Ground state alternative for p-Laplacian with potential term

Pinchover, Yehuda; Tintarev, Kyril
February 2007
Calculus of Variations & Partial Differential Equations;Feb2007, Vol. 28 Issue 2, p179
Academic Journal
Let Ω be a domain in $$\mathbb{R}^d$$ , d ≥ 2, and 1 < p < ∞. Fix $$V \in L_{\mathrm{loc}}^\infty(\Omega)$$ . Consider the functional Q and its Gâteaux derivative Q′ given by $$ Q(u) := \mathop \int_\Omega (|\nabla u|^p+V|u|^p){\rm d}x,\,\, \frac{1}{p}Q^\prime (u) := -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u.$$ If Q ≥ 0 on $$C_0^{\infty}(\Omega)$$ , then either there is a positive continuous function W such that $$\int W|u|^p\,\mathrm{d}x\leq Q(u)$$ for all $$u\in C_0^{\infty}(\Omega)$$ , or there is a sequence $$u_k\in C_0^{\infty}(\Omega)$$ and a function v > 0 satisfying Q′ ( v) = 0, such that Q( u k ) → 0, and $$u_k\to v$$ in $$L^p_\mathrm{loc}(\Omega)$$ . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ ( u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every $$\psi\in C_0^\infty(\Omega)$$ satisfying $$\int \psi v\,{\rm d}x \neq 0$$ there exists a constant C > 0 such that $$C^{-1}\int W|u|^p\,\mathrm{d}x\le Q(u)+C\left|\int u \psi\,\mathrm{d}x\right|^p$$ . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.


Related Articles

  • HIGH-ORDER QUASI-LINEAR ELLIPTIC EQUATIONS AND NONLINEAR SEMIGROUPS OF COMPRESSIONS. Porembsky, Ya. F.; Kukharchuk, M. M. // Naukovi visti NTUU - KPI;2007, Vol. 2007 Issue 2, p142 

    The work is the logical continuation of the works by N.M. Kukharchuk ‘On solvability of the high-order quasi-linear elliptic equations in Rℓ’, ‘Dissipative operators and nonlinear semigroups of compressions in Lp(Rℓ,dℓ x)’. It was established, that...

  • Existence and Non-Existence Result for Singular Quasilinear Elliptic Equations. Mingzhu Wu; Zuodong Yang // Applied Mathematics;Nov2010, Vol. 1 Issue 5, p351 

    We prove the existence of a ground state solution for the qusilinear elliptic equation -div(∣ ∇u ∣p-2 ∇u) =f (x,u) in ω ⊂ RN, under suitable conditions on a locally Holder continuous non-linearity f(x, t), the non-linearity may exhibit a singularity as t →...

  • Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations. Zuodong Yang; Bing Xu // Boundary Value Problems;2007, p1 

    We consider the problem -div(∣Δu∣P-2Δu) = a(x)(um + λun),x ? RN,N ≥ 3, where 0 < m< p-1 < n,a(x) ≥ 0,a(x) is not identically zero. Under the condition that a(x) satisfies (H), we show that there exists λ0 > 0 such that the above-mentioned equation admits at...

  • Construction of solutions of quasilinear parabolic equations in parametric form. Volosov, K. // Differential Equations;Apr2007, Vol. 43 Issue 4, p507 

    The article focuses on the construction of solutions of quasilinear parabolic equations in parametric form. The system of nonlinear algebraic equations for the derivatives has a unique solution which are treated as an overdetermined system for the functions that provides a solvability condition....

  • Existence results for degenerate quasilinear elliptic equations in weighted Sobolev spaces. Cavalheiro, Albo Carlos // Bulletin of the Belgian Mathematical Society - Simon Stevin;Feb2010, Vol. 17 Issue 1, p141 

    In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations -div [v(x)A(x, u, ?u)] + w(x)A0(x, u(x)) = f0 - ?j=1 nDjfj, on O in the setting of the weighted Sobolev spacesW01,p (O,w, ?).

  • ON THE SMOOTHNESS OF SOLUTIONS OF ONE QUASILINEAR EQUATION IN R. Kukcharchuk, M. M.; Yaremenko, M. I. // Naukovi visti NTUU - KPI;2009, Vol. 2009 Issue 2, p148 

    We consider the smoothness of solutions of quasilinear elliptic partial differential equations. In this paper we develop the form method for proving the conditions of Minty-Browder's theorem, as well as for obtaining the research results on the smoothness of the solutions.

  • Existence and Nonexistence Results for a Class of Quasilinear Elliptic Systems. El Manouni, Said; Perera, Kanishka // Boundary Value Problems;2007, p1 

    Using variational methods, we prove the existence and nonexistence of positive solutions for a class of (p,q)-Laplacian systems with a parameter.

  • Existence of Multiple Positive Solutions of Quasilinear Elliptic Problems in ℝN. Afrouzi, G. A.; Khademloo, S. // Advances in Dynamical Systems & Applications;2007, Vol. 2 Issue 1, p1 

    This paper concerns quasilinear elliptic equations of the form --div(|Δ u|p-2Δu) = γa(x)u(x)|u|p-2(1-|u|λ in ℝN with p > 1 and a(x) changes sign. We discuss the question of existence and multiplicity of solutions when a(x) has some specific properties.

  • Multiplicity solutions for a class of quasilinear critical problems in RN involving sign-changing weight function. Miotto, Márcio Luís // Revista Ciência e Natura;2014, p367 

    In this paper, existence and multiplicity results to the following quasilinear critical problem ... are established, where ƛ > 0, 1 < q < p, with 2 ≤ p < N, p* = Np/N-p and the weight function ƒ, among other conditions, can possibly change sign in RN. The study is based on...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics