Bifurcation analysis for a modified quasilinear equation with negative exponent

In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H1(Ω) and H 1 * (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K(y) and N(x). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of W 2 1 (Ω) in L2(Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1 < p < 2), via Morse theory

Let us consider the quasilinear problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] is a parameter and [Formula: see text] is a continuous function with [Formula: see text], having a subcritical growth. We prove that there exists [Formula: see text] such that, for every [Formula: see text], [Formula: see text] has at least [Formula: see text] solutions, possibly counted with their multiplicities, where [Formula: see text] is the Poincaré polynomial of [Formula: see text]. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on [Formula: see text], approximating [Formula: see text].

On one class of solutions of the quasilinear matrix differential equations

In the mathematical description of various phenomena and processes that arise in mathematical physics, electrical engineering, economics, one has to deal with matrix differential equations. Therefore, these equations are relevant both for mathematicians and for specialists in other areas of natural science. Many studies are devoted to them, in which the solvability of matrix equations in various function spaces, boundary value problems for matrix differential equations, and other problems were investigated. In this article, a quasilinear matrix equation is considered, the coefficients of which can be represented in the form of absolutely and uniformly converging Fourier series with coefficients and frequency slowly varying in a certain sense. The problem is posed of obtaining sufficient conditions for the existence of particular solutions of a similar structure for the equation under consideration. For this purpose, the corresponding linear equation is considered first. It is written down in component-wise form, and, based on the assumptions made, the existence of the only particular solution of the specified structure is proved. Then, using the method of successive approximations and the principle of contracting mappings, the existence of a unique particular solution of the indicated structure for the original quasilinear equation are proved.

The Existence of Nontrivial Solutions to a Class of Quasilinear Equations

In this paper, we study the following quasilinear equation: − div ϕ ∇ u ∇ u + ϕ u u = f u in ℝ N , where ϕ ∈ C 1 ℝ + , ℝ + and Φ t = ∫ 0 t s ϕ ∣ s ∣ d s . In the Orlicz-Sobolev space, by variational methods and a minimax theorem, we prove the equation has a nontrivial solution.

H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

In this paper, we aim to study the asymptotic behavior (when $$\varepsilon \;\rightarrow \; 0$$ ε → 0 ) of the solution of a quasilinear problem of the form $$-\mathrm{{div}}\;(A^{\varepsilon }(\cdot ,u^{\varepsilon }) \nabla u^{\varepsilon })=f$$ - div ( A ε ( · , u ε ) ∇ u ε ) = f given in a perforated domain $$\Omega \backslash T_{\varepsilon }$$ Ω \ T ε with a Neumann boundary condition on the holes $$T_{\varepsilon }$$ T ε and a Dirichlet boundary condition on $$\partial \Omega $$ ∂ Ω . We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices $$(x,d)\mapsto A^{\varepsilon }(x,d)$$ ( x , d ) ↦ A ε ( x , d ) is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d, the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to $$A^{\varepsilon }(\cdot ,d)$$ A ε ( · , d ) in the perforated domain. Once the $$H^{0}$$ H 0 -limit $$A^{0}(\cdot ,d)$$ A 0 ( · , d ) of the pair $$(A^{\varepsilon },T^{\varepsilon })$$ ( A ε , T ε ) is determined, in the second step, we deduce that the solution $$u^{\varepsilon }$$ u ε converges in some sense to the unique solution $$u^{0}$$ u 0 in $$H^{1}_{0}(\Omega )$$ H 0 1 ( Ω ) of the quasilinear equation $$-\mathrm{{div}}\;(A^{0}(\cdot ,u^{0})\nabla u )=\chi ^{0}f$$ - div ( A 0 ( · , u 0 ) ∇ u ) = χ 0 f (where $$ \chi ^{0}$$ χ 0 is $$L^{\infty }$$ L ∞ weak $$^{\star }$$ ⋆ limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.

Subdifferential Decomposition of 1D-regularized Total Variation with Nonhomogeneous Coefficients

In this paper, we consider a convex function defined as a 1D-regularized total variation with nonhomogeneous coefficients, and prove the Main Theorem concerned with the decomposition of the subdifferential of this convex function to a weighted singular diffusion and a linear regular diffusion. The Main Theorem will be to enhance the previous regularity result for quasilinear equation with singularity, and moreover, it will be to provide some useful information in the advanced mathematical studies of grain boundary motion, based on KWC type energy.

On the Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be non-unique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy suband super-solutions, in the case when at least one of the initial functions is periodic. The obtained results are generalization of the results known for conservation laws to the parabolic case.