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Journal of Elliptic and Parabolic Equations
Number of Followers: 0  Hybrid journal (It can contain Open Access articles)ISSN (Print) 2296-9020 - ISSN (Online) 2296-9039 Published by Springer-Verlag  [2468 journals]
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- Analysis of solutions for a class of $$(p_1(x),\ldots ,p_n(x))$$
-Laplacian systems with Hardy potentials-
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Abstract: In this current work, we examine a \((p_1(x),\ldots ,p_n(x))\) -Laplacian system with Hardy potentials. We establish the existence of at least one non-zero critical point and at least three distinct critical points to this system from an abstract critical point result of Bonanno et al. (Adv Nonlinear Stud 14(4):915–939, 2014) and a recent three critical points theorem of Bonanno and Marano (Appl Anal 89:1–10, 2010). This article represents, as far as we are aware, one of the first efforts towards the study of the \((p_1(x),\ldots ,p_n(x))\) -Laplacian systems with Hardy potentials, in which the nonlinearity in \(\Omega \) may change sign. Furthermore, we provide an example to illustrate our main conclusions.
PubDate: 2024-07-29
DOI: 10.1007/s41808-024-00293-5
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- The solvability and regularity results for elliptic equations involving
mixed local and nonlocal p-Laplacian-
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Abstract: Abstract Let \(\Omega \subset \mathbb {R}^{N}\) for \(N\ge 2\) be a bounded \(C^{1}\) domain. For \(0<s<1<p<N\) , we consider an elliptic problem involving mixed local and nonlocal p-Laplacian $$\begin{aligned} \left\{ \begin{aligned} -\Delta _p u+(-\Delta )_p^s u&=f(x) \quad{} & {} \text { in } \Omega , \\ u&=0{} & {} \text{ in } \mathbb {R}^N \backslash \Omega , \end{aligned}\right. \end{aligned}$$ where \((-\Delta )_p^s u\) is the fractional p-Laplacian. We make use of Stampacchia Lemma and Solution Obtained as Limit of Approximations (SOLA) approach to prove the solvability and regularity of weak solution and distributional solution under assumption of \(f(x)\in L^{m}(\Omega )\) with different ranges of \(m>1\) , respectively.
PubDate: 2024-07-26
DOI: 10.1007/s41808-024-00291-7
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- Existence results for nonlinear degenerate elliptic problems involving
singular and supercritical nonlinearities-
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Abstract: Abstract In this article, we establish the existence of nonnegative solutions to a class of nonlinear degenerate elliptic equations subject to zero Dirichlet boundary conditions on \(\Omega \) , an open, bounded subset of \({\mathbb {R}}^N (N\ge 3)\) . The problem is modeled by: $$\begin{aligned} {\left\{ \begin{array}{ll} -\text {div}\left( \dfrac{a(x)\nabla u}{(1+u)^{\theta }}\right) = \dfrac{\lambda }{u^\gamma }+ u^p &{}\text { in } \Omega ,\\ u\ge 0 &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\lambda , \theta , \gamma \) and p are positive parameters and a(x) is a real-valued function defined on \(\Omega \) such that for \(\alpha , \beta \in {\mathbb {R}}, \) it satisfies \(0<\alpha \le a(x)\le \beta \) . The main contributions of this paper are the treatment of singular and supercritical nonlinearities on the right-hand side, as well as the lack of coercivity in the principle part.
PubDate: 2024-07-19
DOI: 10.1007/s41808-024-00292-6
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- On the solution of evolution p-Laplace equation with memory term and
unknown boundary Dirichlet condition-
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Abstract: Abstract In the present paper, we are interested in investigating a classe of nonlinear parabolic integro-differential equation with unknown flux on the part of Dirichlet boundary. A discrete scheme for the time approximations is introduced. Existence and uniqueness of a weak solution at each time step are proved. Convergence of the approximate solution to the weak solution is shown with the help of some a priori estimates. At the end, our proposed theoretical approach is supported by computational experiments.
