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Local regularity results to nonlinear elliptic Dirichlet problems with lower order terms

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Abstract

In this paper we study local regularity properties of weak solutions to a class of nonlinear noncoercive elliptic Dirichlet problems with \(L^1\) datum. The model example is

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p(w)+b(x)|Dw|^{p-1}=f(x)&{}\text {in }\Omega ,\\ w=0&{}\text {on }\partial \Omega . \end{array}\right. } \end{aligned}$$

Here \(\Omega \subset {\mathbb {R}}^N\) is a bounded open subset, \(N>1\), \(-\Delta _p\) is the well known p-Laplace operator, \(1<p<N\), b is a function in the Lorentz space \(L^{N,1}(\Omega )\) and f is a function in \(L^1(\Omega )\). We also investigate similar issues for a lower order perturbation of these problems.

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Notes

  1. For every \(1<q<\infty \), \(q'\) denotes the Hölder conjugate of q, that is, \(q'=\frac{q}{q-1}\).

  2. For every \(1\le q<N\), \(q^*\) denotes the Sobolev conjugate of q, that is, \(q^*=\frac{Nq}{N-q}\).

  3. We recall that, by Sobolev inequality, there exists a positive constant \({\mathcal {S}}_0\) which depends only on N and p, such that

    $$\begin{aligned} \Vert v\Vert _{L^{p^*}(\Omega )}\le {\mathcal {S}}_0\Vert |Dv|\Vert _{L^p(\Omega )}\quad \forall \,v\in W_0^{1,p}(\Omega ). \end{aligned}$$

  4. We recall that, by Sobolev inequality, there exists a positive constant \({\mathcal {S}}\) which depends only on N and p, such that (see [1])

    $$\begin{aligned} \Vert v\Vert _{L^{p^*}(U)}\le {\mathcal {S}}\left[ \frac{1}{|U|^{\frac{1}{N}}}\Vert v\Vert _{L^p(U)}+\Vert |Dv|\Vert _{L^p(U)} \right] \quad \forall \,\text {cube }U\subset {\mathbb {R}}^N,\,\forall \, v\in W^{1,p}(U). \end{aligned}$$

  5. For every \(t\in {\mathbb {R}}\), [t] denotes the integer part of t.

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  1. Dipartimento di Matematica ”Guido Castelnuovo”, Sapienza Università di Roma, P.le A. Moro 2, 00185, Rome, Italy

    Francesco Clemente

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Clemente, F. Local regularity results to nonlinear elliptic Dirichlet problems with lower order terms. Nonlinear Differ. Equ. Appl. 27, 9 (2020). https://doi.org/10.1007/s00030-019-0613-3

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