• Korean Journal of Mathematics
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• 제13권1호
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• pp.83-90
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• 2005

A.H.Hamel proved the Ekeland's principle in a locally convex Hausdorff topological vector spaces by constructing the norm and applying the Ekeland's principle in Banach spaces. In this paper we show that the Ekeland's principle in a locally convex Hausdorff topological vector spaces can be proved directly by applying the famous general principle of H.Br$\acute{e}$zis and F.E.Browder.

• 대한수학회보
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• 제54권3호
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• pp.715-729
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• 2017

In this paper, we investigate the existence and multiplicity of solutions for systems driven by two non-local integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tools are the Saddle point theorem, Ekeland's variational principle and the Mountain pass theorem.

• 대한수학회보
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• 제48권5호
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• pp.1023-1032
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• 2011

In this paper, a new Ekeland type variational principle on gauge spaces is established. As applications, we give Caristi-Kirk type fixed point theorems on gauge spaces, and Dane$\check{s}$' drop theorem on seminormed spaces. Also, we show that the Palais-Smale condition implies coercivity on semi-normed spaces.

• 대한수학회보
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• 제51권2호
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• pp.437-442
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• 2014

In this paper we prove Ekeland's variational principle in the setting of locally complete spaces for lower semi continuous functions from above and bounded below. We use this theorem to prove Caristi's fixed point theorem in the same setting and also for lower semi continuous functions.

• 대한수학회보
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• 제54권1호
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• pp.125-144
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• 2017

Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

• 대한수학회논문집
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• 제35권2호
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• pp.517-532
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• 2020

In this paper, we consider a class of nonlocal problems with indefinite weights in Orlicz-Sobolev space. Under some suitable conditions on the nonlinearities, we establish some existence results using variational techniques and Ekeland's variational principle.

• 대한수학회보
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• 제53권6호
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• pp.1753-1769
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• 2016

This paper is concerned with the quasilinear $Schr{\ddot{o}}dinger$ system $$(0.1)\;\{-{\Delta}u+a(x)u-{\Delta}(u^2)u=Fu(u,v)+h(x)\;x{\in}{\mathbb{R}}^N,\\-{\Delta}v+b(x)v-{\Delta}(v^2)v=Fv(u,v)+g(x)\;x{\in}{\mathbb{R}}^N,$$ where $N{\geq}3$. The potential functions $a(x),b(x){\in}L^{\infty}({\mathbb{R}}^N)$ are bounded in ${\mathbb{R}}^N$. By using mountain pass theorem and the Ekeland variational principle, we prove that there are at least two solutions to system (0.1).

• 대한수학회지
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• 제57권6호
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• pp.1551-1571
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• 2020

This paper is concerned with the following Kirchhoff-Schrödinger-Poisson system $$\begin{cases} -(a+b{\displaystyle\smashmargin{2}\int\nolimits_{\mathbb{R}^3}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+{\mu}{\phi}u={\lambda}f(x){\mid}u{\mid}^{p-2}u+g(x){\mid}u{\mid}^{p-2}u,&{\text{ in }}{\mathbb{R}}^3,\\-{\Delta}{\phi}={\mu}{\mid}u{\mid}^2,&{\text{ in }}{\mathbb{R}}^3, \end{cases}$$ where a > 0, b, µ ≥ 0, p ∈ (1, 2), q ∈ [4, 6) and λ > 0 is a parameter. Under some suitable assumptions on V (x), f(x) and g(x), we prove that the above system has at least two different nontrivial solutions via the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. Some recent results from the literature are improved and extended.

• Journal of applied mathematics & informatics
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• 제32권1_2호
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• pp.113-128
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• 2014

Consider a class of p(x)-Kirchhoff type equations of the form $$\left\{-M\left({\int}_{\Omega}\;\frac{1}{p(x)}{\mid}{\nabla}u{\mid}^{p(x)}\;dx\right)\;div\;({\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)={\lambda}V(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u=0\;on\;{\partial}{\Omega},$$ where p(x), $q(x){\in}C({\bar{\Omega}})$ with 1 < $p^-\;:=inf_{\Omega}\;p(x){\leq}p^+\;:=sup_{\Omega}p(x)$ < N, $M:{\mathbb{R}}^+{\rightarrow}{\mathbb{R}}^+$ is a continuous function that may be degenerate at zero, ${\lambda}$ is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in two cases when the weight function V (x) may change sign or not.

• 대한수학회지
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• 제58권6호
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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 ℝ^N , where N ≥ 2, 1 < p < q < N, 1 < α ≤ p^*q'/p', α < q, 1 < r < min{p, α}, φ(t) behaves like t^q/2 for small t and t^p/2 for large t, and p' and q' the conjugate exponents of p and q, respectively. Here, V : ℝ^N → (0, ∞) is a potential function and h : ℝ^N × 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.