PubDate: 2024-07-14
DOI: 10.1007/s41808-024-00290-8
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- On the superlinear Kirchhoff problem involving the double phase operator
with variable exponents-
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Abstract: Abstract The paper deals with the following Kirchhoff-double phase problem $$\begin{aligned} \left\{ \begin{array}{ll} m \left( L(u) \right) D(u) = \vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2} u &{}\quad \text {in } \Omega ,\\ m \left( L(u) \right) \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu = \lambda g(x,u) &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$ where \(L(u)=\int _{\Omega } ( \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)}+ \frac{b(x)}{q(x)} \vert \nabla u \vert ^{q(x)}) dx\) and D is the double phase operator with variable exponents. The goal is to determine the precise positive interval of \(\lambda \) for which the above problem admits at least two nontrivial weak solutions without assuming the Ambrosetti–Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain Theorem with Cerami condition.
PubDate: 2024-07-01
DOI: 10.1007/s41808-024-00289-1
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- On the regular solutions for a generalized compressible
Landau–Lifshitz–Bloch equation-
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Abstract: Abstract In this paper, first we will prove that the regular solution for the compressible Landau–Lifshitz–Bloch equation found in (Ayouch, C., Benmouane, M. & Essoufi, E. H. Regular solution for the compressible Landau–Lifshitz–Bloch equation in a bounded domain of \(\mathbb {R}^{3}\) . J Elliptic Parabol Equ 8, 419-441 (2022)) is global in time in dimension two when the initial data is small enough. Next, we deal with a two generalized version of the compressible LLB equation and we prove existence and uniqueness of regular solution for this new models.
PubDate: 2024-06-17
DOI: 10.1007/s41808-024-00287-3
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- A new kind of double phase problems governed by anisotropic matrices
diffusion-
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Abstract: Abstract We investigate a class of double phase elliptic problems governed by matrices diffusion. We establish the existence and uniqueness of weak solutions within the Musielak–Orlicz framework. Our primary approach involves combining variational methods with Schaefer’s fixed point theorem.
PubDate: 2024-06-17
DOI: 10.1007/s41808-024-00288-2
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- Existence and uniqueness of minimizing solution for a nonlinear clamped
cylindrical shell model-
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Abstract: Abstract In this paper, by using the inverse function theorem, we will establish the existence of a solution to the nonlinear clamped cylindrical shell model around particular solutions associated with specific applied forces. Furthermore, we will also prove that the solution found in this fashion is also the unique minimizer to the associated nonlinear energy functional.
PubDate: 2024-06-12
DOI: 10.1007/s41808-024-00286-4
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- Asymptotics for the infinite Brownian loop on noncompact symmetric spaces
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Abstract: Abstract The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length T around a fixed origin when \(T \rightarrow +\infty \) . The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces G/K of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely \(L^1\) asymptotic convergence without requiring bi-K-invariance for initial data, and strong \(L^{\infty }\) convergence.
PubDate: 2024-06-01
DOI: 10.1007/s41808-023-00250-8
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- Existence and multiplicity of non-trivial solutions for fractional
Schrödinger–Poisson systems with a combined nonlinearity-
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Abstract: Abstract In this paper, we are concerned with the following fractional Schrödinger–Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle (-\Delta )^s u + V(x)u +\lambda \phi u=K(x){ u }^ {q-2}u+ f(x,u), &{}\qquad \ x \in \mathbb {R}^3 \\ \displaystyle (-\Delta )^t\phi = u^2,&{} \qquad \ x \in \mathbb {R}^3\\ \end{array}\right. \end{aligned}$$ where \(\lambda >0\) is a constant, \(s,t \in (0,1]\) , \(2t+4s>3\) , \(1<q<2\) and f(x, u) is linearly bounded in u at infinity. With some assumptions on K, V and f we get the existence and multiplicity of non-trivial solutions with the help of the variational methods.
PubDate: 2024-06-01
DOI: 10.1007/s41808-023-00258-0
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- Local integrability of $$G(\cdot )$$ -superharmonic functions in Lebesgue
and Musielak–Orlicz spaces-
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Abstract: Abstract In this article, we study local integrability properties of superharmonic functions related to partial differential equations with Musielak-Orlicz growth conditions in Lebesgue and Musielak-Orlicz spaces.
PubDate: 2024-06-01
DOI: 10.1007/s41808-023-00253-5
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- Critical nonhomogeneous fourth-order Schrödinger–Kirchhoff-type
equations-
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Abstract: Abstract In this paper we study the following class of stationary fourth-order Schrödinger–Kirchhoff-type equations: $$\begin{aligned} \Delta ^{2} u-M\left( \Vert \nabla u\Vert ^2_2 \right) \Delta u+V(x)u=h(x) u ^{q-2}u+ u ^{2_*-2}u+ g(x) u ^{\tau -2}u, ~~x \in \mathbb {R}^{N}, \end{aligned}$$ where \(N\ge 8,\) and \(2_*=\frac{2N}{N-4}\) is the critical Sobolev exponent. Under some assumptions on the Kirchhoff function M, the potential V(x) and g(x), by using Ekeland’s Variational Principle and the Mountain Pass Theorem, we obtain the existence of multiple solutions for the above problem. These results are new even for the local case, which corresponds to nonlinear fourth order Schrödinger equations.
PubDate: 2024-06-01
DOI: 10.1007/s41808-023-00249-1
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- A variational approach for some singular elliptic problems with Hardy
potential-
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Abstract: Abstract In this paper, we are interested in applying a three critical points theorem by B. Ricceri, to establish the existence of three weak solutions for some singular elliptic problems involving a p-Laplace operator, subject to Dirichlet boundary conditions in a smooth bounded domain in \(\mathbb {R}^N.\)
PubDate: 2024-06-01
DOI: 10.1007/s41808-023-00260-6
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- Global existence of solutions for a parabolic systems with logarithmic
nonlinearity-
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Abstract: Abstract The primary goal of this paper is to study the global existence and blow-up of solutions in a finite time for a class of parabolic systems that are characterized by the fractional p-Laplacian operator with logarithmic nonlinearity. By employing the classical Galerkin method, we show the existence and decay exponential estimates of the global weak solutions with critical initial energy \(J(u_0,v_0)<d\) . Moreover, we discuss the finite time blow up of weak solutions.
PubDate: 2024-04-16
DOI: 10.1007/s41808-024-00265-9
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- On the existence of capacity solution for a perturbed thermistor problem
in anisotropic Orlicz–Sobolev spaces-
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Abstract: Abstract In this paper, in the context of anisotropic Orlicz–Sobolev spaces, we analyze the existence of a capacity solution to a system of two coupled perturbed elliptic equations, one of which has a quadratic growth in the gradient and the second one is an non-uniformly elliptic equation. The system describes the heat produced in a semiconductor device by an electric current which may be considered as a generalization of the well-known thermistor problem. We assume that the N-functions do not satisfy the \(\Delta _2\) -condition.
PubDate: 2024-04-12
DOI: 10.1007/s41808-024-00275-7
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- Existence and regularity results for nonlinear anisotropic degenerate
parabolic equations-
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Abstract: Abstract In this paper, we study the existence and regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with degenerate coercivity and \(L^m\) data, where \(m>1\) . Our approach is based on approximating the initial problem with a non-degenerate problem that is well-posed. We then establish the necessary estimates to pass to the limit.
PubDate: 2024-04-09
DOI: 10.1007/s41808-024-00277-5
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- Existence and uniqueness results for an elliptic equation with blowing-up
coefficient and singular lower order term-
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Abstract: Abstract In this paper, we study a class of nonlinear elliptic problems whose model is the following $$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&-\textrm{div}\Big (b(u) \nabla u ^{p-2}\nabla u\Big )=f\Big (1+\frac{1}{ u ^\gamma }\Big )\ \ \textrm{in}\ \Omega , \\ {}&u=0\ \ \textrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned} \end{aligned}$$ where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N (N\ge 2)\) , \(\gamma > 0\) , b is a positive continuous function which blows up for a finite value of the unknown u. We will prove existence and uniqueness of a renormalized nonnegative solution in the case where the nonnegative source f belongs to \(L^1(\Omega )\) .
PubDate: 2024-03-22
DOI: 10.1007/s41808-024-00272-w
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- Well-posedness of a fully nonlinear evolution inclusion of second order
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Abstract: We consider the well-posedness of the abstract Cauchy problem for the doubly nonlinear evolution inclusion equation of second order given by $$\begin{aligned} \left\{ \begin{array}{ll} u''(t)+\partial \Psi (u'(t))+B(t,u(t))\ni f(t), &{} t\in (0,T),\, T>0,\\ u(0)=u_0, &{} u'(0)=v_0 \end{array}\right. \end{aligned}$$ where the function \( u \) takes values in a real separable Hilbert space, denoted by \(\mathscr {H}\) . Here, \( u_0 \) lies in \(\mathscr {H}\) , \( v_0 \) is in the intersection \(\overline{\textrm{dom}(\partial \Psi )}\cap \textrm{dom}(\Psi )\) , and \( f \) belongs to \( {\mathrm L}^2(0,T;\mathscr {H}) \) . The functional \(\Psi : \mathscr {H}\rightarrow (-\infty ,+\infty ] \) is assumed to be proper, lower semicontinuous, and convex. Additionally, the nonlinear operator \( B:[0,T]\times \mathscr {H}\rightarrow \mathscr {H} \) is assumed to satisfy either a global or a local Lipschitz condition. In the case where \( B \) satisfies a global Lipschitz condition, we can establish the existence and uniqueness of strong solutions \( u \) belonging to \( {\mathrm H}^2(0,T^*;\mathscr {H}) \) . Furthermore, these solutions continuously depend on the data. We derive these results using the theory of nonlinear semigroups combined with the Banach fixed-point theorem. On the other hand, when \( B \) satisfies a local Lipschitz condition, we can guarantee the existence of strong local solutions.
PubDate: 2024-03-21
DOI: 10.1007/s41808-024-00270-y
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- Monotone iterations of two obstacle problems with different operators
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Abstract: Abstract In this paper we analyze iterations of the obstacle problem for two different operators. We solve iteratively the obstacle problem from above or below for two different differential operators with obstacles given by the previous functions in the iterative process. When we start the iterations with a super or a subsolution of one of the operators this procedure generates two monotone sequences of functions that we show that converge to a solution to the two membranes problem for the two different operators. We perform our analysis in both the variational and the viscosity settings.
PubDate: 2024-03-07
DOI: 10.1007/s41808-024-00268-6
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- Decay estimates for quasilinear elliptic equations and a
Brezis–Nirenberg result in $$D^{1,p}({\mathbb R}^N)$$-
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Abstract: Abstract We prove decay estimates for solutions of quasilinear elliptic equations in the whole \({\mathbb R}^N\) of the form $$\begin{aligned} u\in X: -\text{ div }\,A(x,\nabla u)=a(x) f(x,u), \end{aligned}$$ where \(X=D^{1,p}({\mathbb R}^N)\) is the Beppo-Levi space (also called homogeneous Sobolev space). Based on these decay estimates we are able to prove a Brezis–Nirenberg type result for the energy functional \(\Phi : X\rightarrow {\mathbb R}\) related to the p-Laplacian equation in \({\mathbb R}^N\) in the form $$\begin{aligned} u\in X: -\Delta _p u=a(x) g(u), \end{aligned}$$ saying that for the subspace V of bounded continuous functions with weight \(1+ x ^{\frac{N-p}{p}},\) a local minimizer of \(\Phi \) in the finer V topology is also a local minimizer in the X-topology. Global \(L^\infty \) -estimates on the one hand and pointwise estimates for solutions of quasilinear elliptic equations in terms of nonlinear Wolff potentials on the other hand play a crucial role in the proofs.
PubDate: 2024-02-21
DOI: 10.1007/s41808-024-00263-x
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