Nontrivial Weak Solutions Research Articles

Existence of Solutions for the Fractional (p, q)-Laplacian Problems Involving a Critical Sobolev Exponent

In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent: $$({P_{\mu ,\lambda }})\left\{ {\begin{array}{*{20}{l}} {( - \Delta )_p^{{s_1}}u + ( - \Delta )_q^{{s_2}}u = \mu |u{|^{q - 2}}u + \lambda |u{|^{p - 2}}u + |u{|^{p_{{s_1}}^* - 2}}u,}&{\text{in}\;\Omega ,} \\ {u = 0,}&{\text{in}\;{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN is a smooth and bounded domain, λ, μ > 0, 0 < S2 < s1 < 1, $$1 < q < p < {\textstyle{N \over {{s_1}}}}.$$ We establish the existence of a non-negative nontrivial weak solution to (Pμ,λ) by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.

The Existence of a Nontrivial Weak Solution to a Double Critical Problem Involving a Fractional Laplacian in ℝN with a Hardy Term

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: \begin{equation} \label{eq0.1} (-\Delta)^{s}u-{\gamma} {\frac{u}{|x|^{2s}}}= {\frac{{|u|}^{ {2^{*}_{s}}(\beta)-2}u}{|x|^{\beta}}}+ \big [ I_{\mu}* F_{\alpha}(\cdot,u) \big](x)f_{\alpha}(x,u), \ \ u \in {\dot{H}}^s(\R^{n}) \end{equation} where $s \in(0,1)$, $0\leq \alpha,\beta<2s<n$, $\mu \in (0,n)$, $\gamma<\gamma_{H}$, $I_{\mu}(x)=|x|^{-\mu}$, $F_{\alpha}(x,u)=\frac{ {|u(x)|}^{ {2^{\#}_{\mu} }(\alpha)} }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $f_{\alpha}(x,u)=\frac{ {|u(x)|}^{{ 2^{\#}_{\mu} }(\alpha)-2}u(x) }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $2^{\#}_{\mu} (\alpha)=(1-\frac{\mu}{2n})\cdot 2^{*}_{s} (\alpha)$, $\delta_{\mu} (\alpha)=(1-\frac{\mu}{2n})\alpha$, ${2^{*}_{s}}(\alpha)=\frac{2(n-\alpha)}{n-2s}$ and $\gamma_{H}=4^s\frac{\Gamma^2(\frac{n+2s}{4})} {\Gamma^2(\frac{n-2s}{4})}$. We show that problem (\ref{eq0.1}) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings \begin{equation} \label{eq0.2} {\dot{H}}^s(\R^{n}) \hookrightarrow {L}^{2^*_{s}(\alpha)}(\R^{n},|y|^{-\alpha}) \hookrightarrow L^{p,\frac{n-2s}{2}p+pr}(\R^{n},|y|^{-pr}) \end{equation} where $s \in (0,1)$, $0<\alpha<2s<n$, $p\in[1,2^*_{s}(\alpha))$, $r=\frac{\alpha}{ 2^*_{s}(\alpha) }$; We also establish an improved Sobolev inequality. By using mountain pass lemma along with an improved Sobolev inequality, we obtain a nontrivial weak solution to problem (\ref{eq0.1}) in a direct way. It is worth while to point out that the improved Sobolev inequality could be applied to simplify the proof of the main results in \cite{NGSS} and \cite{RFPP}.

Biharmonic equations with totally characteristic degeneracy

In this paper, we study the biharmonic operator on a manifold with conical singularities and consider the existence of non-trivial weak solution for the corresponding semilinear degenerate elliptic equation with critical cone Sobolev exponents by the cone Sobolev inequality and Poincaré inequality. Furthermore, we show an interesting integral identity, which can be used to discuss the non-existence results for some biharmonic equations under Navier boundary conditions.

On Local Weak Solutions for Fractional in Time SOBOLEV-Type Inequalities

We consider two fractional in time nonlinear Sobolev-type inequalities involving potential terms, where the fractional derivatives are defined in the sense of Caputo. For both problems, we study the existence and nonexistence of nontrivial local weak solutions. Namely, we show that there exists a critical exponent according to which we have existence or nonexistence.

On nonlinear Schrödinger equations on the hyperbolic space

We study existence of weak solutions for certain classes of nonlinear Schrödinger equations on the Poincaré ball model BN, N≥3. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.

A note on nonlinear Schrödinger equations: The relation between spectral gaps and the nonlinearity

We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schrödinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term. Our goal is to weaken the necessary assumptions to obtain a linking structure to the problem, for instance to allow zero being in the spectrum or the nonlinearity being sign-changing. Our main difficulty is to overcome the lack of monotonicity on the nonlinear term, as well, as the lack of compactness since the domain is unbounded. With this purpose, we require periodicity on V.

On a class of nonlocal problems in new fractional Musielak-Sobolev spaces

ABSTRACT In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following form where , Ω is an open bounded subset in with Lipschitz boundary , and is a Carathéodory function with suitable growth conditions.

A model of capillary phenomena in $\mathbb R^N$ with subcritical growth

This paper deals with the nonlinear Dirichlet problem of capillary phenomena involving an equation driven by the p -Laplacian-like di¤erential operator in \mathbb R^N . We prove the existence of at least one nontrivial nonnegative weak solution, when the reaction term satisfies a sub-critical growth condition and the potential term has certain regularities. We apply the energy functional method and weaker compactness conditions.

