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Ground state alternative for p-Laplacian with potential term
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2007 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 28, no 2, p. 179-201Article in journal (Refereed) Published
##### Abstract [en]

Let Omega be a domain in R-d, d >= 2, and 1 < p < infinity. Fix V. is an element of L-loc(infinity)(Omega). Consider the functional Q and its G teaux derivative Q' given by Q(u) := integral(vertical bar del u vertical bar(p) + V vertical bar u vertical bar(p)) dx, 1/p Q'(u) := - del. (vertical bar del u vertical bar(p-2)del u) + V vertical bar u vertical bar(p-2)u. If Q >= 0 on C-0(infinity)(Omega), then either there is a positive continuous function W such that integral W vertical bar u vertical bar(p) dx = Q'(u) for all u is an element of C-0(infinity) (Omega), or there is a sequence u(k) is an element of C-0(infinity)(Omega) and a function v > 0 satisfying Q'(v) = 0, such that Q(u(k)) -> 0, and u(k) -> v in L-loc(p)(Omega). In the latter case, v is ( up to a multiplicative constant) the unique positive supersolution of the equation Q'(u) = 0 in Omega, and one has for Q an inequality of Poincare type: there exists a positive continuous function W such that for every psi is an element of C-0(infinity) (Omega) satisfying integral psi v dx not equal 0 there exists a constant C > 0 such that C-1 integral W vertical bar u vertical bar(p) dx <= Q(u) + C vertical bar integral u psi dx vertical bar(p) for all u is an element of C-0(infinity) (Omega). 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.

##### Place, publisher, year, edition, pages
2007. Vol. 28, no 2, p. 179-201
##### Keyword [en]
quasilinear elliptic operator, p-Laplacian, ground state, positive solutions, green function, isolated singularity
Mathematics
##### Identifiers
ISI: 000242295000003OAI: oai:DiVA.org:uu-153341DiVA, id: diva2:416319
Available from: 2011-05-11 Created: 2011-05-11 Last updated: 2017-12-11Bibliographically approved

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Department of Mathematics
##### In the same journal
Calculus of Variations and Partial Differential Equations
Mathematics

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Cite
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