in}C^1([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ and ${\nabla}W(t,u)$ is the gradient of W(t, u) at u. The novelty of this paper is that, when the nonlinearity W(t, u) involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.

MULTIPLE SOLUTIONS OF A PERTURBED YAMABE-TYPE EQUATION ON GRAPH

  • Liu, Yang
    • Journal of the Korean Mathematical Society
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    • v.59 no.5
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    • pp.911-926
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    • 2022
  • Let u be a function on a locally finite graph G = (V, E) and Ω be a bounded subset of V. Let 𝜀 > 0, p > 2 and 0 ≤ λ < λ1(Ω) be constants, where λ1(Ω) is the first eigenvalue of the discrete Laplacian, and h : V → ℝ be a function satisfying h ≥ 0 and $h{\not\equiv}0$. We consider a perturbed Yamabe equation, say $$\{\begin{array}{lll}-{\Delta}u-{\lambda}u={\mid}u{\mid}^{p-2}u+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω and ∂Ω denote the interior and the boundary of Ω, respectively. Using variational methods, we prove that there exists some positive constant 𝜀0 > 0 such that for all 𝜀 ∈ (0, 𝜀0), the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation $$\{\begin{array}{lll}-{\Delta}u=f(u)+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ and prove similar result for certain nonlinear term f(u).

EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

  • Zou, Henghui
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.547-572
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    • 2010
  • We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

EXISTENCE AND MULTIPLICITY RESULTS FOR SOME FOURTH ORDER SEMILINEAR ELLIPTIC PROBLEMS

  • Jin, Yinghua;Wang, Xuechun
    • Korean Journal of Mathematics
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    • v.17 no.4
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    • pp.473-480
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    • 2009
  • We prove the existence and multiplicity of nontrivial solutions for a fourth order problem ${\Delta}^2u+c{\Delta}u={\alpha}u-{\beta}(u+1)^-$ in ${\Omega}$, ${\Delta}u=0$ and $u=0$ on ${\partial}{\Omega}$, where ${\lambda}_1{\leq}c{\leq}{\lambda}_2$ (where $({\lambda}_i)_{i{\geq}1}$ is the sequence of the eigenvalues of $-{\Delta}$ in$H_0^1({\Omega})$) and ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$. The results are proved by applying minimax arguments and linking theory.

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ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

  • Bin, Ge
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.409-421
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    • 2014
  • This paper deals with the superlinear elliptic problem without Ambrosetti and Rabinowitz type growth condition of the form: $$\{-div\((1+\frac{|{\nabla}u|^{p(x)}}{\sqrt{1+|{\nabla}u|^{2p(x)}}}})|{\nabla}u|^{p(x)-2}{\nabla}u\)={\lambda}f(x,u)\;a.e.\;in\;{\Omega}\\u=0,\;on\;{\partial}{\Omega}$$ where ${\Omega}{\subset}R^N$ is a bounded domain with smooth boundary ${\partial}{\Omega}$, ${\lambda}$ > 0 is a parameter. The purpose of this paper is to obtain the existence results of nontrivial solutions for every parameter ${\lambda}$. Firstly, by using the mountain pass theorem a nontrivial solution is constructed for almost every parameter ${\lambda}$ > 0. Then we consider the continuation of the solutions. Our results are a generalization of that of Manuela Rodrigues.

ON EXISTENCE OF WEAK SOLUTIONS OF NEUMANN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING p-LAPLACIAN IN AN UNBOUNDED DOMAIN

  • Hang, Trinh Thi Minh;Toan, Hoang Quoc
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1169-1182
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    • 2011
  • In this paper we study the existence of non-trivial weak solutions of the Neumann problem for quasilinear elliptic equations in the form $$-div(h(x){\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)+b(x){\mid}u{\mid}^{p-2}u=f(x,\;u),\;p{\geq}2$$ in an unbounded domain ${\Omega}{\subset}\mathbb{R}^N$, $N{\geq}3$, with sufficiently smooth bounded boundary ${\partial}{\Omega}$, where $h(x){\in}L_{loc}^1(\overline{\Omega})$, $\overline{\Omega}={\Omega}{\cup}{\partial}{\Omega}$, $h(x){\geq}1$ for all $x{\in}{\Omega}$. The proof of main results rely essentially on the arguments of variational method.

