• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1269-1283
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• 2011

For a class of quasilinear elliptic equations we establish the existence of multiple solutions by variational methods.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.191-202
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• 2007

By using the weighted averaging techniques, we establish oscillation criteria for the second order damped quasilinear elliptic differential equation $$\sum_{i,j=1}^{N}D_i(a_{ij}(x){\parallel}Dy{\parallel}^{p-2}D_jy)+{\langle}b(x),\;{\parallel}Dy{\parallel}^{p-2}Dy{\rangle}+c(x)f(y)=0,\;p>1$$. The obtained theorems include and improve some existing ones for the undamped halflinear partial differential equation and the semilinear elliptic equation.

• Journal of the Korean Mathematical Society
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• v.49 no.6
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• pp.1273-1299
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• 2012

We prove the global regularity of weak solutions to a conormal derivative boundary value problem for quasilinear elliptic equations in divergence form on Lipschitz domains under the controlled growth conditions on the low order terms. The leading coefficients are in the class of BMO functions with small mean oscillations.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.73-79
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• 2008

The purpose of this paper is to study the relations between quasilinear elliptic equations on Riemannian manifolds and differential forms. Two classes of differential forms are introduced and it is shown that some differential expressions are connected in a natural way to quasilinear elliptic equations.

• 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 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.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.969-980
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• 2011

In this paper, our main purpose is to consider the quasilinear elliptic equation $$div(|{\nabla}u|^{p-2}{\nabla}u)=(p-1)f(u)$$ on a bounded smooth domain ${\Omega}\;{\subset}\;R^N$, where p > 1, N > 1 and f is a smooth, increasing function in [0, ${\infty}$). We get some estimates of a solution u satisfying $u(x){\rightarrow}{\infty}$ as $d(x,\;{\partial}{\Omega}){\rightarrow}0$ under different conditions on f.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.1011-1016
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• 2011

In this paper, our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equation on $R^N$. we obtained the results under different suitable conditions on the locally H$\"{o}$lder continuous nonlinearity f(x, u), we needn't any mono-tonicity condition about the nonlinearity.

• 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.

• The Pure and Applied Mathematics
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• v.8 no.1
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• pp.15-23
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• 2001

This paper gives the sufficient conditions for the existence of positive solution of a quasilinear elliptic with homogeneous Dirichlet boundary conditon.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1229-1243
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• 2011

In this paper we consider a system of N-Laplacian elliptic equations with critical exponential growth. The existence and multiplicity results of solutions are obtained by a limit index method and Trudinger-Moser inequality.