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Eigenvalue problems with weights in Lorentz spaces. (English) Zbl 1188.35124

Summary: Given \(V, w\) locally integrable functions on a general domain \(\Omega \) with \(V \geq 0\) but \(w\) is allowed to change sign, we study the existence of ground states for the nonlinear eigenvalue problem:
\[ -\Delta u + V u = \lambda w |u|^{p-2} u, \quad u|_{\partial \Omega} =0, \]
with \(p\) subcritical. These are minimizers of the associated Rayleigh quotient whose existence is ensured under suitable assumptions on the weight \(w\). In the present paper we show that an admissible space of weight functions is provided by the closure of smooth functions with compact support in the Lorentz space \({L(\tilde p,\infty)}\) with \({\frac{1}{\tilde p} + \frac{p}{2^*} =1}\). This generalizes previous results and gives new sufficient conditions ensuring existence of extremals for generalized Hardy-Sobolev inequalities. The existence in such a generality of a principal eigenfunction in the linear case \(p=2\) is applied to study the bifurcation for semilinear problems of the type
\[ -\Delta u= \lambda (a(x)u + b(x) r(u)), \]
where \(a, b\) are indefinite weights belonging to some Lorentz spaces, and the function \(r\) has subcritical growth at infinity.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J61 Semilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B32 Bifurcations in context of PDEs
49J40 Variational inequalities
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