Existence of solutions for quasilinear problem with Neumann boundary conditions

My abstract is:This paper is devoted to the study of the existence of weak solutionsfor quasilinear systems of a partial dierential equations which are the combinationof the Perona-Malik equation and the Heat equation. The proof of the main resultsare based on the compactness method and the motonocity arguments.

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.

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].

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.

Multiplicity of Positive Solutions for a Quasilinear Schrödinger Equation with an Almost Critical Nonlinearity

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem:\left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.where Ω is a smooth and bounded domain in {\mathbb{R}^{N},N\geq 3}. More specifically we prove that, for p near the critical exponent {22^{*}=4N/(N-2)}, the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

The aim of this paper is analyzing the positive solutions of the quasilinear problem-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign once in {(0,1)} and satisfies {\int_{0}^{1}a(x)\,dx<0}, and {f\in\mathcal{C}^{1}(\mathbb{R})} is positive and increasing in {(0,+\infty)} with a potential, {F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at {+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions, {\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from {(\lambda,0)} at some {\lambda_{0}>0} and from {(\lambda,\infty)} at some {\lambda_{\infty}>0}. It also establishes that {\mathscr{C}_{\lambda_{0}}^{+}} consists of regular solutions if and only if\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty.Equivalently, the small positive regular solutions of {\mathscr{C}_{\lambda_{0}}^{+}} become singular as they are sufficiently large if and only if \Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1).This is achieved by providing a very sharp description of the asymptotic profile, as {\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions of {\lambda_{0}} and {\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.

Three Solutions for a Singular Quasilinear Elliptic Problem

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.