Boundary value problems having natural growth terms and Hardy potential

I. Well-posedness of boundary-value problems.- to Part I.- 1. Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems.- 1. Introductory notes.- 2. Joint spectrum of operators.- 3. Position of the joint spectrum.- 4. Estimates for roots of characteristic polynomials.- 5. Definitions of well-posedness and weak well-posedness of boundary-value problems for equation (1).- 6. Spaces of boundary data.- 7. (Weak) well-posedness and uniform (weak) well-posedness.- 2. Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B.- 1. The Cauchy problem.- 2. The Dirichlet problem.- 3. The Neumann problem.- 4. The inverse Cauchy problem.- 3. The Cauchy problem.- 1. Distinction of the general case of commuting normal operators A and B.- 2. A criterion for the weak well-posedness.- 3. Proof of Theorem 3.1.- 4. A criterion for the well-posedness.- 5. The (weak) well-posedness in particular cases.- 6. Spectrum of the associated operator pencil.- 7. Fattorini's definitions of well-posedness of the Cauchy problem.- 4. Boundary-value problems on a finite segment.- 1. The (weak) well-posedness of the Dirichlet problem.- 2. The (weak) well-posedness of the Dirichlet problem in particular cases.- 3. Boundary conditions for the Dirichlet problem.- 4. The weak well-posedness and the well-posedness of the Neumann problem.- 5. Boundary conditions for the Neumann problem.- 6. The inverse Cauchy problem.- II. Initial data of solutions.- to Part II.- 5. Boundary behaviour of an integral transform R(t) as t ? 0 depending on the sub-integral measure.- 1. Analogy with Tauberian theorems.- 2. A model example.- 3. Proof of Lemma 5.1.- 4. Further results.- 5. Continuity of R(t) on R+. in extreme cases.- 6. Continuity of R(t) on R+.- 7. Continuity, boundedness, and integrability of R(t) on a finite segment [0,T].- 8. Equivalence of conditions on a sub-integral measure.- 6. Initial data of solutions.- 1. The set of initial data of solutions.- 2. When FC = D(B) x (D(A) ? D(|B|1/2))?.- 3. When FC = D(B) x D(A)?.- 4. E-sequences of vectors and the general expression for weak solutions.- 5. The set of initial data of weak solutions.- 6. When $${F_C}^\prime = H \times {H_{ - 1}}$$?.- 7. Fatou-Riesz property.- III. Extension, stability, and stabilization of weak solutions.- to Part III.- 7. The general form of weak solutions.- 1. Another general expression for weak solutions.- 2. Continuity, boundedness, and integrability of R(t) on [0.T] in a more general case.- 3. The general form of weak solutions where (2.2) holds.- 4. Initial data of weak solutions where (2.2) holds.- 5. Weak well-posedness of the Cauchy problem in a special space of initial data.- 8. Fatou-Riesz property.- 1. Fatou-Riesz property.- 2. Two-sided Fatou-Riesz property.- 3. First order equation and incomplete second order equations.- 4. The case of self-adjoint A and B.- 5. Spectrum of the associated operator pencil.- 9. Extension of weak solutions.- 1. Extension of weak solutions on a finite interval.- 2. Boundedness of weak solutions on a finite interval.- 3. Exponential growth of weak solutions.- 4. Two-sided extension of weak solutions.- 5. Spectrum of the associated operator pencil.- 6. Comparison of the results on extension of weak solutions and bounded weak solutions.- 7. Intermediate classes of weak solutions.- 8. Extension of weak solutions and weak well-posedness of boundary-value problems.- 10. Stability and stabilization of weak solutions.- 1. Stability and uniform stability of an equation.- 2. Stabilization of the Cesaro means for weak solutions.- 3. Stabilization of a weak solution.- 4. Stabilization of weak solutions and asymptotic stability of an equation.- 5. Exponential stability and uniform exponential stability of an equation.- 6. Stabilization of $$\frac{{y(t)}}{t}$$ for weak solutions of an equation.- 7. The case of self-adjoint A and B.- IV. Boundary-value problems on a half-line.- to Part IV.- 11. The Dirichlet problem on a half-line.- 1. Classes of (weak) uniqueness.- 2. Existence of (weak) solutions.- 3. A criterion for the (weak) well-posedness.- 4. Boundary data of solutions.- 12. The Neumann problem on a half-line.- 1. Classes of uniqueness (weak uniqueness).- 2. Existence of solutions and weak solutions.- 3. Criteria for the well-posedness and the weak well-posedness.- 4. Boundary data of solutions and weak solutions.- Commentaries on the literature.- List of symbols.