Nontrivial weak solutions for nonlocal nonhomogeneous elliptic problems

This article concerns the following class of nonlocal nonhomogeneous elliptic problems: where Ω is a bounded domain of , α, , is a matrix with variable coefficients and is a positive function which are defined almost everywhere with respect to the variable x. By using the Schauder and Tychonoff fixed-point theorems, we establish two existence theorems of weak solutions for this problem according to two bundles of hypotheses on α, β and g.

Nontrivial Solutions of the Kirchhoff-Type Fractional p-Laplacian Dirichlet Problem

In this article, we consider the new results for the Kirchhoff-type p-Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.

On𝓅(x)-Kirchhoff-type equation involving𝓅(x)-biharmonic operator via genus theor

UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the𝓅(x)-Kirchhoff-type problem,u=Δu=0on∂Ω.By using variational approach and Krasnoselskii’s genus theory, we prove the existence and multiplicity of solutions for the𝓅(x)-Kirchhoff-type equation.

Landesman-Lazer type (p, q)-equations with Neumann condition

We consider a Neumann problem driven by the (p, q)-Laplacian under the Landesman-Lazer type condition. Using the classical saddle point theorem and other classical results of the calculus of variations, we show that the problem has at least one nontrivial weak solution.

Nontrivial Solutions for a Class ofp-Kirchhoff Dirichlet Problem

This paper is devoted to the followingp-Kirchhoff type of problems−a+b∫Ω∇updxΔpu=fx,u,x∈Ωu=0,x∈∂Ωwith the Dirichlet boundary value. We show that thep-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.

Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space

This paper is concerned with a class of fractional a-Laplace type problem with Dirichlet boundary data of the following form $$\begin{aligned} (P_a) \left\{ \begin{array}{clclc} (-\varDelta )^s_a u +a(|u|)u = \lambda f(x,u) &{} \text { in }&{} \varOmega , \\ u = 0 &{} \text { in } &{} {\mathbb {R}} ^N\setminus \varOmega . \end{array} \right. \end{aligned}$$ By means of Ekeland’s variational principal and direct variational approach, we investigate the existence of nontrivial weak solutions for the above problem.

A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions

In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.

A problem of capillarity under Neumann condition

We formulate a Neumann variational problem, which characterizes the capillary phenomena, and disLaplacian‐like operator defined bycuss the existence of nontrivial weak solutions. We get the results using sufficient hypotheses on the reaction term.

The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold

In this paper, we consider the following elliptic problem: −divg(|∇gu|N−2∇gu)+V(x)|u|N−2u=f(x,u)a(x)in M,(Pa)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\mathtt{div}_{g}\\bigl( \\vert \\nabla_{g} u \\vert ^{N-2}\\nabla_{g} u \\bigr)+V(x) \\vert u \\vert ^{N-2}u = \\frac{f(x, u)}{a(x)}\\quad \\mbox{in } M, \\qquad (P_{a}) $$\\end{document} where (M, g) be a complete noncompact N-dimensional Riemannian manifold with negative curvature, Ngeq2, V is a continuous function satisfying V(x) geq V_{0 }> 0, a is a nonnegative function and f(x, t) has exponential growth with t in view of the Trudinger–Moser inequality. By proving some estimates together with the variational techniques, we get a ground state solution of (P_{a}). Moreover, we also get a nontrivial weak solution to the perturbation problem.

Existence of solutions for critical fractional p&q-Laplacian system

ABSTRACT In this article, we are concerned with the existence of solutions for the following critical fractional p&q-Laplacian system (1) where Ω is a smooth bounded set in , are two parameters, , , , and satisfy with is the fractional Sobolev critical exponent, and is the fractional t-Laplacian operator. By using variational methods, we show that there exists a nontrivial weak solution for system (1).

An extended variational characterization of the Fučík Spectrum for the p-Laplace operator

We consider the boundary value problem $$\begin{aligned} \begin{array}{rcl} -\Delta _p u &{} = &{} \alpha |u|^{p-2}u^+-\beta |u|^{p-2}u^- \text{ in } \Omega ,\\ u &{} = &{} 0 \text{ on } \partial \Omega , \end{array} \end{aligned}$$where $$\Delta _p u:=\nabla \cdot (|\nabla u|^{p-2}\nabla u)$$ for $$1<p<\infty $$, $$\Omega $$ is a smooth bounded domain in $$\mathbb {R}^N,u^{\pm }:=\max \{\pm u,0\}$$, and $$(\alpha ,\beta )\in \mathbb {R}^2$$. If this problem has a nontrivial weak solution, then $$(\alpha ,\beta )$$ is an element of the Fucik Spectrum, $$\Sigma \subset \mathbb {R}^2$$. Our main result is to provide a variational characterization for a class of curves in $$\Sigma $$.

Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter λ . By mountain pass theorem, we obtain that system has at least one non-trivial weak solution u λ under some local conditions for each given large parameter λ . We get a concrete lower bound of the parameter λ , and then obtain two estimates of weak solutions u λ . We also obtain that u λ → 0 if λ tends to ∞. Finally, we present an example as an application of our results.