MULTIPLICITY RESULTS OF CRITICAL LOCAL EQUATION RELATED TO THE GENUS THEORY

  • Mohsen Alimohammady;Asieh Rezvani;Cemil Tunc
    • Communications of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.1045-1061
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    • 2023
  • Using variational methods, Krasnoselskii's genus theory and symmetric mountain pass theorem, we introduce the existence and multiplicity of solutions of a parameteric local equation. At first, we consider the following equation $$\{-div[a(x,{\mid}{\nabla}u{\mid}){\nabla}u]\,=\,{\mu}(b(x){\mid}u{\mid}^{s(x)-2}-{\mid}u{\mid}^{r(x)-2})u\;in\;{\Omega},\\u\,=0\,on\;{\partial}{\Omega},$$ where Ω⊆ ℝN is a bounded domain, µ is a positive real parameter, p, r and s are continuous real functions on ${\bar{\Omega}}$ and a(x, ξ) is of type |ξ|p(x)-2. Next, we study boundedness and simplicity of eigenfunction for the case a(x, |∇u|)∇u = g(x)|∇u|p(x)-2∇u, where g ∈ L(Ω) and g(x) ≥ 0 and the case $a(x,\,{\mid}{\nabla}u{\mid}){{\nabla}u}\,=\,(1\,+\,{\nabla}u{\mid}^2)^{\frac{p(x)-2}{2}}{\nabla}u$ such that p(x) ≡ p.

EXISTENCE OF SOLUTIONS FOR FRACTIONAL p&q-KIRCHHOFF SYSTEM IN UNBOUNDED DOMAIN

  • Bao, Jinfeng;Chen, Caisheng
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1441-1462
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    • 2018
  • In this paper, we investigate the fractional p&q-Kirchhoff type system $$\{M_1([u]^p_{s,p})(-{\Delta})^s_pu+V_1(x){\mid}u{\mid}^{p-2}u\\{\hfill{10}}={\ell}k^{-1}F_u(x,\;u,\;v)+{\lambda}{\alpha}(x){\mid}u{\mid}^{m-2}u,\;x{\in}{\Omega}\\M_2([u]^q_{s,q})(-{\Delta})^s_qv+V_2(x){\mid}v{\mid}^{q-2}v\\{\hfill{10}}={\ell}k^{-1}F_v(x,u,v)+{\mu}{\alpha}(x){\mid}v{\mid}^{m-2}v,\;x{\in}{\Omega},\\u=v=0,\;x{\in}{\partial}{\Omega},$$ where ${\Omega}{\subset}{\mathbb{R}}^N$ is an unbounded domain with smooth boundary ${\partial}{\Omega}$, and $0<s<1<p{\leq}q$ and sq < N, ${\lambda},{\mu}>0$, $1<m{\leq}k<p^*_s$, ${\ell}{\in}R$, while $[u]^t_{s,t}$ denotes the Gagliardo semi-norm given in (1.2) below. $V_1(x)$, $V_2(x)$, $a(x):{\mathbb{R}}^N{\rightarrow}(0,\;{\infty})$ are three positive weights, $M_1$, $M_2$ are continuous and positive functions in ${\mathbb{R}}^+$. Using variational methods, we prove existence of infinitely many high-energy solutions for the above system.

MULTIPLICITY OF SOLUTIONS FOR QUASILINEAR SCHRÖDINGER TYPE EQUATIONS WITH THE CONCAVE-CONVEX NONLINEARITIES

  • Kim, In Hyoun;Kim, Yun-Ho;Li, Chenshuo;Park, Kisoeb
    • Journal of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1461-1484
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    • 2021
  • We deal with the following elliptic equations: $\{-div({\varphi}^{\prime}(\left|{\nabla}z\right|^2){\nabla}z)+V(x)\left|z\right|^{{\alpha}-2}z={\lambda}{\rho}(x)\left|z\right|^{r-2}z+h(x,z),\\z(x){\rightarrow}0,\;as\;\left|x\right|{\rightarrow}{\infty},$ in ℝN , where N ≥ 2, 1 < p < q < N, 1 < α ≤ p*q'/p', α < q, 1 < r < min{p, α}, φ(t) behaves like tq/2 for small t and tp/2 for large t, and p' and q' the conjugate exponents of p and q, respectively. Here, V : ℝN → (0, ∞) is a potential function and h : ℝN × ℝ → ℝ is a Carathéodory function. The present paper is devoted to the existence of at least two distinct nontrivial solutions to quasilinear elliptic problems of Schrödinger type, which provides a concave-convex nature to the problem. The primary tools are the well-known mountain pass theorem and a variant of Ekeland's variational principle.