scholarly journals Existence of Nontrivial Solutions for Perturbedp-Laplacian Equation inℝNwith Critical Nonlinearity

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huixing Zhang ◽  
Xiaoqian Liu
Keyword(s):  
Variational Method ◽  
Positive Solution ◽  
Critical Nonlinearity ◽  
Nontrivial Solutions ◽  
Laplacian Equation

We consider a perturbedp-Laplacian equation with critical nonlinearity inℝN. By using variational method, we show that it has at least one positive solution under the proper conditions.

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

scholarly journals On the Existence of Solutions for the Critical Fractional Laplacian Equation inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zifei Shen ◽  
Fashun Gao
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Lower Order ◽  
Fractional Laplacian ◽  
Critical Power ◽  
Existence Of Solutions ◽  
Order Perturbation ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Laplacian Equation

We study existence of solutions for the fractional Laplacian equation-Δsu+Vxu=u2*s-2u+fx, uinℝN,u∈Hs(RN), with critical exponent2*s=2N/(N-2s),N>2s,s∈0, 1, whereVx≥0has a potential well andf:ℝN×ℝ→ℝis a lower order perturbation of the critical poweru2*s-2u. By employing the variational method, we prove the existence of nontrivial solutions for the equation.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals Existence of Nontrivial Solutions for Perturbed Elliptic System inℝN

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Juan Jiang
Keyword(s):  
Variational Method ◽  
Critical Exponent ◽  
Elliptic System ◽  
Existence Result ◽  
Nonlinear Elliptic System ◽  
Nontrivial Solutions ◽  
Nonlinear Elliptic ◽  
Sobolev Critical Exponent

We consider the perturbed nonlinear elliptic system-ε2Δu+V(x)u=K(x)|u|2*-2u+Hu(u,v),  x∈ℝN,-ε2Δv+V(x)v=K(x)|v|2*-2v+Hv(u,v),  x∈ℝN, whereN≥3,2*=2N/(N-2)is the Sobolev critical exponent. Under proper conditions onV,H, andK, the existence result and multiplicity of the system are obtained by using variational method providedεis small enough.

scholarly journals The behavior of solutions of a parametric weighted $ (p, q) $-Laplacian equation

2021 ◽  
Vol 7 (1) ◽  
pp. 499-517
Author(s):  
Dušan D. Repovš ◽  
◽  
Calogero Vetro ◽  
Keyword(s):  
Bounded Domain ◽  
Variational Approach ◽  
Critical Parameter ◽  
Differential Operators ◽  
Dirichlet Condition ◽  
Parametric Equation ◽  
Nontrivial Solutions ◽  
Smooth Solutions ◽  
Laplacian Equation ◽  
Existence And Nonexistence

<abstract><p>We study the behavior of solutions for the parametric equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda &gt;0, $\end{document} </tex-math></disp-formula></p> <p>under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $, $ a_1, a_2 \in L^\infty(\Omega) $ with $ a_1(z), a_2(z) &gt; 0 $ for a.a. $ z \in \Omega $, $ p, q \in (1, \infty) $ and $ \Delta_{p}^{a_1}, \Delta_{q}^{a_2} $ are weighted versions of $ p $-Laplacian and $ q $-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $ f(z, x) $ asymptotically as $ x \to \pm \infty $ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When $ \lambda $ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.</p></abstract>

scholarly journals Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Chunmei Yuan ◽  
Shujuan Guo ◽  
Kaiyu Tong
Keyword(s):  
Variational Method ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Real Parameter ◽  
Existence Of Positive Solutions

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities-Δu=λa(x)up+b(x)uqinΩ,u>0inΩ,u=0on∂Ω, whereλ>0is a real parameter,0<p<1<q.Ωis a bounded domain inRN  (N≥3), anda(x)andb(x)are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

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