In this article, we focus on triple weak solutions for some p-Laplacian-type elliptic equations with Hardy potential, two parameters, and mixed boundary conditions. We show the existence of at least three distinct weak solutions by using variational methods, the Hardy inequality, and the Bonanno–Marano-type three critical points theorem under suitable assumptions, and the existence of solutions to some particular cases of this type of elliptic equations are also obtained.

In this paper we consider nonlinear boundary value problems whose simplest model is the following: $${\left\{ {\begin{array}{*{20}c} {\Delta u + = \gamma |\nabla u|^2 + \frac{A} {{|x|^2 }}} & {in\;\Omega } & {\left( {\gamma ,\;A\; \in \;\mathbb{R}} \right)} \\ {\;\quad \quad \quad \quad \quad u = 0} & {\quad on\;\partial \Omega .} & {} \\ \end{array} } \right.}$$ where Ω is a bounded open set in \(\mathbb{R}^N \), N > 2.Keywordsquasi-linear elliptic equationsnatural growth termsquadratic growth with respect to the gradientHardy inequality

Existence of regular solutions for nonlinear Signorini's problems

This paper investigates the existence of at least two distinct weak solutions using Bonanno’s theorem [G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220, Theorem 3.2], and establishes the existence of no fewer than two nontrivial weak solutions through the Bonanno–D’Aguì theorem [G. Bonanno and G. D’Aguì, Two nonzero solutions for elliptic Dirichlet problems, Z. Anal. Anwend. 35 (2016), no. 4, 449–464, Theorem 2.1] applied to a general class of fourth-order Leray–Lions problems that include an s ⁢ ( x ) {s(x)} -Hardy potential and a nonlocal singular term.

On weak and strong solutions of [formula omitted]-implicit generalized variational inequalities with applications

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Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 September 1969Published online: 17 February 2012Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

The 3D-primitive equations with only horizontal viscosity are considered on a cylindrical domain $\Omega=(-h,h) \times G$, $G\subset \mathbb{R}^2$ smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering a vanishing vertical viscosity limit, we apply a direct approach which in particular avoids unnecessary boundary conditions on top and bottom. For the initial value problem, we obtain existence and uniqueness of local $z$-weak solutions for initial data in $H^1((-h,h),L^2(G))$ and local strong solutions for initial data in $H^1(\Omega)$. If $v_0\in H^1((-h,h),L^2(G))$, $\partial_z v_0\in L^q(\Omega)$ for $q>2$, then the $z$-weak solution regularizes instantaneously and thus extends to a global strong solution. This goes beyond the global well-posedness result by Cao, Li and Titi (J. Func. Anal. 272(11): 4606-4641, 2017) for initial data near $H^1$ in the periodic setting. For the time-periodic problem, existence and uniqueness of $z$-weak and strong time periodic solutions is proven for small forces. %These solutions are in the set of solutions with small norms. Since this is a model with hyperbolic and parabolic features for which classical results are not directly applicable, such results for the time-periodic problem even for small forces are not self-evident.

We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h0 ≥ 0 and f(h) = |h|n, 0 < n < 3. We show that there exists a weak nonnegative solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T*, and asymptotically approaches its means as t → ∞. The weak solution is in the classical sense of distributions for 3/8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n ≤ 3/8. Furthermore, the solutions have high enough regularity to just include the unique source-type solutions [2] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Second, we consider strictly positive initial data h0 ≥ m > 0 and f(h) = |h|n, 0 < n < ∞. For this problem we show that even if a finite-time singularity of the form h → 0 does occur, there exists a weak nonnegative solution for all time t. This weak solution becomes strong and positive again after some critical time T*. As in the first problem, we show that the solution approaches its mean as t → ∞. The main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity. We show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result. © 1996 John Wiley & Sons, Inc.

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

The paper analyzes the influence on the meaning of natural growth in the gradient, of a perturbation by a Hardy potential in some elliptic equations. We obtain a linear differential operator that, in a natural way, is the corresponding gradient for the perturbed elliptic problem. The main results are: i) Optimal summability of the data to have weak solutions; ii) Optimal linear operator associated, and, iii) Multiplicity and characterization of all solutions in terms of some measures. The results also are new for the Laplace operator perturbed for an inverse-square potential.

In this paper, we prove existence results for entropy solutions of a nonlinear boundary value problems represented by a class of nonlinear elliptic anisotropic equations with variable exponents and natural growth terms. The functional setting involves variable exponents anisotropic Sobolev spaces.

The main purpose of this paper is to investigate a nonlinear elliptic problem with a natural growth term under Robin boundary conditions. Using approximation techniques and surjectivity criteria of an operator mapping from a Banach space into its dual, we prove the existence of a sequence of weakly approximated solutions and take its limit to establish the existence of a renormalized or entropy solution for the initial problem.

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