Some remarks for one-dimensional mean curvature problems through a local minimization principle

In this paper, we revisit the following nonlocal Kirchhoff diffusion problem:∂tu+M([u]s2)LKu=|u|p−2u,inΩ×R+,u(x,t)=0,in(RN\\Ω)×R+,u(x,0)=u0(x),inΩ,where is a bounded domain with Lipschitz boundary, [u]s is the Gagliardo seminorm of u, 0 < s < min{1, N/2}, is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator (−Δ)s, u0 : Ω → [0, +∞) is the initial function, M : [0, +∞) → [0, +∞) is a continuous function and there exist two constants θ > 1 and m0 > 0 such thatM(σ)⩾m0σθ−1,∀σ∈[0,+∞).This problem has been investigated by Xiang, Rădulescu and Zhang in [], and Ding and Zhou in [] by using potential well method. If , in [], the authors showed the existence of a nontrivial, nonnegative global weak solution, where . However, if , these two papers only studied the model in special cases, the details are as follows: in [], the blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) < 0; in [], the global existence and blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) ⩽ d and M(σ) = m0σθ−1, where J(u0) denotes the initial energy and d > 0 denotes the depth of the potential well (see ()). The main purpose of this paper is to extend the above results to the general case M(σ) ⩾ m0σθ−1, , and the conditions on global existence and finite time blow-up are obtained. Furthermore, the decay estimates for global weak solutions, the growth estimates for blow-up solutions, the upper and lower bounds of blow-up time to blow-up solutions, the behavior of the energy functional as t → T (where T denotes the blow-up time) are studied. Moreover, some blow-up conditions independent of d and some equivalent conditions for the weak solutions existing globally or blowing up in finite time are investigated. Finally, the global existence and finite time blow-up results with high initial energy (i.e., J(u0) > d) are obtained.

In this paper, a p-Laplacian boundary-value problem with impulsive effects is considered. The existence of at least one non-trivial weak solution and at least three non-negative weak solutions via variational methods and critical point theory is obtained. Some recent results are extended and improved. Some examples are presented to demonstrate the application of our main results.

Existence and multiplicity of solutions for Kirchhoff–Schrödinger type equations involving [formula omitted]-Laplacian on the entire space [formula omitted

This paper deals with the existence of weak solutions to a Dirichlet problem for a semilinear elliptic equation involving the difference of two main nonlinearities functions that depends on a real parameter λ . According to the values of λ , we give both nonexistence and multiplicity results by using variational methods. In particular, we first exhibit a critical positive value such that the problem admits at least a nontrivial non-negative weak solution if and only if λ is greater than or equal to this critical value. Furthermore, for λ greater than a second critical positive value, we show the existence of two independent nontrivial non-negative weak solutions to the problem.

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1,1}$ boundary. We consider problems of the form $-\Delta u=\chi_{\left\{ u&gt;0\right\}}\left( au^{-\alpha}-g\left( .,u\right) \right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u\geq0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left( \Omega\right) ,$ $\alpha\in\left( 0,1\right) ,$ and $g:\Omega\times\left[ 0,\infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g,$ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega,$ of the found solution $u$, is also proved.

In this paper, we study the existence of a weak solution for a class of elliptic systems involving nonhomogeneous operators. By using the Mountain-Pass theorem and establishing a compactness result, we prove the existence of nontrivial and non-negative weak solution. Our results generalize previous work by considering broader classes of operators and nonlinearities that satisfy superlinear growth conditions.

Multiple solutions for a nonlinear and non-homogeneous problem in Orlicz–Sobolev spaces

Quasilinear elliptic systems in [formula omitted] with multipower forcing terms depending on the gradient

In this paper, we investigate a parametric nonlinear problem driven by the variable exponent double phase operator and subject to nonlinear mixed boundary conditions. Both in the differential equation and the boundary condition, the power-type nonlinear terms can exhibit critical growth in the sense that the exponents may coincide, at some points or everywhere, with the corresponding Sobolev critical exponents. Under general assumptions on the reaction term, we identify a range of parameters for which the problem admits one or two nontrivial weak solutions. The analysis relies on variational methods and critical point theory through a local minimum theorem and a two critical point theorem. We also establish a Poincaré-type inequality for the Musielak-Orlicz Sobolev space $ W^{1, \mathcal{H}}_{0, \Gamma_1}(\Omega) $ related to the mixed boundary configuration, that is, the set of functions belonging to $ W^{1, \mathcal{H}}(\Omega) $ with null trace on a submanifold $ \Gamma_1 $ of the boundary $ \partial\Omega $.

In this paper, the Sturm-Liouville boundary value problem is studied for fractional differential equation with generalized (<i>p</i>, <i>q</i>)-Laplacian operator. By imposing mild assumptions on nonlinearity <i>f</i>, several new existence results of at least one or two nontrivial weak solutions are established through variational methods and critical point theorems. Furthermore, the criteria is also investigated for the nonexistence result.

In this paper, we investigate abstract critical point theorems for continuously Gâteaux differentiable functionals satisfying the Cerami condition via the generalized Ekeland variational principle developed by C.-K. Zhong. As applications of our results, under certain assumptions, we show the existence of at least one or two weak solutions for nonlinear elliptic equations with variable exponents \\[ -\\operatorname{div} (\\varphi(x, \\nabla u)) + V(x) |u|^{p(x)-2} u = \\lambda f(x,u) \\quad \\textrm{in } \\mathbb{R}^{N}, \\] where the function $\\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with a continuous function $p \\colon \\mathbb{R}^{N} \\to (1,\\infty)$, $V \\colon \\mathbb{R}^{N} \\to (0,\\infty)$ is a continuous potential function, $\\lambda$ is a real parameter, and $f \\colon \\mathbb{R}^{N} \\times \\mathbb{R} \\to \\mathbb{R}$ is a Carathéodory function. Especially, we localize precisely the intervals of $\\lambda$ for which the above equation admits at least one or two nontrivial weak solutions by applying our critical points results.

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m0 > 0 and such that As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows: The local existence of nontrivial, nonnegative weak solution for , where . The blow-up conditions for nontrivial, nonnegative weak solution when J(u0) < 0, where J(u0) denotes the initial energy.The main purpose of this paper is to extend the above results and we get: The global existence of nontrivial, nonnegative weak solution for any . The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in ().

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

On entire solutions of degenerate elliptic differential inequalities with nonlinear gradient terms

We prove the existence of a non-trivial non-negative radial weak solution to the problem \begin{equation*} \begin{cases} (-\Delta) ^{\alpha} u+bu=\lambda \dfrac{u}{|x|^{2\alpha}} +|u|^{p-1}u+\mu |u|^{r-1}u & \mathrm{in} \ \mathbb{R}^N,\\ \lim\limits_{|